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MEASUREMENT AND UNITS & DIMENSIONS Synopsis : 1. Every measurement has two parts. The first is a number (n) and the next is a unit (u). Q = nu.

Eg : Length of an object = 40 cm. 2. The number expressing the magnitude of a physical quantity is inversely proportional to the unit

selected. 3. If n1 and n2 are the numerical values of a physical quantity corresponding to the units u1 and u2,

then n1u1 = n2u2. Eg : 2.8 m = 280 cm; 6.2 kg = 6200 g 4. The quantities that are independent of other quantities are called fundamental quantities. The units

that are used to measure these fundamental quantities are called fundamental units. 5. There are four systems of units namely C.G.S, M.K.S, F.P.S and SI 6. The quantities that are derived using the fundamental quantities are called derived quantities. The

units that are used to measure these derived quantities are called derived units. 7. The early systems of units : 8. Fundamental and supplementary physical quantities in SI system (Systeme Internationale d’units) :

Physical quantity Unit Symbol Length Metre m Mass kilogram kg Time second s Electric current ampere A Thermodynamic temperature kelvin K Intensity of light candela cd Quantity of substance mole mol

Supplementary quantities:

Plane angle radian rad Solid angle steradian sr

SI units are used in scientific research. SI is a coherent system of units. 13. A coherent system of units is one in which the units of derived quantities are obtained as multiples

or submultiples of certain basic units. SI system is a comprehensive, coherent and rationalised M.K.S. Ampere system (RMKSA system) and was devised by Prof. Giorgi.

System of units Fundamental Quantity

C.G.S. M.K.S. F.P.S. Length centimetre Metre foot

Mass Gram Kilogram pound

Time second Second second

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Measurement and Units & $ dimensions

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14. Metre : A metre is equal to 1650763.73 times the wavelength of the light emitted in vacuum due to electronic transition from 2p10 state to 5d5 state in Krypton–86. But in 1983, 17th General Assembly of weights and measures, adopted a new definition for the metre in terms of velocity of light. According to this definition, metre is defined as the distance travelled by light in vacuum during a time interval of 1/299, 792, 458 of a second.

15. Kilogram : The mass of a cylinder of platinum–iridium alloy kept in the International Bureau of weights and measures preserved at Serves near Paris is called one kilogram.

16. Second : The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of caesium–133 atom is called one second.

17. Ampere : The current which when flowing in each of two parallel conductors of infinite length and negligible cross–section and placed one metre apart in vacuum, causes each conductor to experience a force of 2x10–7 newton per metre of length is known as one ampere.

18. Kelvin : The fraction of 1/273.16 of the thermodynamic temperature of the triple point of water is called kelvin.

19. Candela : The luminous intensity in the perpendicular direction of a surface of a black body of area 1/600000 m2 at the temperature of solidifying platinum under a pressure of 101325 Nm–2 is known as one candela.

20. Mole : The amount of a substance of a system which contains as many elementary entities as there are atoms in 12x10 3 kg of carbon–12 is known as one mole.

21. Radian : The angle made by an arc of the circle equivalent to its radius at the centre is known as radian. 1 radian = 57o17l45ll.

22. Steradian : The angle subtended at the centre by one square metre area of the surface of a sphere of radius one metre is known as steradian.

23. The quantity having the same unit in all the systems of units is time. 24. Angstrom is the unit of length used to measure the wavelength of light. 1 Å = 10–10 m. 25. Fermi is the unit of length used to measure nuclear distances. 1 fermi = 10–15 metre. 26. Light year is the unit of length for measuring astronomical distances. 27. Light year = distance travelled by light in 1 year = 9.4605x1015 m. 28. Astronomical unit = Mean distance between the sun and earth = 1.5x1011 m. 29. Parsec = 3.26 light years = 3.084x1016 m 30. Barn is the unit of area for measuring scattering cross–section of collisions. 1 barn = 10–28 m2. 31. Chronometer and metronome are time measuring instruments. 32. PREFIXES : (or) Abbreviations for multiples and sub–multiples of 10.

MACRO Prefixes MICRO Prefixes

Kilo K 103 Mega M 106

Giga G 109 Tera T 1012 Peta P 1015 Exa E 1018 Zetta Z 1021 Yotta y 1024

Milli m 103 micro μ 10 6 nano n 109 pico p 1012 femto f 1015 atto a 1018 zepto z 1021 yocto y 10 24





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Note : The following are not used in SI system. deca 101 deci 101 hecta 102 centi 102

33. Full names of the units, even when they are named after a scientist should not be written with a capital letter. Eg : newton, watt, ampere, metre.

34. Unit should be written either in full or in agreed symbols only. 35. Units do not take plural form. Eg : 10 kg but not

10 kgs, 20 w but not 20 ws 2 A but not 2 As 36. No full stop or punctuation mark should be used within or at the end of symbols for units. Eg : 10

W but not 10 W. 37. Dimensions of a physical quantity are the powers to which the fundamental units are raised to

obtain one unit of that quantity. 38. The expression showing the powers to which the fundamental units are to be raised to obtain one

unit of a derived quantity is called the dimensional formula of that quantity. 39. If Q is the unit of a derived quantity represented by Q = MaLbTc, then MaLbTc is called

dimensional formula and the exponents a,b and c are called the dimensions. 40. Dimensional Constants : The physical quantities which have dimensions and have a fixed value

are called dimensional constants. Eg : Gravitational constant (G), Planck’s constant (h), Universal gas constant (R), Velocity of light in vacuum (C) etc.

41. Dimensionless quantities are those which do not have dimensions but have a fixed value. a) Dimensionless quantities without units. Eg : Pure numbers, π e, sinθ cosθ tanθ …. etc., b) Dimensionless quantities with units. Eg : Angular displacement – radian, Joule’s constant – joule/calorie, etc.,

42. Dimensional variables are those physical quantities which have dimensions and do not have fixed value. Eg : velocity, acceleration, force, work, power… etc.

43. Dimensionless variables are those physical quantities which do not have dimensions and do not have fixed value. Eg : Specific gravity, refractive index, coefficient of friction, Poisson’s ratio etc.

44. Dimensional formulae are used to a) verify the correctness of a physical equation, b) derive relationship between physical quantities and c) to convert the units of a physical quantity from one system to another system.

45. Law of homogeneity of dimensions : In any correct equation representing the relation between physical quantities, the dimensions of all the terms must be the same on both sides. Terms separated by ‘+’ or ‘–’ must have the same dimensions.

46. A physical quantity Q has dimensions a, b and c in length (L), mass (M) and time (T) respectively, and n1 is its numerical value in a system in which the fundamental units are L1, M1 and T1 and n2 is the numerical value in another system in which the fundamental units are L2, M2 and T2 respectively, then

c

2

1b

2

1a

2

112 T

TMM

LL

nn ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

47. Fourier laid down the foundations of dimensional analysis. 48. Limitations of dimensional analysis :





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1. Dimensionless quantities cannot be determined by this method. Constant of proportionality cannot be determined by this method. They can be found either by experiment (or) by theory.

2. This method is not applicable to trigonometric, logarithmic and exponential functions. 3. In the case of physical quantities which are dependent upon more than three physical

quantities, this method will be difficult. 4. In some cases, the constant of proportionality also possesses dimensions. In such cases we

cannot use this system. 5. If one side of equation contains addition or subtraction of physical quantities, we can not use

this method to derive the expression. 50. Some important conversions : 51. 1 bar = 06 dyne/cm2 = 105 Nm� = 105 pascal

76 cm of Hg = 1.013x106 dyne/cm2

= 1.013x105 pascal = 1.013 bar. 1 toricelli or torr = 1 mm of Hg = 1.333x103 dyne/cm2

= 1.333 millibar. 1 kmph = 5/18 ms 1

1 dyne = 10 5 N, 1 H.P = 746 watt 1 kilowatt hour = 36x105 J 1 kgwt = g newton 1 calorie = 4.2 joule 1 electron volt = 1.602x1019 joule 1 erg = 10 7 joule

52. Some important physical constants : Velocity of light in vacuum (c) = 3x108 ms1

Velocity of sound in air at STP = 331 ms 1

Acceleration due to gravity (g) = 9.81 ms2

Avogadro number (N) = 6.023x1023 /mol Density of water at 4oC = 1000 kgm3 or 1 g/cc. Absolute zero = 273.15oC or 0 K Atomic mass unit = 1.66x1027 kg Quantum of charge (e) = 1.602x1019 C Stefan’s constant( σ ) = 5.67x10–8 W/m2/K4

Boltzmann’s constant (K) = 1.381x1023 JK1

One atmosphere = 76 cm Hg = 1.013x105 Pa Mechanical equivalent of heat (J) = 4.186 J/cal Planck’s constant (h) = 6.626x10 34 Js Universal gas constant (R) = 8.314 J/mol–K Permeability of free space ( oμ ) = 4 π x107 Hm1





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Permittivity of free space ( oε ) = 8.854x1012 Fm1

Density of air at S.T.P. = 1.293 kgm3

Universal gravitational constant = 6.67x1011 Nm2kg 2

53. Derived SI units with special names :

Physical quantity SI unit Symbol Frequency hertz Hz Energy joule J Force newton N Power watt W Pressure pascal Pa Electric charge or quantity of electricity coulomb C Electric potential difference and emf volt V Electric resistance ohm Ω Electric conductance siemen S Electric capacitance farad F Magnetic flux weber Wb Inductance henry H Magnetic flux density tesla T Illumination lux Lx Luminous flux lumen Lm

Dimensional formulae for some physical quantities :

Physical quantity Unit Dimensional formula

Acceleration or acceleration due to gravity ms–2 LT–2 Angle (arc/radius) rad MoLoTo Angular displacement rad MoloTo Angular frequency (angular displacement / time) rads–1 T–1 Angular impulse (torque x time) Nms ML2T–1 Angular momentum (I ω ) kgm2s–1 ML2T–1 Angular velocity (angle / time) rads–1 T–1 Area (length x breadth) m2 L2 Boltzmann’s constant JK–1 ML2T–2 θ –1

Bulk modulus (V

V.PΔ

Δ ) Nm–2, Pa M1L–1T–2

Calorific value Jkg–1 L2T–2





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Coefficient of linear or areal or volume expansion oC–1 or K–1 θ –1 Coefficient of surface tension (force/length) Nm–1 or Jm–2 MT–2 Coefficient of thermal conductivity Wm–1K–1 MLT–3 θ –1

Coefficient of viscosity (F = dxdvAη ) poise ML–1T–1

Compressibility (1/bulk modulus) Pa–1, m2N–2 M–1LT2 Density (mass / volume) kgm–3 ML–3 Displacement, wavelength, focal length m L Electric capacitance (charge / potential) CV–1, farad M–1L–2T4I2

Electric conductance (1 / resistance) Ohm–1 or mho or siemen M–1L–2T3I2

Electric conductivity (1 / resistivity) siemen/metre or Sm–1 M–1L–3T3I2

Electric charge or quantity of electric charge (current x time) coulomb IT

Electric current ampere I Electric dipole moment (charge x distance) Cm LTI Electric field strength or Intensity of electric field (force / charge) NC–1, Vm–1 MLT–3I–1

Electric resistance (current

difference potential ) ohm ML2T–3I–2

Emf (or) electric potential (work / charge) volt ML2T–3I–1 Energy (capacity to do work) joule ML2T–2

Energy density (volumeenergy ) Jm–3 ML–1T–2

Entropy ( T/QS Δ=Δ ) J θ –1 ML2T–2 θ –1 Force (mass x acceleration) newton (N) MLT–2 Force constant or spring constant (force / extension) Nm–1 MT–2 Frequency (1 / period) Hz T–1 Gravitational potential (work / mass) Jkg–1 L2T–2 Heat (energy) J or calorie ML2T–2

Illumination (Illuminance) lux (lumen/metre2) MT–3

Impulse (force x time) Ns or kgms–1 MLT–1

Inductance (L) (energy = 2L21 I ) or

coefficient of self induction henry (H) ML2T–2I–2





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Intensity of gravitational field (F / m) Nkg–1 L1T–2 Intensity of magnetisation (I) Am–1 L–1I Joule’s constant or mechanical equivalent of heat Jcal–1 MoLoTo Latent heat (Q = mL) Jkg–1 MoL2T–2 Linear density (mass per unit length) kgm–1 ML–1 Luminous flux lumen or (Js–1) ML2T–3 Magnetic dipole moment Am2 L2I Magnetic flux (magnetic induction x area) weber (Wb) ML2T–2I–1 Magnetic induction (F = Bil) NI–1m–1 or T MT–2I–1 Magnetic pole strength (unit: ampere–metre) Am LI Modulus of elasticity (stress / strain) Nm–2, Pa ML–1T–2 Moment of inertia (mass x radius2) kgm2 ML2 Momentum (mass x velocity) kgms–1 MLT–1

Permeability of free space (21

2

o mmFd4π

=μ ) Hm–1 or NA–2 MLT–2I–2

Permittivity of free space (221

oFd4QQ

π=ε ) Fm–1 or C2N–1m–2 M–1L–3T4I2

Planck’s constant (energy / frequency) Js ML2T–1 Poisson’s ratio (lateral strain / longitudinal strain) –– MoLoTo Power (work / time) Js–1 or watt (W) ML2T–3 Pressure (force / area) Nm–2 or Pa ML–1T–2 Pressure coefficient or volume coefficient oC–1 or θ –1 θ –1 Pressure head m MoLTo

Radioactivity disintegrations per second MoLoT–1

Ratio of specific heats –– MoLoTo Refractive index –– MoLoTo Resistivity or specific resistance Ω –m ML3T–3I–2 Specific conductance or conductivity (1 / specific resistance)

siemen/metre or Sm–1 M–1L–3T3I2

Specific entropy (1/entropy) KJ–1 M–1L–2T2 θ Specific gravity (density of the substance / density of water) –– MoLoTo

Specific heat (Q = mst) Jkg–1 θ –1 MoL2T–2 θ –1 Specific volume (1 / density) m3kg–1 M–1L3 Speed (distance / time) ms–1 LT–1





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Stefan’s constant ⎟⎟⎠

⎞⎜⎜⎝

⎛4etemperaturxtimexarea

energyheat Wm–2 θ –4 MLoT–3 θ –4

Strain (change in dimension / original dimension) –– MoLoTo Stress (restoring force / area) Nm–2 or Pa ML–1T–2 Surface energy density (energy / area) Jm–2 MT–2 Temperature oC or θ MoLoTo θ

Temperature gradient (distance

etemperatur in change ) oCm–1 or θ m–1 MoL–1To θ

Thermal capacity (mass x specific heat) J θ –1 ML2T–2 θ –1 Time period second T Torque or moment of force (force x distance) Nm ML2T–2 Universal gas constant (work / temperature) Jmol–1 θ –1 ML2T–2 θ –1

Universal gravitational constant (F = G.2

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dmm ) Nm2kg–2 M–1L3T–2

Velocity (displacement/time) ms–1 LT–1

Velocity gradient (dxdv ) s–1 T–1

Volume (length x breadth x height) m3 L3 Water equivalent kg MLoTo Work (force x displacement) J ML2T–2

54. Quantities having the same dimensional formulae :

a) impulse and momentum. b) work, energy, torque, moment of force, energy c) angular momentum, Planck’s constant, rotational impulse d) stress, pressure, modulus of elasticity, energy density. e) force constant, surface tension, surface energy. f) angular velocity, frequency, velocity gradient g) gravitational potential, latent heat. h) thermal capacity, entropy, universal gas constant and Boltzmann’s constant. i) force, thrust. j) power, luminous flux.





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ERRORS AND SIGNIFICANT FIGURES

ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENT : 1. The measured value of a physical quantity is usually different from its true value. The result of

every measurement by any measuring instrument is an approximate number, which contains some uncertainty. This uncertainty is called error.

2. Every calculated quantity which is based on measured values also has an error. We distinguish between two terms accuracy and precision.

3. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured.

4. In general, the errors in measurement can be broadly classified as (a) systematic errors and (b) random errors.

5. Systematic errors : The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are : a) Instrumental errors that arise from the errors due to imperfect design or calibration of the

measuring instrument, etc. For example, in a Vernier calipers the zero mark of vernier scale may not coincide to the zero mark of the main scale, or simply an ordinary metre scale may be worn off at one end.

b) Imperfection for experimental technique or procedure. For example, to determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature.

c) Personal errors that arise due to an individual’s bias, lack of proper setting of the apparatus of individuals, carelessness in taking observations without observing proper precautions, etc. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax.

6. Random errors : The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply).

7. Least count error : a) The least count error is the error associated with the resolution of the instrument. For example,

a vernier calipers has a least count as 0.001 cm. It occurs with both systematic and random errors. The smallest division on the scale of the measuring instrument is called its least count.

b) Systematic errors can be minimized by improving experimental techniques, selecting better instruments and removing personal bias as far as possible.

c) Random errors are minimized by repeating the observations several times and taking the arithmetic mean of all the observations. The mean value would be very close to the true value of the measured quantity.

8. Absolute Error, Relative Error and Percentage Error : a) Suppose the values obtained in several measurements are a1, a2, a3 … an. The arithmetic mean

of these values is taken as the best possible value of the quantity under the given conditions of measurement as :

mean 1 2 3 na (a a a ... a ) /n= + + + + (or) n

mean ii 1

a a /n=

= ∑





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• The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement. This is denoted by |Δt| (As we do not know the true value of a quantity, let us accept the arithmetic mean of all measurements as the true value of the measured quantity). Then the absolute errors in the individual measurement values are Δa1 = amean – a1 ; Δa2 = amean – a2 ; … ; Δan = amean – an.

• The arithmetic errors may be positive in certain cases and negative in some other cases. b) The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the

value of the physical quantity ‘a’. It is represented by Δamean. Thus, Δamean = ( |Δa1| + |Δa2| + |Δa3| + … + |Δan| ) / n

= n

ii 1

| a | /n=

Δ∑

• If we do a single measurement, the value we get may be in the range amean ± Δamean. i.e. a = amean ± Δamean. or

amean – Δamean ≤ a ≤ amean + Δamean. • This implies that any measurement of the physical quantity ‘a’ is likely to lie between

(amean + Δamean) and (amean – Δamean) c) Instead of the absolute error, we often use the relative error or the percentage error (δa). The

relative error is the ratio of the mean absolute error Δamean to the mean value amean of the quantity measured. Relative error = Δamean / amean.

• When the relative error is expressed in percent, it is called the percentage error (δa) • Thus, Percentage error

δa = (Δamean / amean) × 100% 9. Combination of Errors : If we do an experiment involving several measurements, we

must know how the errors in all the measurements combine. a) Error of a sum or a difference : Suppose two physical quantities A and B have measured

values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors. We wish to find the error ΔZ in the sum

Z = A + B We have by addition, Z ± ΔZ = (A ± ΔA) + (B ± ΔB). The maximum possible error in Z = ΔZ = ΔA + ΔB For the difference Z = A – B, we have

Z ± ΔZ = (A ± ΔA) – (B ± ΔB) = (A – B) ± ΔA ± ΔB. or

± ΔZ = ± ΔA ± ΔB The maximum value of the error ΔZ is again ΔA + ΔB.

• When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities. b) Error of a product or a quotient : Suppose Z = AB and the measured values of A and B are A

± ΔA and B ± ΔB. Then Z ± ΔZ = (A ± ΔA) (B ± ΔB).





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= AB ± B ΔA ± A ΔB ± ΔA ΔB. Dividing LHS by Z and RHS by AB we have, 1 ± (ΔZ / Z) = 1 ± (ΔA / A) ± (ΔB / B) ± (ΔA / A) (ΔB / b).

Z = ΔZ / Z = (ΔA / A) + (ΔB / B) • When two quantities are multiplied or divided, the fractional error in the result is the sum of the

fractional errors in the multipliers. c) Error due to the power of a measured quantity :

Z = A2, then ΔZ / Z = (ΔA / A) + (ΔA / A) = 2 (ΔA / A)

If Z = Ap Bq / Cr, then ΔZ / Z = p (ΔA / A) + q (ΔB / B) + r (ΔC / C)

• The fractional error in a physical quantity raised to the power is the power times the fractional error in the individual quantity.

SIGNIFICANT FIGURES : 10. Every measurement involves errors. Thus, the result of measurement should be reported

in a way that indicates the precision of measurement. 11. Normally, the reported result of measurement is a number that includes all digits in the

number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.

12. If we say the period of oscillation of a simple pendulum is 1.62 s, the digits 1 and 6 are reliable and certain, while the digit 2 is uncertain. Thus, the measured value has three significant figures. The length of an object reported after measurement to be 287.5 cm has four significant figures, the digits 2, 8, 7 are certain while the digit 5 is uncertain.

13. Then a length of 16.2 cm means l = 16.20 ± 0.05 cm, i.e. it lies between 16.15 cm and 16.25 cm.

14. A choice of change of different units does not change the number of significant digits or figures in a measurement.

a) For example, the length 2.308 cm has four significant figures. But in different units, the same value can be written as 0.02308 m or 23.08 mm or 23080 μm. The example gives the following rules : i) All the non-zero digits are significant. ii) All the zeros between two non-zero digits are significant, no matter where the decimal point is,

if at all. iii) If the number is less than 1, the zeros on the right of decimal point but to the left of the first

non-zero digit are not significant. (In 0.002308, the underlined zeros are not significant) iv) The terminal or trailing zeros in a number without a decimal point are not significant. v) (Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zeroes being

not significant). However, you can also see the next observation. vi) The trailing zeros in a number with a decimal point are significant. (The numbers 3.500 or

0.06900 have four significant figures each). b) There can be some confusion regarding the trailing zeros. Suppose a length is reported to be 4.700

m. It is evident that the zeros here are meant to convey the precision of measurement and are, therefore, significant. (If these were not, it would be superfluous to write them explicitly, the reported measurement would have been simply 4.7 m). 4.700 m = 470.0 cm = 4700 mm = 0.004700 km. Since the last number has trailing zeroes in a number with no decimal, we would conclude erroneously from observation (1) above that the





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number has two significant figures, while infact it has four significant figures and a mere change of units cannot change the number of significant figures.

c) To remove such ambiguities in determining the number of significant figures, the best way is to report every measurement m scientific notation (in the power of 10). In this notation, every number is expressed as a × 10b, where ‘a’ is a number between 1 and 10, and b is any positive or negative exponent of 10.

4.700 m = 4.700 × 102 cm ⇒ 4.700 × 103 mm = 4.700 × 10–3 km Thus, in the scientific notation, no confusion arises about the trailing zeros in the base number ‘a’. They are always significant.

d) The scientific notation is ideal for reporting measurement. But if this is not adopted, we use the rules adopted in the preceding example : i) For a number greater than 1, without any decimal, the trailing zeros are not significant. ii) For a number with a decimal, the trailing zeros are significant.

15. Rules for Arithmetic Operations with Significant Figures : a) The result of a calculation involving approximate measured values of quantities (i.e. values with

limited number of significant figures) must reflect the uncertainties in the original measured values.

b) It cannot be more accurate than the original measured values themselves on which the result is based. In general, the final result should not have more significant figures than the original data from which it was obtained. Thus, if mass of an object is measured to be, say, 4.237 g (four significant figures) and its volume is measured to be 2.51 cm3, then its density, by mere arithmetic division, is 1.68804780876 g/cm3 upto 11 decimal places.

c) It would be clearly absurd and irrelevant to record the calculated value of density to such a precision when the measurements on which the value is based, have much less precision. The following rules for arithmetic operations with significant figures ensure that the final result of a calculation is shown with the precision that is consistent with the precision of the input measured values : i) In multiplication or division, the final result should retain as many significant figures as are

there in the original number with the least significant figures.

Density = 33

4.237g 1.69g cm2.51 cm

−=

Similarly, if the speed of light is given as 3.00 × 108 ms–1 (three significant figures). ii) In addition or subtraction, the final result should retain as many decimal places as are there in

the number with the least decimal places. iii) For example, the sum of the numbers 436.32 g, 227.2 g and 0.301 g by mere arithmetic

addition, is 663.821 g. iv) But the least precise measurement (227.2 g) is correct to only one decimal place. The final

result should, therefore, be rounded off to 663.8 g.

v) Similarly, the difference in length can be expressed as : 0.307 m – 0.304 m = 0.003 m = 3 × 10–3 m.

vi) They do not convey the precision of measurement properly. For addition and subtraction, the rule is in terms of decimal places.

16. Rounding off the Uncertain Digits :





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a) The result of computation with approximate numbers, which contain more than one uncertain digit, should be rounded off.

b) Preceding digit is raised by 1 if the insignificant digit to be dropped (the underlined digit in this case) is more than 5, and is left unchanged if the latter is less than 5.

c) But what if the number is 2.745 in which the insignificant digit is 5. Here the convention is that if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1.

17. Rules for Determining the Uncertainty of Number in Arithmetic Operations : a) The uncertainty or error in the measured value, as already mentioned, is normally taken to be half

of the least count of the measuring instrument. The rules for determining the uncertainty of number in arithmetic operations can be understood from the following examples. i) If the length and breadth of a thin rectangular sheet are measured as 16.2 cm and 10.1 cm

respectively, there are three significant figures in each measurement. It means that the true length l may be written as l = 16.20 ± 0.05 cm = 16.20 cm ± 0.3% Similarly, the breadth b may be written as b = 10.10 ± 0.05 cm = 10.10 cm ± 0.5%

To determine the uncertainty of the product of two (or more) experimental values, we often following a rule that is founded upon probability. If we assume that uncertainties combine randomly, we have the rule : When two or more experimentally obtained numbers are multiplied, the percentage uncertainty of the final result is equal to the square root of the sum of the squares of the percentage uncertainties of the original numbers. Following the square root of the sum of the squares rule, we may write for the product of length l and breadth b as

l b = 163.62 cm2 ± 2 2(0.3%) (0.5%)+ = 163.62 cm2

± 0.6% = 163.62 ± 1.0 cm2

The result leads us to quote the final result as l b = 163.62 ± 1.0 cm2

b) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures. i) However, if data are subtracted, the number of significant figures can be reduced. ii) For example : 12.9 g – 7.06 g, both specified to three significant figures, cannot properly be

evaluated as 5.84 g but only as 5.8 g, as uncertainties in subtraction or addition combine in a different fashion (smallest number of decimal places rather than the number of significant figures in any of the number added or subtracted).

c) The fractional error of a value of number specified to significant figures depends not only on n but also on the number itself. i) For example, the accuracy in measurement of mass 1.02 g is ± 0.005 g whereas another

measurement 9.89 g is also accurate to ± 0.005g. ii) The fractional error in 1.02 g is

= (± 0.005 / 1.02) × 100% ⇒ ± 0.5% iii) On the other hand, the fractional error in 9.89 g is = (± 0.005 / 9.89) × 100% ⇒ ± 0.05% iv) Finally, remember that intermediate results in a multi-step computation should be

calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.





14

v) Theses should be justified by the data and then the arithmetic operations may be carried out; otherwise rounding errors can build up. For example, the reciprocal of 9.58, calculated (after rounding off) to the same number of significant figures (three) is 0.104, but the reciprocal of 0.104 calculated to three significant figures is 9.62. However, if we had written 1/9.58 = 0.1044 and then taken the reciprocal to three significant figures, we would have retrieved the original value of 9.58.

iv) This example justifies the idea to retain one more extra digit (than the number of digits in the least precise measurement) in intermediate steps of the complex multi-step calculations in order to avoid additional errors in the process of rounding off the numbers.




23

ELEMENTS OF VECTORS

1. Scalar : A physical quantity having only magnitude but not associated with any direction is called a scalar. eg: time, mass, distance, speed, work, energy, power, pressure, temperature, electric current, gravitational potential, pole strength, magnetic flux, entropy, electric capacity, velocity of light, large angular displacement, electric charge, etc.

2. Scalars are added and subtracted by algebraic method. 3. Vector : A physical quantity having magnitude as well as associated direction and which obeys

vector laws is called a vector. eg: displacement, velocity, acceleration, force, momentum, impulse, moment of force, small angular displacement, angular velocity, angular acceleration, magnetic moment, dipole moment, current density, intensity of electric field or magnetic field, shearing stress, weight, centrifugal force, infinitesimally small area, etc.

4. Vectors are completely described by a number with a unit followed by a statement of direction.

5. Angle can be considered as vector if it is small. Large angles can not be treated as vectors as they do not obey laws of vector addition.

6. Surface area can be treated both as a scalar and a vector. A is magnitude of surface area which is a scalar. This area is enclosed by a closed curve as shown if n is a unit vector normal to the surface, we can write ˆA n as a vector. ⇒ Surface area is a vector. [If the four fingers of right hand curl along the direction of arrow of enclosing curve, thumb indicates direction of area vector]

7. Tensor is a physical quantity which will have different values along different directions. e.g. Moment of inertia, stress.

8. A vector is represented by a directed line segment. The length of the line segment is proportional to the magnitude of the vector.

9. The magnitude or modulus of a vector (| r | or r) is a scalar. 10. Electric current, velocity of light has both magnitude and direction but they do not obey the laws of

vector addition. Hence they are scalars. 11. Equal vectors : Two vectors are said to be equal if they have the same magnitude and direction

irrespective of their initial points. 12. Negative vectors : A and A are vectors having the same magnitude and opposite direction. A is

called the negative of A . 13. Proper vector : A vector whose magnitude is not zero is known as proper vector. 14. Null Vector (Zero Vector): It is a vector whose magnitude is zero and direction is unspecified.

Examples: a) Displacement after one complete revolution. b) Velocity of vertically projected body at the highest point

15. Parallel vectors : Vectors in the same direction are called parallel vectors. 16. Antiparallel vectors : Vectors in opposite direction are called antiparallel vectors. 17. Like vectors or co-directional vectors : The vectors directed in the same direction,

irrespective of their magnitudes are called co-directional vectors or like vectors.

n




Elements of Vectors

24

18. Collinear vectors : Two or more vectors parallel or antiparallel to each other are called collinear

vectors. 19. Coplanar vectors : Vectors lying on the same plane are called coplanar vectors and the plane in

which they lie is called the plane of the vectors. 20. Unit vector : It is a vector whose magnitude is unity. A unit vector parallel to a given vector.

21. If A is a vector, the unit vector in the direction of A is written as|A|

AA of modulus

A vectorAora == .

kandj,i are units vectors along x, y and z axis. 22. Position vector : The vector which is used to specify the position of a point

‘P’ with respect to some fixed point ‘O’ is represented by OP and is known as the position vector of ‘P’ with respect to ‘O’.

23. Real Vector or Polar Vector: If the direction of a vector is independent of the coordinate system., then it is called a polar vector. Example: linear velocity, linear momentum, force, etc.,

24. Pseudo or axial vectors : Axial vectors or pseudo vectors are those whose direction is fixed by convention and reverses in a mirror reflection. Cross product of two vectors gives an axial vector. eg : Torque, angular velocity, etc.

25. A vector remains unchanged when it is moved parallel to itself. 26. If m is a scalar and A a vector, then m A is a vector. Its magnitude is m times that of magnitude of

A . Its direction is the same as that of A , if m is positive and opposite if m is negative, ; amF ; VmP ==

BmF ; qEF == . If m is zero, m A is a null vector. 27. Vector multiplication obeys commutative law when multiplied by a scalar. s A = A s where s is

scalar. 28. Vector multiplication obeys associative law when multiplier by a scalar i.e. m(n A )=mn A (m, n

are scalars) 29. Vector multiplication obeys distributive law when multiplied by a scalar. s( A +B )=s A +sB . ADDITION OF VECTORS: 30. Addition of vectors is also called resultant of vectors. 31. Resultant is a single vector that gives the total effect of number of vectors. Resultant can be found by using a) Triangle law of vectors

b) Parallelogram law of vectors c) Polygon law of vectors

32. Two vectors can be added either by triangle law or parallelogram law of vectors. 33. Triangle law : If two vectors are represented in magnitude and direction by the

two sides of a triangle taken in order, then the third side taken in the reverse order represents their sum or resultant in magnitude as well as in direction.

34. Parallelogram law : If two vectors Q and P are represented by the two sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

R = θ++ cosPQ2QP 22

O

P

αθ

P

RQβ

P

QPR +=

Q




Elements of Vectors

25

αβ

A

B

R

Tanθ+

θ=α

cosQPsinQ ;

θ+θ

=βcosPQ

sinPtan

35. The resultant of two vectors is the vectorial addition of two vectors. 36. The resultant of any two vectors makes lesser angle with the greater vector. 37. If |B||A| > β<α 38. The magnitude of the resultant of two vectors of magnitudes a and b with arbitrary directions must

be in the range (a – b) to (a + b). 39. b and a are two vectors which when added give a vector cba )(i.e., c =+ and if

i) b and a then |c| |b||a| =+ are parallel vectors ( θ = 0°) ii) b and a then |c| |b||a| 222 =+ are

perpendicular vectors ( θ = 90°) iii) b and a then |c| |b||a| =− are antiparallel

vectors ( θ = 180°) iv) b and a then |c| |b| |a| == are inclined to

each other at 120° v) If °=θ=+= 120 then |,A| |BA| and |B| |A| . vi) If °=θ=−= 60 then |,A| |BA| and |B| |A| . vii) If °=θ=+ 90 then |,B-A| |BA| .

40. If two vectors each of magnitude F act at a point, the magnitude of their resultant (R) depends on the angle θ between them. R = 2Fcos( θ /2).

Angle between forces ( θ ) Magnitude of resultant 0° 2F 60° 3 F 90° 2 F

120° F 180° 0

41. Minimum number of equal vectors to give a zero resultant is 2. 42. The minimum number of unequal vectors to give a zero resultant is 3. 43. There are three laws of addition of vectors.

a) Commutative law: ABBA +=+ b) Associative law: ( ) ( ) CBACBA ++=++ c) Distributive law: ( ) BmAmBAm +=+ where m is a scalar.

44. If the number of vectors is more than two, polygon law of vectors is used.

45. Polygon law : If a number of vectors are represented by the sides of a polygon taken in the same order, the resultant is represented by the closing side of the polygon taken in the reverse order.

DCBAR +++=

A

R B

CD




Elements of Vectors

26

46. Resolution of a vector in two dimensions : If A is a vector making an angle θwith x–axis, then X–component = Acos θ , Y–component = Asin θ .

47. If j and i are unit vectors along X and Y axes, any vector lying in XOY plane can be

represented as ; jAiAA yx +=

48. AAA |A| 2y

2x +== ;

x

y

AA

Tan =θ

49. The component of a vector can have a magnitude greater than that of the vector itself. 50. The rectangular component cannot have magnitude greater than that of the vector itself. 51. If a number of vectors ,.....D ,C ,B ,A acting at a point are resolved along

X–direction as Ax, Bx, Cx, Dx, … along Y–direction as Ay, By, Cy, Dy….. and if R is the resultant of all the vectors, then the components of R along X–direction and Y–direction are given by Rx = Ax + Bx + Cx + Dx + … and Ry = Ay + By + Cy + Dy + … respectively, and

R =x

y2y

2x R

Rtan ;RR =θ+ where θ is the angle made by the resultant with

X–direction. 52. If k and j ,i are unit vectors along X, Y and Z–axes, any vector in 3 dimensional space can be

expressed as 2z

2y

2xzyx AAAA |A| ;kAjAiAA ++==++= Here Ax, Ay, Az are the components of A and B are

scalars. A is body diagonal of the cube. 53. If α, β and γ are the angles made by A with X–axis, Y–axis and Z–axis respectively, then

1coscoscos

and A

Acos ;

AA

cos ; A

Acos

222

zyx

=γ+β+α

=γ=β=α

2sinsinsin 222 =γ+β+α 54. If cosα= l, cosβ = m and cos γ = n, then l, m, n are called direction cosines of the vector.

l2 + m2 + n2 = 1.

55. If vectors kAjAiAA zyx ++= and x y zˆ ˆ ˆB B i B j B k= + + are parallel, then

z

z

y

y

x

xBA

BA

BA

== and A =KB

where K is a scalar. 56. The vector kji ++ is equally inclined to the coordinate axes at an angle of 54.74°. 57. The position vector of a point P(x,y,z) is given by

222 zyx |OP| and kzjyixOP ++=++= 58. The vector having initial point P(x1, y1, z1) and final point Q(x2, y2, z2) is given by

k)zz(j)yy(i)xx(PQ 121212 −+−+−= . 59. Equilibrium is the state of a body in which there is no acceleration i.e., net force acting on a body

is zero. 60. The forces whose lines of action pass through a common point (called the point of concurrence)

are called concurrent forces.

O (origin)

P(x,y,z)

Y

O ∧

∧ X

Aθ

Axi

Ayj

Y

O X

Aθ

Acosθ

Asinθ

A

X

Y

Z

jAy

kAz

iAx




Elements of Vectors

27

61. Resultant force is the single force which produces the same effect as a given system of forces acting simultaneously.

62. A force which when acting along with a given system of forces produces equilibrium is called the equilibrant.

63. Resultant and equilibrant have equal magnitude and opposite direction. They act along the same line and they are themselves in equilibrium.

64. Triangle law of forces : If a body is in equilibrium under the action of three coplanar forces, then these forces can be represented in magnitude as well as

direction by the three sides of a triangle taken in order. |R|

r|Q|

q|P|

p== where p, q,

r are sides of a triangle. R,Q,P are coplanar vectors. 65. Lami’s theorem : When three coplanar forces R and Q ,P keep a body in

equilibrium, then γ

=β

=α sin

Rsin

Qsin

P .

66. When a number of forces acting on a body keep it in equilibrium, then the algebraic sum of the components along the X–direction is equal to zero and the algebraic sum of the components along the Y–direction is also equal to zero. i.e., ∑ ∑ == .0F and 0F yx

67. If 0A....AAA n321 =++++ and A1 = A2 = A3 = … An, then the adjacent vectors are inclined to each

other at an angle N2π or

N360° .

68. N forces each of magnitude F are acting on a point and angle between any two adjacent forces is θ,

then resultant force Fresultant =

NFsin2

sin( / 2)

θ⎛ ⎞⎜ ⎟⎝ ⎠θ

.

69. BODY PULLED HORIZONTALLY : i) A body is suspended by a string from a rigid support. It is pulled aside so that it makes an angle

θ with the vertical by applying a horizontal force F. When the body is in equilibrium, ii) Horizontal force,

F= mgTanθ iii) Tension in the string,

T = θCos

mg

iv) T= 22 F)mg( +

v) xF

xl

mglT

22=

−=

70. If a body simultaneously possesses two velocities v and u , the resultant velocity is given by the following formulae. a) If v and u are in the same direction, the resultant velocity will be |v||u| + in the direction of u

or v . b) If v and u are in opposite direction, the magnitude of the resultant velocity will be |v|~|u| and

acts in the direction of the greater velocity. c) If v and u are mutually perpendicular, the resultant velocity will be 22 vu + making an angle

tan–1(v/u) with the direction of u .

αβ

γ

Q

R

P

R r pq

P

Q

mg

FT

lθ

x

l-x

22




Elements of Vectors

28

71. If 1v is the velocity of the flow of water in a river and 2v is the velocity of a boat (relative to still water), then the velocity of the boat w.r.t the ground is 21 vv + . i. If the boat is going down stream, the velocity of the boat relative to the ground,

|v||v||v| 21 += . ii. If the boat is going upstream, the velocity of the boat relative to the ground, |v||v||v| 21 −= .

iii. If the boat is moving at right angles to the stream, 22

21 vvv += making an angle of tan 1(v1/v2)

with the original direction of the motion. 72. MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST TIME : If RB V and V are the velocities of a boat and river flow respectively then to

cross the river in shortest time, the boat is to be rowed across the river i.e., along normal to the banks of the river.

i) The direction of the resultant is θ = tan–1 ⎟⎟⎠

⎞⎜⎜⎝

⎛

B

RVV with the normal or tan

θ=dx

VV

B

R =

ii) Magnitude of the resultant velocity 2R

2B vvv +=

iii) Time taken to cross the river, t = Bvd where

d=width of the river or t=R2

R2

B

22

Vx

VV

xd=

+

+

iv) This time is independent of velocity of the river flow. v) The distance travelled down stream = BC

x = RB

VVd

×

73. MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST DISTANCE : i) The boat is to be rowed upstream making some angle θ with normal to the

bank of the river which is given by θ = sin–1 ⎟⎟⎠

⎞⎜⎜⎝

⎛

B

RVV or sinθ=

B

RVV

ii) The angle made by boat with the bank or river current is (90° + θ) iii) Resultant velocity has a magnitude of

V = 2R

2B VV −

iv) The time taken to cross the river is t = 2R

2B VV

d

−

Subtraction of two vectors : 74. If Q and P are two vectors, then P – Q is defined as P + )Q(− where Q− is the negative vector of Q .

If QPR −= , then θ−+= PQCos2QPR 22 In the parallelogram OMLN, the diagonal OL represents BA + and the diagonal NM represents BA − a) subtraction of vectors does not obey commutative law ABBA −≠− b) subtraction of vectors does not obey Associative law C)BA()CB(A −−≠−−

θ

A

d

BC

VB

VR

x

θ

A

d

B C

BV RV

x

N L

AO

B

M




Elements of Vectors

29

c) subtraction of vectors obeys distributive law m BmAm)BA( −=− 75. If two vectors each of magnitude ‘F’ act at a point, the magnitude of their difference depends on

the angle ‘θ’ between then

Magnitude of difference of vectors = 2Fsin ⎟⎠

⎞⎜⎝

⎛ θ2

76. Relative velocity : When the distance between two bodies is altering either in magnitude or direction or both, then each is said to have a relative velocity with respect to the other.

Relative velocity is vector difference of velocities. a. The relative velocity of body 'A' w.r.t. 'B' is given by BAR VVV −= b. The relative velocity of body 'B' w.r.t. 'A' is given by ABR VVV −= c. ABBA VV andVV −− are equal in magnitude but opposite in direction d. θ−+=−= cos.VV.2VVVVV BA

2B

2ABAR

e. For two bodies moving in the same direction, relative velocity is equal to the difference of velocities. (θ = 0°.cos 0 = 1)

RV = VA – VB

f. For two bodies moving in opposite direction, relative velocity is equal to the sum of their velocities. (θ =180°;cos180 = –1) ∴ RV =VA + VB

g. If they move at right angle to each other, then the relative velocity = 2

221 vv + .

77. Rain is falling vertically downwards with a velocity RV and a person is travelling with a velocity .VP Then the relative velocity of rain with respect to the person is PR VVV −= .

Relative velocity = 2P

2R VV|V| += .

78. The direction of relative velocity (or) the angle with the vertical at which an umbrella is to be held

is given by Tanθ = R

P

VV .

79. If the product of two vectors is another vector, such a product is called vector product or cross product.

80. If the product of two vectors is a scalar, then such a product is called scalar product or dot product.

DOT PRODUCT : 81. Dot product is the product of one vector and the component of another vector in its direction.

Eg : Magnetic flux, instantaneous power, work done, potential energy. 82. Dot product of two vectors A and B = A .

B = | A ||B | cos θ = AB cos θ a) Scalar product is commutative i.e., a.bb.a = b) Scalar product is distributive i.e., c.ab.a)cb.(a +=+

83. If A and B are parallel vectors, then A .B = AB.

-VP

VR

VP

Vα

θ




Elements of Vectors

30

84. If A andB are perpendicular to each other, then A .B =0. 85. If A andB are antiparallel vectors, then A .B = –AB. 86. Dot product of two vectors may be positive or negative. If θ<90°, it is positive and 90°<θ<270° it

is negative. 87. In the case of unit vectors,

1k kj ji i === . . . and 0i kk jj i === . . . 88. If x y z

ˆ ˆ ˆA A i A j A k= + + x y zˆ ˆ ˆand B B i B j B k= + + , then x x y y z zA B A B A B A B= + +. , 2 2 2

x y zA A A A A= + +. . APPLICATIONS OF DOT PRODUCT : 89. W = S.F (dot product of force and displacement is work) 90. P = V.F (dot product of force and velocity is power) 91. Ep = h.gm (dot product of gravitational force and vertical displacement is P.E) 92. Magnetic flux, φ = B.A (dot product of area vector and magnetic flux density vector)

93. Angle between the two vectors b and a is given by Cosθ = |b||a|

b.a

94. The magnitude of component of vectorB along vector |A|

B.AA = .

95. The magnitude of component of vector A along vector |B|

B.AB = .

96. Component of vector A along vector B = A.B ˆ.B| B |

.

CROSS PRODUCT : 97. The vector or cross product of two vectors B and A is a vector C whose magnitude is AB sin θ where

θ is the angle between the vectors B and A and the direction of C is perpendicular to both B and A such that C and B ,A form a right hand triple.

Eg: angular momentum ( ω×= rL ), torque ( Fr ×=τ ), angular velocity ( rV ×ω= ) etc.

In the case of unit vectors 0. k kj ji i

and j- k x i ;i- j x k ;k- i x j

; ji k ; i k j ; kj i

=×=×=×

===

=×=×=×

98. If kBjBiBB and kAjAiAA zyxzyx ++=++= , then BA × = zyx

zyx

BBBAAAkji

=

k)BABA(j)BABA(i)BABA( xyyxxzzxyzzy −+−−−

99. If B x A = 0 and B and A are not null vectors, then they are parallel to each other. B x A = – AxB

)B(m AB )A(m)B Am(

; C AB A)CB( A

×=×=×

×+×=+×

a) ABBA ×≠× (commutative law is not obeyed) b) C)BA()CB(A ××≠××




Elements of Vectors

31

(Associate law is not obeyed) c) CxABA)CB(A +×=+×

(Distributive law is obeyed) APPLICATIONS OF CROSS PRODUCT : 100. Torque is the cross product of radius vector and force vector, Fr ×=τ 101. Angular momentum is the cross product of radius vector and linear momentum, prL ×= 102. Linear velocity in circular motion may be defined as the cross product of angular velocity and

radius vector. rV ×ω=

103. The area of the triangle formed by B andA BA21 is sides adjacent as × .

104. Area of triangle ABC if position vector of A is a , position vector of B is b and position vector of

C is c , then area = |accbba|21

×+×+× .

105. The area of the parallelogram formed by B andA as adjacent sides is .BA ×

106. If QandP are diagonals of a parallelogram, then area of parallelogram= )QxP(21 .

107. Unit vector parallel to C or normal to A and |BxA|

BxAnisB = .

108. If 0)CxB.(A = , then CandB,A are coplanar. 109. If CBA =+ , then 0)CxB.(A = . 110. Division by a vector: is not defined because it is not possible to divide a direction by a

direction.




37

KINEMATICS

1. The branch of physics that deals with the motion of a body due to the application of force is called mechanics.

2. Mechanics is divided into two branches namely dynamics and statics. 3. The branch of mechanics that deals with the state of rest of a body is called statics. 4. The branch of mechanics that deals with the state of motion of a body is called dynamics. 5. Dynamics is classified into kinematics and kinetics. 6. Kinematics is the study of motion which relates to the motion of bodies without reference to either

the mass or the force causing it. 7. Kinetics is the study of motion which relates to the action of forces causing the motion and the

mass that is moved. 8. A body is said to be at rest if its position remains constant with respect to its surroundings or frame

of reference. 9. A body is said to be in motion if its position is changing with respect to its surroundings or frame

of reference. 10. The line joining the successive positions of a moving body is called its path. The length of the

path gives the distance travelled by the body. 11. Displacement is the directed line segment joining the initial and final positions of a moving body.

It is a vector. 12. If every particle of a moving body traverses the same distance along parallel paths, which may be

straight or curved, while the body is moving, then the motion of the body is called translatory motion.

13. When the path traversed by each particle of a body is a straight line, then its motion is said to be rectilinear.

14. When the path traversed by the particles are parallel paths, then the motion is said to be curvilinear.

SPEED : 15. Speed of a body is the rate at which it describes its path. Its SI unit is ms−1. 16. Speed is a scalar quantity.

17. Speed =taken timetravelled distance .

18. A body is said to be moving with uniform speed if it has equal distances in equal intervals of time, however small these intervals may be.

19. A body is said to be moving with non uniform speed if it has unequal distances in equal intervals of time or equal distances in unequal intervals of time, however small these intervals may be.

20. Average speed =taken time totaltravelled distance total

21. Instantaneous speed =dtds

tsLt

0t=

∆∆

→∆.




Kinematics

38

22. If a particle covers the 1st half of the total distance with a speed ‘a’ and the second half with a speed ‘b’.

Average speed =ba

ab2+

.

23. If a particle covers 1st 1/3rd of a distance with a speed ‘a’, 2nd 1/3rd of the distance with speed ‘b’ and 3rd 1/3rd of the distance with speed ‘c’

Average speed =cabcab

abc3++

.

24. 1 kmph = 185 ms−1 ; 1 mph =

1522 fts−1

25. For a body with uniform speed, distance travelled = speed x time.

Velocity : 26. The rate of change of displacement of a body is called velocity. Its SI unit is ms−1. 27. Velocity is a vector quantity. 28. A body is said to be moving with uniform velocity, if it has equal displacements in equal intervals

of time, however small these intervals may be. 29. For a body moving with uniform velocity, the displacement is directly proportional to the time

interval. 30. If the direction or magnitude or both of the velocity of a body change, then the body is said to be

moving with non-uniform velocity.

31. Average velocity = taken time total

ntdisplaceme net

32. For a body moving with uniform acceleration, the average velocity = 2

vu + .

33. The velocity of a particle at any instant of time or at any point of its path is called instantaneous

velocity. V =dtds

tsLt

0t=

∆∆

→∆

34. Average velocity : a. If a particle under goes a displacement s1 along a straight line t1 and a displacement s2 in time t2

in the same direction, then

Average velocity=21

21ttss

++

b. If a particle undergoes a displacement s1 along a straight line with velocity v1 and a displacement s2 with velocity v2 in the same direction, then

Average velocity =1221

2121vsvsvv)ss(

++

c. If a particle travels first half of the displacement along a straight line with velocity v1 and the next half of the displacement with velocity v2 in the same direction, then




Kinematics

39


21vv

vv2+

(in the case (b) put s1 = s2)

d. If a particle travels for a time t1 with velocity v1 and for a time t2 with velocity v2 in the same direction, then


2211tt

tvtv++

e. If a particle travels first half of the time with velocity v1 and the next half of the time with velocity v2 in the same direction, then

Average velocity =2

vv 21 + (in the case d put t1 = t2)

35. Velocity of a particle is uniform if both its magnitude and direction remains unchanged. 36. Velocity of a body changes when magnitude or direction or both change. 37. Acceleration :

a. If the velocity of a body changes either in magnitude or in direction or both, then it is said to have acceleration.

b. For a freely falling body, the velocity changes in magnitude and hence it has acceleration. c. For a body moving round a circular path with a uniform speed, the velocity changes in

direction and hence it has acceleration. d. For a projectile, whose trajectory is a parabola, the velocity changes in magnitude and in

direction, and hence it has acceleration. e. The acceleration and velocity of a body need not be in the same direction. eg : A body thrown

vertically upwards. f. If equal changes of velocity takes place in equal intervals of time, however small these

intervals may be, then the body is said to be in uniform acceleration. g. Negative acceleration is called retardation or deceleration. h. The acceleration of a particle at any instant or at any point is called instantaneous

acceleration.

a =2

2

0t dtsd

dtvd

tvLt ==

∆∆

→∆

i. For a body moving with uniform acceleration, the average velocity = 2

vu + .

j. A body can have zero velocity and non-zero acceleration. Eg : for a particle projected vertically up, velocity at the highest point is zero, but acceleration is −g.

k. If a body has a uniform speed, it may have acceleration. Eg : uniform circular motion l. If a body has uniform velocity, it has no acceleration. m. When a body moves with uniform acceleration along a straight line and has a distance ‘x’ travelled

in the nth second, in the next second it travels a distance x+a, where ‘a’ is the acceleration. n. Acceleration of free fall in vacuum is uniform and is called acceleration due to gravity (g) and

it is equal to 980 cms−2 or 9.8 ms−2.




Kinematics

40

38. displacement-time graph :

1) Straight lines represents uniform velocity 2) Slope of straight line gives velocity 3) Smooth curves represents uniform acceleration 4) Zig zag curve represents non-uniform acceleration

s s s

t t t

uniformvelocity zero velocity

variablevelocity

s

s st

t t

1) 2)

3) 4)

uniformacceleration

uniformdeceleration

uniformacceleration

uniformdeceleration

s

t

39. Velocity-time graph :

1) Slope gives the acceleration. 2) Area under the graph gives the distance travelled 3) Curve represents non-uniform acceleration. 4) Straight line represents uniform acceleration.

v v v

v

v

v

v

t t t

t

t

t

t

zero acceleration

uniformretardation

uniformacceleration

A body projected up

variableacceleration area=s ut

(at2)/2

40. The equations of motion for uniform acceleration :

1) v = u + at 2) s = ut + 21 at2

3) v2 - u2 = 2as 4) sn = u + 2a (2n − 1)

5) s =

+

2vu t




Kinematics

41

One dimensional motion : 41. If a body starts from rest and moves with uniform acceleration ‘a’ and if Dm is the distance

travelled by it in mth second and Dn is the distance travelled in nth second, then a = n mD Dn m

−−

.

42. For a particle moving with uniform velocity a = 0 ∴ S = U t or S α t

43. For a particle moving with uniform retardation along a straight line distance travelled before

coming to rest (v = 0) is s = a2

u2

∴ sα U2 ∴ 22

21

2

1

uu

ss

=

44. If a particle travels along a straight line with uniform acceleration and travels distances Sn and Sn+1 in two successive seconds, the acceleration of the particle is

a = Sn+1 - Sn 45. If a particle travels along a straight line with uniform acceleration and travels distances S and SI in

two successive intervals of n seconds each, the acceleration of the particle is

a = 2

I

nSS −

46. Moving with uniform acceleration from rest, a body attains a velocity 'v' after a displacement 'x', then its velocity becomes 'nv' after a further displacement (n2 - 1)x.

47. If a bullet loses (1/n)th of its velocity while passing through a plank, then the no. of such planks

required to just stop the bullet is = 1n2

n2

−

48. The velocity of a body becomes th

n1

of its initial velocity after a displacement of 'x', then it will

come to rest after a further displacement of 1n

x2 −

.

49. The displacement of a body is proportional to the square of time, then its initial velocity is zero. 50. Starting from rest a body travels with an acceleration 'α' for some time and then with deceleration

'β' and finally comes to rest. If the total time of journey is 't', then the maximum velocity and displacement are given by

( ) ( ))(2

ts,tV2

max β+ααβ

=β+α

αβ= and

average velocity =

2

Vmax

51. If R is the range of head lights and ‘a’ is the maximum retardation of an automobile, then its maximum safe speed = aR2 .

52. If a body starts from rest and moves with uniform acceleration, then the ratio of the times to cover 1st, 2nd, 3rd, 4th …… nth metres of the distance is )1nn)....(23(:)12(:)01( −−−−−

53. A body is projected vertically up from a topless car relative to the car which is moving horizontally relative to earth. a. If car velocity is constant, ball will be caught by the thrower.




Kinematics

42

b. If car velocity is constant, path of ball relative to the ground is a parabola and relative to this car is straight up and then straight down.

c. If the car accelerates, ball falls back relative to the car. d. If acceleration or retardation of the car is constant path relative to car is a straight line and

relative to ground is a parabola. 54. The equations of motion for a body

a) projected up b) freely falling v = u −gt v = gt s = ut −

21 gt2 s =

21 gt2

v2 −u2 =−2gs sn = 2g (2n −1)

time of rise =gu time of fall =

gu =

g2h

55. Freely falling body : a. Any freely falling body travels g/2 metres or 4.9 m in the first second. b. During the free fall of a body, after a certain interval of time, if gravity disappears then the

body continues to move with uniform velocity acquired during its free fall. c. The displacements of a freely falling body in successive seconds or in equal intervals of time

are in the ratio of 1 : 3 : 5 : 7 : …… This also holds good for a body starting from rest and moving with uniform acceleration.

d. The displacements of a freely falling body in 1 s, 2 s, 3 s, 4 s, ……………….n s are in the ratio 12 : 22 : 32 : 42 : 52 : 62……

e. In the presence of air resistance, the acceleration of a denser body is greater.

f. The acceleration of a body in a medium is given by gI = g

−

b

mdd1 =

−

b

mdd

1g

(dm = density of the medium) (db = density of the body) If dm = db; gI = 0. So it will remain at rest or in uniform motion.

g. A freely falling body' passes through two points A and B in time intervals of t1 and t2 from the start, then the distance between the two points is given by AB = ( )2

122 tt

2g

−

h. A freely falling body passes through two points A and B distant h1 and h2 from the start, then

the time taken by it to move from A to B is given by T = )hh(g2

gh2

gh2

1212 −=−

i. Two bodies are dropped from heights h1 and h2 simultaneously. Then after any time the distance between them is equal to (h2 – h1)

j. stone is dropped into a well of depth 'h', then the sound of splash is heard after a time of 't'

given by t= soundvh

gh2

+

k. A stone is dropped into a river from the bridge and after 'x' second another stone is projected down into the river from the same point with a velocity of 'u'. If both the stones reach the water simultaneously, then




Kinematics

43

22 )xt(g21)xt(ugt

21

−+−=

Body thrown vertically upwards : 56. The time of flight of a body thrown up vertically with velocity u is 2u/g.

maximum height =g2

u2

57. Hmax = 2max

2uH

g2u

α⇒ (independent of the mass of the body)2

2

122

11

2

1uu

u

uHH

==⇒

58. If a body projected vertically upwards takes ‘t’ seconds to reach the maximum height, then in the first t/2 seconds it travels 3/4th of the maximum height.

59. ta = td = gu ⇒ T =ta + td =

gu2 ⇒T or ta ∝ u

60. At any point of the journey, a body possesses the same speed while moving up and while moving down.

61. The height reached in the first second of ascent is equal to the height of fall in the last second of descent.

62. Irrespective of velocity of projection, all the bodies pass through a height 2g in the last second of

ascent. 63. The change in velocity over the complete journey is '2u' (downwards). 64. If a vertically projected body rises through a height 'h' in nth s, then in (n-1)th s it will rise through a

height h + g and in (n+1)th s it will rise through height h-g. 65. If a body is projected vertically up from the top a tower of height h with a

velocity u and takes "t" seconds to reach the ground

h = −u t + 2gt21 utgt

21h 2 −=⇒ ;

h = g2uv 22 −

66. If an object is dropped from a balloon rising up with a velocity u at a height h. a) Equation of motion relative to earth is

h = 2gt21 – ut

b) Equation of motion relative to balloon is

h = 2B t)ag(

21

+ , where aB is acceleration of the balloon.

c) Relative to earth body goes up and then falls d) Relative to the balloon it falls vertically downward. 67. If a body is projected vertically up with a velocity u from a tower and it reaches the ground with a

velocity nu, the height of the tower is

h = )1n(g2

u 22

−

v

h u

u




Kinematics

44

68. A body is thrown vertically upwards with a velocity ‘u’ from the ground reaches a point ‘P’ on its path at height ‘h’ after a time t1 and t2 from the beginning, then

u = )tt(2g

21 + ; h = 21tgt21 .

69. A particle projected vertically up from the top of a tower takes t1 sec to reach the ground. Another particle thrown downwards with the same velocity from the top of the tower takes t2 seconds to reach the ground. If the particle is dropped from the top of the tower, time taken is t, then

a) t = 21tt

b) height of the tower is h = 21tgt21

c) Velocity of projection is u = )tt(2g

21 −

d) In the first and second case body reaches the ground with the same velocity 70. A body is dropped from the top edge of a tower of height 'h' and at the same time another body is

projected vertically up from the foot of the tower with a velocity 'u', then they will meet after a time of

t = uh and at a distance of

2

22

1u2

ghgt21h == from the top of the tower (or)

h2 = h - 2

2

u2gh from the foot of the tower.

Their velocities at the meeting point are

V1 = gt = ugh (freely falling body)

V2 = u - ugh (vertically projected body)

71. In the absence of air resistance, time of ascent and time of descent are equal. 72. Time of rise < time of fall, if air resistance is taken into account. 73. A elevator is accelerating upwards with an acceleration a. If a person inside the elevator throws a

particle vertically up with a velocity u relative to the elevator, time of flight is t = ag

u2+

74. In the above case if elevator accelerates down, time of flight is t = ag

u2−

PROJECTILES : 75. A body thrown with an angle with the horizontal is called a projectile. 76. The path traced by a projectile is called trajectory and is a parabola. 77. Oblique projectile :

1) For a projectile, the horizontal component of velocity is (ucos θ ). It remains constant throughout the motion

2) The vertical component (usin θ ) is subjected to acceleration due to gravity.




Kinematics

45

78. Equations for an oblique projectile :

a) Maximum height reached=g2

sinu 22 θ

b) Time of flight =gsinu2 θ

Time of rise = time of fall =g

sinu θ

c) Range =g

2sinu2 θ ; Rmax = gu2

for θ = 45°

d) tanθ =R

H4 max ; tanθ =R2

gT2

79. At the maximum height

a) Kinetic Energy = θ= 222x Cosmu

21mu

21

b) Potential Energy = mgHmax θ= 22Sinmu21

c) Total Energy = K.E + P.E = 2mu21

d) If K.E = P.E then θ = 45°

e) E.KE.Ptan2 =θ

80. Velocity after time "t": a) ax = 0; ay = – g b) Horizontal component of velocity through out the motion is constant. c) vertical component of velocity changes with time d) Horizontal component of velocity Vx=Ucosθ e) Vertical component of velocity Vy = U Sinθ−gt f) Velocity of the particle V = 22 VyVx +

g) Velocity of a projectile after t seconds v = 22 )gtsinu()cosu( −θ+θ .

h) Velocity of a projectile when it is at a height h is v = ]gh2)sinu[()cosu( 22 −θ+θ or gh2u2 − .

i) Direction of motion w. r. t. to horizontal.

x

y1VV

tan−=α

j) If α is the angle made by a projectile after t seconds, then θ

−θ=α

cosugtsinutan .

k) If α is the angle made by a projectile after traveling a height of h, then

θ

−θ=α

cosugh2)sinu(

tan2




Kinematics

46

l) Velocity at highest point is UCosθ in the horizontal direction. m) Vertical component of velocity at the highest point Vy = 0 n) Velocity and acceleration are perpendicular to each other at the

highest point. o) If projected from level ground velocity of landing and angle of landing is same in magnitude as

during projection. p) If projected from level ground velocity is maximum during projection and during landing and

minimum at the highest point. q) If projected from level ground

a) Change in velocity till it reaches highest point = Usinθ b) Change in velocity for complete trajectory = 2Usinθ

r) Velocity of the projectile when it moves perpendicular to its initial velocity is U cotθ.

s) Time taken for the velocity to become perpendicular to the initial velocity is θsing

u

t) If V1 and V2 are the magnitudes of velocities at heights h1 and h2; g2VV 21

22 −=− (h2-h1)

81. Position of the projectile after time 't': a) If x and y represent the horizontal and vertical displacements with respect the point of

projection 't' seconds after projection x = (U cosθ) t

y = (U sinθ) t − 2gt21

b) Equation of trajectory is

Y = (tanθ) x−θ22

2

cosuxg

21

= 222

x.cosu2

gxtanθ

−θ

c) y = Ax–Bx2; A = tanθ, B = θ22Cosu2

g

maxH2A

4B= ; Range R =

BA

and 4A

RHmax = or =θtan

RH4 max

82. Complementary angles of projection a) For a given velocity of projection angles of projection are θ and 90–θ then they are called as

complementary angles of projection. Ex: 30°, 60°

b) Range is equal for complementary angles of projection (u = constant) R = Constant for θ and 90−θ.

c) If h1 and h2 are the maximum heights attained for complementary angles of projection

h1 + h2 = g2

u2; R = 4 21hh

θ θ

u

u




Kinematics

47

2

12hh

tan =θ ; Rmax = 2 (h1 +h2)

d) If t1 and t2 are the times of flight for complementary angles of projection

R = 21tgt21

83. If a man throws a body to a maximum distance R then he can project the body to vertical height R/2.

84. If a man throws a body to a maximum distance R then the greatest height attained by the body is R/4.

a) The angle between velocity and acceleration during the rise of projectile is 1800 < θ < 90°. b) The angle between velocity and acceleration during the fall of projectile is 0° < θ < 90°

85. If a body is projected up at an angle θ with the horizontal from the top of a tower of height 'h', then h = – (u sinθ) t + 2gt

21

86. If a body is projected down at an angle θ with the horizontal from the top of a tower then h = – (u sinθ) t + 2gt

21

Horizontal projection : 87. When a body is projected, horizontally from the top of a tower path of the body is parabola relative

to ground. a) it reaches the ground tracing a parabolic path. b) its time of descent is g/h2 . c) during its journey it suffers a horizontal displacement of u g/h2 . d) the angle α with which it strikes the ground is given by

tan α =u

g/h2 = ugt

e) the velocity with which it hits the ground is given by v = gh2u2 + or v = 22 )gt(u + .

88. If a body is projected horizontally and another is dropped from the same height, both the bodies will take same time to reach the ground. Position after time t :

Horizontal displacement after time t. x = u t

V R

O u

x

y

Y

V

X

V Vx = u θ

h




Kinematics

48

Distance fallen in time "t" y = 2gt21

Velocity after time t :

v = gh2u)gt(u 222 +=+

If angle made with the horizontal is α ugttan =α =

gh2gt .

Equation of path:

y = 2

2

uxg

21

89. Form a certain height. If two bodies are projected horizontally with velocities u1 and u2 in opposite directions.

a) Time after which velocity vectors are perpendicular is t = guu 21

b) Time after which displacement vectors are perpendicular is t = g

uu2 21

c) Distance between the two bodies when velocity vectors are perpendicular is

)uu(guu

2121 +

d) Horizontal distance between the two bodies when displacement vectors' are perpendicular is

)uu(guu

2 2121 +

90. Two bodies are projected from a tower horizontally with velocities 'u1' and 'u2' then

gh2u

gh2uvv

anduu

xx

nda1tt

22

21

2

1

2

1

2

1

2

1

+

+===

91. A bomb dropped from a plane moving horizontally with a uniform velocity reaches the ground following a parabolic path. (As seen by the pilot the bomb takes a vertically downward path).

92. From the top of a tower a stone is dropped and simultaneously another stone is projected horizontally with a uniform velocity. Both of them reach the ground simultaneously.

93. In the case of a projectile, velocity varies both in magnitude and direction but the acceleration remains constant both in magnitude and direction.

94. If air resistance is considered, trajectory departs from parabola; time of flight increases; striking angle increases; range decreases; maximum height decreases; striking velocity decreases and the time of ascent is less than the time of descent.

95. For projectiles like Inter Continental Ballistic Missiles (ICBM) the trajectory is a portion of an ellipse (due to large variation in altitude).

Motion of a body along an inclined plane : 96. Acceleration of a body sliding down a smooth inclined plane of angle of inclination θ with the

horizontal is given by a = g sin θ 97. If a body travells from rest on a smooth inclined plane

Velocity v= gh2singl2 =θ .




Kinematics

49

Time taken t=θ

=θ 2sing

h2sing

l2 .

98. A body on an inclined plane reaches the bottom with the same velocity as that of a freely falling body but in a different direction after a time of

θSin1 times that of a freely falling body.




57

DYNAMICS Synopsis : NEWTON’S LAWS OF MOTION : 1. Momentum is the quantity of motion possessed by a body by virtue of which it can set other

bodies in motion by collision. 2. Momentum is the product of mass and velocity ( vmP = ). SI unit is kg ms 1. It is a vector having the

same direction as that of velocity. 3. In finding the change in momentum, vector subtraction must be used. 4. If a ball of mass m moving with a speed v strikes a wall at right angle to it and rebounds with the

same speed, then the change in momentum is 2mv. 5. If a body of mass m thrown vertically upwards with a velocity u returns to the starting point, then

the change in its momentum is 2mu. 6. If a ball of mass m moving with a velocity u is struck by a bat and retraces its path with a velocity

v, then the change in momentum is m(v + u). 7. Newton’s first law of motion : Every body continues to be in the state of rest or of uniform

motion unless it is compelled by an external force to change that state. i.e., the momentum of a body remains constant as long as no external force acts on it.

8. The first law of motion leads to the concepts of force and inertia. 9. Inertia is the tendency of a body to preserve its state of rest or of uniform motion along a straight

line in the absence of any external force 10. The three types of inertia are:

i) inertia of rest ii) inertia of motion and iii) Inertia of direction

11. Inertia of rest: The inability of a body to change its state of rest by itself is called inertia of rest. Eg: When a bus at rest starts suddenly passengers fall back

12. Inertia of motion: the inability of a body to change its uniform motion by itself is called as inertia of motion. Eg: when a bus in uniform motion suddenly stops , the passengers fall forward.

13. Inertia of direction: The inability of a body to change its direction of motion by itself is called inertia of direction. Eg: When a bus takes a turn passengers will be pressed outwards.

14. Force is that which changes or tends to change the state of rest or of uniform motion of a body along a straight line.

15. Newton’s second law of motion : The rate of change of momentum of a body is directly proportional to the impressed force and takes place in the direction of force.

16. The second law of motion gives the direction and magnitude of force.

17. Force = time

momentum in change , i.e.,t

)uv(mF −= .

18. Force = mass x acceleration ; amF =

19. t

mumvF −=

t)uv(mF −

=

dtdvmF =

dtdm.vF =

F = ma ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

s2uvmF

22

2t

)uts(2mxF −=

s

mu21mv

21

F22 −

=




Dynamics

58

s

workF = velocitypowerF =

F = pressure × area F = mg 20. A unit force : is one which when acting on unit mass produces unit acceleration in its direction.

Units : SI unit is newton and cgs unit is dyne; 1 N = 105 dynes. 21. Gravitational unit of force :

1 kgwt = g N = 9.8 newton 1 gwt = g dynes = 980 dynes. Force = rate of change of mass x change in velocity

F=t

)uv(m − [rocket, conveyor belt problems, etc.

can be solved by this formula]

22. If a rocket ejects the exhaust gases with a velocity u relative to the rocket at the rate of dtdm , the

force F acting on the rocket is F = ⎟⎠

⎞⎜⎝

⎛dtdmu .

23. If gravel is dropped on a conveyor belt at the rate of dtdm , the extra force required to keep the belt

moving with velocity is F = ⎟⎠

⎞⎜⎝

⎛dtdmu .

24. A jet of water of density d from a tube of area of cross section a comes out with a velocity v. a) Average force exerted by tube on water is dAv2

b) Force required to hold the tube in a fixed position = dAv2 c) If the water traveling horizontally strikes a vertical wall normally and then flows down along

the wall, the normal force exerted on the wall is dAv2. d) In the above case if water rebounds with the same speed, force exerted on the wall is 2dAv2 e) In the above case if water strikes the surface at angle θ with the normal and reflects with the

same speed, force exerted on the wall is 2dAv2 Cosθ. 25. If a gun fires n bullets each of mass m per second each with a velocity u, the force F necessary to

hold the gun is F = mnu. 26. A very large force acting for a short interval of time is called impulsive force. Eg : Blow of a

hammer on the head of a nail. 27. The impulse of a force is defined as the product of the average force and the time interval for

which it acts. Impulse J = FAV Δ t = m v - mu

Impulse momentum theorem = ∫2

1

t

tF dt=m v –mu

28. Impulse due to a variable force is given by the area under F–t graph.

Impulse = dt.F2

1

t

t∫

29. While catching a fast moving cricket ball the hands are lowered, there by increasing the time of catch which thus decreases the force on hands.

30. A person jumping on to sand experiences less force than a person jumping on to a hard floor, because sand stops the person in more time.




Dynamics

59

31. If a force F1 acts on a body at rest for a time t1 and after that another force F2 brings it to rest again in a time t2, then F1t1 = F2t2.

32. The gravitational force that acts on a body is called its weight (W = mg). It is a vector always pointing in a vertically downward direction.

33. A bird is in a wire cage hanging from a spring balance when the bird starts flying in the cage, the reading of the balance decreases.

34. In the above case, if the bird is in a closed cage or air - tight cage and it hovers in the cage, the reading of the spring balance does not change.

35. In the above case for a closed cage if the bird accelerates upward reading of the balance is R = Wbird + ma, where m is the mass of the bird and a its acceleration.

36. If a block of mass m hangs at the end of a massless string and the string is pulled up, the tension in string is T = m (g + a) if the block accelerates in the upward direction. T = m (g - a) if the block is accelerated in the downward direction. T = mg if the block is moved up or down with uniform speed.

37. When a man stands on a weighing machine, the weighing machine measures the normal force between the man and the machine.

38. When the man and the weighing machine are at rest relative to the earth, reading of the weighing machine is N = mg = Weight of the man. N is also called apparent weight.

39. Man inside an elevator: a) Elevator accelerates up: Relative to earth N-mg=ma (M = mass of man) Apparent weight=N=m(g+a) Tension in the cable T = (Melevator + Mman) (g + a) Same effect is felt when elevator retards while going down. b) Elevator accelerates downward: Relative to earth mg-N=ma Apparent weight=N=m(g-a) Tension in the cable T = (Melevator + Mmam) (g - a)

Same effect is felt when elevator goes up and retards. If elevator falls freely (cable breads) N = 0 i.e. apparent weight of in a free fall = 0 40. Man inside an artificial satellite a) An artificial satellite orbiting the earth in a circular orbit is a freely falling body because its

centripetal acceleration is equal to the acceleration due to gravity in that orbit. 41. With a car at rest, mark the position of the stationary pendulum bob on the table under it, with the

car in motion a) If the bob remains over the mark only when the car is moving in a straight line at a constant

speed. (inertial frame of reference) b) If the car is gaining or losing the speed or is negotiating a bend, the bob moves from its mark

and the car is a non-inertial frame. 42. If you put a ball at rest on a rotating merry-go round, no identifiable force acts on the ball but it

does not remain at rest, it is non - inertial reference frame. 43. Newton’s third law : For every action there is an equal and opposite reaction.

Cable

m

a

mg

N

Cable

m

a

mg

N




Dynamics

60

44. Newton’s first and third laws are only special cases of second law. 45. Thrust is the total force applied on a given area. It is measured as the product of pressure and the

area on which the pressure is applied. 46. In nature forces always occur in pairs (action and reaction) 47. When a body exerts a force on another body, the second body exerts a force on the first body of the

same magnitude but in opposite direction. 48. If we tie one end of a string to any point of a body and pull at the other end of the string, we exert a

force on the body. Such a force exerted by means of a string is called tension. 49. When two objects are connected by an inextensible massless string passing over a smooth pulley or

peg, then (i) both will have the same acceleration and ii) the tension is the same on both the sides of the pulley.

50. Motion on a smooth inclined plane : (i) a = gsin θ (ii) final velocity at the bottom of the inclined plane v = gh2singL2 =θ (iii) time taken to reach the bottom

t =gh2

sin1

singL2

θ=

θ

51. acceleration (a) = 21

21mm

)sinmsinm(g+

β−α

tension = T 21

21mm

g)sin(sinmm+

β+α

52. Two masses m1 and m2 connected by a string pass over a pulley. m2 is suspended and m1slides up over a frictionless inclined plane of angle θ

a) Acceleration a = 21

12mm

g)sinmm(+

θ−

b) Tension in the string

T = m2g - m2a = )mm(

g]sin1[mm

21

21+

θ+

53. A body of mass m1 is placed on a smooth table. A string attached to m1 passes over a light pulley and carries a mass m2. Acceleration of the system.

i) 21

2mmgma

+=

ii) Tension in the string T = gmm

mm

21

21⎟⎟⎠

⎞⎜⎜⎝

⎛+

iii) Thrust on the pulley P= 2 T MOTION OF CONNECTED BODIES : 54. When two bodies are connected by a light string passing over a frictionless pulley a) m1 and m2 will have the same acceleration 'a' If m2 > m1

21

12mm

)mm(a

+−

= g.

T

m2

TR

mgsin1

θ

m g1

m1

m cos1 θ

m2

T Tm1

T

m g2

m g1

m1

m

TP

a




Dynamics

61

b) The tension is same on both sides of pulley 21

21mm

gmm2T

+=

c) Thrust (p) on the pulley is 2 T

21

21mm

gmm4P

+=

55. A block of mass M is pulled by a rope of mass m by a force P on a smooth horizontal plane.

a) Acceleration of the block a = mM

p+

b) Force exerted by the rope on the block

)mM(

MpF+

=

56. Masses m1, m2, m3 are inter connected by light string and are pulled with a string with tension T3 on a smooth table.

a) Acceleration of the system a = )mmm(

T

321

3++

b) Tension in the string

T1 = m1 a = 321

31mmm

Tm++

( ) ( )321

321212 mmm

TmmammT

+++

=+=

T3 = (m1 + m2 + m3) a 57.

Acceleration of the system

(a) =)mmmm(

F

4321 +++

Contact force between m1 and m2,

F12 = )mmmm(Fm

4321

1+++


F23 = )mmmm(F)mm(

4321

21+++

+


F34 = )mmmm(F)mmm(

4321

321+++

++

m1 m2 m3 m4Fig (vi)

F

P m M

T1 T2 T3 m1 m2 m3




Dynamics

62

58. A rope of length ‘L’ is pulled by a constant force ‘F’. The tension in the rope at a distance ‘x’ from

the end where it is applied is F(1 Lx ).

59. Limitations of Newton’s law of motion: a) It is applicable only for speeds V << C (C = speed of light) b) It is not applicable in the domain of atoms, molecules, sub atomic particles. c) It is not applicable when there is a very strong gravitational field. d) The concept of Newton III law is not applicable, when particles interact with each other by

means of a force field. e) Newton’s laws are not applicable for very small accelerations. (a < 10–1° ms–2) WORK–POWER–ENERGY : 1. Work is said to be done when the point of application of force has some displacement in the

direction of the force. 2. The amount of work done is given by the dot product of force and displacement. θ= cosFss.F 3. Work is independent of the time taken and is a scalar. 4. If the force and displacement are perpendicular to each other, then the work done is zero. 5. A person rowing a boat upstream is at rest with respect to an observer on the shore. According to

the observer the person does not perform any work. However, the person performs work against the flow of water. If he stops rowing the boat, the boat moves in the direction of flow of water and work is performed by the force due to flow, as there is displacement in the direction of flow.

6. If the work is done by a uniformly varying force such as restoring force in a spring, then the work

done is equal to the product of average force and displacement. 7. If the force is varying non–uniformly, then the work done = ∫ ds.F = ∫ θcos.ds.F . 8. The area of F–s graph gives the work done. 9. SI unit of work is joule. Joule is the work done when a force of one newton displaces a body

through one metre in the direction of force. 10. CGS unit of work is erg; 1 J = 107 ergs. 11. The work done in lifting an object of mass m through a height ‘h’ is equal to mgh. 12. When a body of mass m is raised from a height h1 to height h2, then the work done = mg(h2 – h1). 13. Let a body be lifted through a height 'h' vertically upwards by a force 'F' acting upwards. Then, the

work done by the resultant force is W = (F – mg)h. 14. The work done on a spring in stretching or compressing it through a distance x is given

W =21 kx2 where k is the force constant or spring constant.

15. Work done in changing the elongation of a spring from x1 to x2 is W = 21 k )xx( 2

122 − .

16. a) The work done in pulling the bob of a simple pendulum of length L through an angle θ as shown in the figure is W=mgL(1–cos θ ) = 2mgLsin2(θ/2) b) the velocity acquired by it when released from that position is )cos1(gl2v θ−=

F

s




Dynamics

63

17. The work done in lifting a homogeneous metal rod lying on the ground such that it makes an angle

'θ' with the horizontal, is W = 2sinmgl θ .

18. The work done in rotating a rod or bar of mass m through an angle θ about a point of

suspension is W = 2

mgL (1 – cos θ ) = mgLsin2(θ/2) where L is the distance of the centre

of gravity from the point of suspension. 19. The work done in lifting a body of mass 'm' and density 'ds' in a liquid of density 'dl' through a

height 'h' under gravity is

W = m g h ⎟⎟⎠

⎞⎜⎜⎝

⎛−

s

ldd

1

20. Work done in pulling back a th

n1 part of length of a chain hanging from the edge onto a smooth

horizontal table completely is W = 2n2

mgl .

21. Inclined plane : i) Work done in moving a block of mass 'm' up a smooth inclined plane of inclination 'θ' through a

distance 's' is W = Fs = mg sinθ s ii) if the plane is rough, then W = mg (sinθ+μk cosθ)s 22. Work done by a gas during expansion at constant pressure 'P' is given by Work done = (pressure) (change in volume)

W = P (dv) = P (V2 - V1) Note: The above formula can also be used to calculate the work done by the heart in pumping the

blood. i) If pressure also varies. then W = dVP2

1

vv∫

Work is positive if V2 > V1 i.e. when gas expands and negative if V2 < V1 i.e when gas is compressed.

ii) Area of P - V graph gives work done by the gas. 23. Rate of doing work is called power.

Power = timework = Force x velocity.

24. SI unit of power is watt and CGS unit is erg/second. 25. One horse power = 746 watt. 26. If a vehicle travels with a speed of v overcoming a total resistance of F, then the power of the

engine is given by P = v.F . 27. If a body is rotated in circular path, the power exerted is given by

P = τω=θ

τdtd

28. If a block of mass 'm' is pulled along the smooth inclined plane of angle 'θ', with constant velocity 'v', then the power exerted is, p = (mg sin θ)v

29. If the block is pulled up a rough inclined plane then the power is P = mg (sinθ + μk cosθ)v 30. If the block is pulled down a rough inclined plane then the power is P = mg (sinθ - μk cosθ)v 31. When water is coming out from a hose pipe of area of cross section 'A' with a velocity 'v' and hits a

wall normally and

L.

.




Dynamics

64

i) stops dead, then force exerted by the water on the wall is Av2 ρ. And the power exerted by water is P = A v3 ρ (ρ = density of water)

ii) If water rebounds with same velocity (v) after striking the wall, P = 2Av3 ρ

32. When sand drops from a stationary dropper at a rate of dtdm on to a conveyer belt moving with a

constant velocity, then the extra force required to keep the belt moving with a constant speed

V is given by F = v.dtdm

and the power required = P = 2vdtdm

33. If a pump lifts the water from a well of depth 'h' and imparts some velocity 'v' to the water, then the power of pump

P = t

mv21mgh 2+

34. Power exerted by a machine gun which fires 'n' bullets in time 't' is P = 1

mv21n

tW

2 ⎟⎠⎞

⎜⎝⎛

=

P = t2

mnv 2

35. If a pump delivers V litres of water over a height of h metres in one minute, then the power of the

engine (P) = 60

Vgh .

36. A motor sends a liquid with a velocity 'V' in a tube of cross section 'A' and 'd' is the density of the liquid, then the power of the motor is

P = 21 AdV3

37. The power of the lungs = K.E. of air blown per second.

=21 (mass of air blown per second)×(velocity)2

= 2v)t

m(21

38. The power of the heart = pressure × volume of blood pumped per second. 39. The capacity to do work is called energy. Work and energy have the same units. 40. Potential energy of a body or system is the capacity for doing work, which is possessed by the

body or system by virtue of the relative positions of its parts. 41. Water stored in a dam, stretched rubber cord, wounded spring of a clock or toy, etc. possess

potential energy.

42. A wounded spring (such as in a clock or toy car) has potential energy U=21 Kθ2 (K is a torque

constant and θ is the number of radians through which it is wound). SPRINGS: 43. Stretched or compressed spring possesses P.E.

a) Elastic potential energy of a stretched spring = Fx21

kF

21kx

21 2

2 ==




Dynamics

65

Where k = Force constant = xF

(S.I unit of 'K' is Nm–1)

b) Work done in increasing the elongation of a spring from x1 and x2 is 21 k ( 2

122 xx − )

44. The energy possessed by a body by virtue of its motion is called kinetic energy. It is measured by the amount of work which the body can do before coming to rest.

45. Running water, a released arrow, a bullet fired from a gun, blowing wind, etc. possess kinetic energy.

46. If a body of mass m is moving with a velocity v, then its kinetic energy = 21 mv2.

47. A flying bird possesses both K.E. and P.E. 48. The work done on a body at rest in order that it may acquire a certain velocity is a measure of its

kinetic energy.

49. If the kinetic energy of a body of mass m is E and its momentum is P, then E = m2

P2.

50. If the momentum of the body increased by ‘n’ times, K.E increase by n2 times. 51. If the K.E of the body increases by ‘n’ times, the momentum increases by n times.

52. a) If the momentum of the body increases by p%, % increase in K.E.= ⎟⎠

⎞⎜⎝

⎛ +100

p2 p%

b) If the momentum of the body decreases by p%, % decrease in K.E.= ⎟⎠

⎞⎜⎝

⎛ −100

p2 p%.

53. a) If the K.E of the body increases by e%, % increase in momentum= %1001100

e1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−+ .

b) If the K.E of the body decreases by e%, % decrease in momentum= %100100

e11 ⎟⎟⎠

⎞⎜⎜⎝

⎛+− .

54. If two bodies, one heavier and the other lighter are moving with the same momentum, then the lighter body possesses greater kinetic energy.

55. If two bodies, one heavier and the other lighter have the same K.E. then the heavier body possesses greater momentum.

56. Two bodies, one is heavier and the other is lighter are moving with the same momentum. If they are stopped by the same retarding force, then

i) the distance travelled by the lighter body is greater. (s∝m1 )

ii) They will come to rest within the same time interval 57. Two bodies, one is heavier and the other is lighter are moving with same kinetic energy. If they are

stopped by the same retarding force, then i) The distance travelled by both the bodies are same. ii) The time taken by the heavier body will be more. (t ∝ m ) 58. Two bodies, one is heavier and the other is lighter are moving with same velocity. If they are

stopped by the same retarding force, then. i) The heavier body covers greater distance before coming to rest. (s∝m) ii) The heavier body takes more time to come to test. (t ∝m)




Dynamics

66

58. Simple pendulum : If the bob (mass m) of a pendulum of length (l) is raised to a vertical height (h) and then released, it executes SHM for smaller angles. The total energy is constant at all positions.

a) At the mean position, KE = 21 mv2 (max), PE=0(min)

b) At the extreme position, KE = 0 (min), PE = mgl(1 – cosθ) (max)

c) KE at the mean position = PE at the extreme position

( )θ−= cos1mgmv21 2

velocity at equilibrium position, v = ( )θ− cos1gl2 , d) When a pendulum of length l is held horizontal and relased. Velocity at mean position, v = gl2 e) The graphs for PE and KE are parabolic in shape. Rebounding body : f) If a body falling from height h1 loses x% of energy during the collision with the ground, the

height to which it rebounds is

h2 ⎟⎠

⎞⎜⎝

⎛ −=

100x100 h1 = 1h

100x1 ⎟

⎠

⎞⎜⎝

⎛ −

g) If a ball strikes a floor from a height h1 and rebounds to a height h2.

% loss of energy = 100xh

hh

1

21 −

59. Projectile :

a) The PE at maximum height is maximum , PEH = mgH = ⎟

⎟⎠

⎞⎜⎜⎝

⎛ θg2

sinumg22

=21 mu2sin2θ = E

sin2θ b) The KE at the highest point is minimum.

KEH =

21 m(u cosθ)2 =

21 mu2cos2θ = Ecos2θ

c) Total energy = PEH + KEH =

θ22 sinmu21 + .cosmu

21 22 θ ⇒ 2mu

21E =

d) The ratio of potential and kinetic energies of a projectile at the highest point is tan2θ.

θ= 2

H

H tanE.KE.P

RECOIL OF A GUN: 60. It a bullet of mass 'm' travelling with a muzzle velocity, is fired from a rifle of mass 'M', then

i) Velocity of recoil of the gun is V = mv/M ii) K.E of the bullet is greater than the K.E of the rifle.

iii) Vv

mM

KEKE

r

b ==

iv) When a gun of mass ‘M’ fire a bullet of mass ‘m’ releasing a total energy ‘E’.

Energy of bullet Eb= mMM.E+




Dynamics

67

Energy of gun EG=mM

m.E+

BALLISTIC PENDULUM: 61. A block of mass 'M' is suspended by a string and a bullet of mass 'm' is fired into the block with a

velocity 'v'. If the bullet embeds in the block, then

i) The common velocity of the system after the impact is V = mM

mv+

ii) The height to which it will rise is h = 2

mMmv

g21

⎟⎠⎞

⎜⎝⎛

+

62. Work–energy theorem : The work done by the resultant force acting on a body is equal to the change in its kinetic energy.

; FsW = 2222 mu

21mv

21)uv(m

21W −=−=

63. In general, the work done = change in energy. 64. Stopping distance of a vehicle is directly proportional to the square of its velocity and inversely

proportional to the braking force. 65. If a body is thrown on a horizontal plane and comes to rest after travelling a distance 's', then

μ m g s = ½ mv2 'μ' coefficient of friction

distance travelled before coming to rest

s = force retarding

K.E. Initalmg

mv21 2

=μ

66. When a body of mass m falls freely from a height, its total energy is mgh. When it falls through a distance x, its K.E. is mgx and P.E. is mg(h x).

67. A stone of mass ‘m’ falls from a height ‘h’ and buries deep into sand through a depth ‘x’ before

coming to rest. The average force of resistance offered by sand is F= ⎟⎠

⎞⎜⎝

⎛ +=+

xh1mg

x)xh(mg .

68. For a freely falling body or for a body thrown up K.E. at the ground is equal to the P.E. at the maximum height.

69. The total energy of a system is constant. Energy can neither be created nor destroyed. But it can be converted from one form to the other. Examples on conversion of energy : 1. Electrical Heat, Eg. Iron, geyser, over 2. Electrical Light, Eg. Filament bulb,

Fluorescent tube 3. Electrical Sound, Eg. Loud speaker,

Telephone receiver 4. Electrical Mechanical. Eg. Fan, Motor 5. Heat Electrical. Eg : Thermal power plant 6. Heat Mechanical, Eg. Steam locomotive 7. Mechanical Electrical. Eg : Dynamo (Generator) 8. Sound Electrical. Eg: Microphone 9. Light Electrical. Eg: Photoelectric effect




Dynamics

68

10. Chemical Electrical. Eg. Primary cell 70. Rest mass energy : Every body or matter possesses a certain inherent amount of energy called rest

energy even if it is at rest (so that K.E.= 0) and is not being acted on by a force (so that P.E.= 0). This rest mass energy is given by E = mc2.




79

COLLISIONS

Synopsis: 1. Collision is an interaction between two or more bodies in which sudden changes of momenta

take place. e.g. : Striking a ball with a bat. 2. Newton’s third law of motion leads to the law of conservation of momentum (momentum can

neither be created nor destroyed). 3. The momentum of a system m1u1 + m2u2 = m1v1 + m2v2 remains constant so long as no external

forces act on it. 4. Rocket, jet plane, etc. work on the law of conservation of linear momentum. 5. When a bullet of mass m is fired from a gun of mass M with a velocity v, then the gun recoils

with a velocity mv/M. 6. When a shot is fired from a gun, the momentum of the shot and the momentum of the gun are

equal in magnitude but opposite in direction. 7. K.E. of gun : K.E. of bullet = mass of bullet : mass of gun. The velocity of a bullet is

determined by ballistic pendulum. 8. When a shot of mass m with a velocity v gets embedded in a block of mass M free to move on

a smooth horizontal surface, then their common velocity = mv/(m+M). 9. A bullet of mass m moving with a velocity v strikes a block of mass M hanging vertically.

After striking, the bullet gets embedded in the block and both rise to a height ‘h’. Then the velocity of the bullet is given by the formula v = gh2

mMm

⎟⎠

⎞⎜⎝

⎛ + .

10. If a boy of mass ‘m’ walks a distance ‘s’ on a stationary boat of mass ‘M’, then the boat moves back through a distance of

mMms+

.

11. When a moving shell explodes, its total (vector sum) momentum remains constant but its total kinetic energy increases.

12. If the velocities of colliding bodies before and after infact are confined to a straight line, it is called head on collision or one dimensional collision.

13. Elastic collisions : 1. Both kinetic energy and linear momentum are conserved. 2. Total energy is constant. 3. Bodies will not be deformed. 4. The temperature of the system does not change. e.g. Collisions between ivory balls; molecular, atomic and nuclear collisions.

14. Perfect elastic collisions : 1. Two bodies of equal masses suffering one dimensional elastic collision, exchange their

velocities after collision. i.e., if m1 = m2 then v1 = u2 and v2 = u1. 2. If a body suffers an elastic collision with another body of the same mass at rest, the first is

stopped dead, whereas the second moves with the velocity of the first. i.e, if m1 = m2 and u2 = 0 then v1 = 0; v2 = u1.

3. When a very light body strikes another very massive one at rest, the velocity of the lighter body is almost reversed and the massive body remains at rest. i.e., if m2 >> m1 and u2 = 0, then v1 = − u1 and v2 = 0.

4. When a massive body strikes a lighter one at rest, the velocity of the massive body remains practically unaffected where as the lighter one begins to move with a velocity nearly double as much as that of the massive one. i.e., if m1 >> m2 and u2 = 0, then v1 = u1 and v2 = 2u1.

5. When m1, m2 are moving with velocities u1, u2 and v1, v2 before and after collisions, then




Collisions

80

a) v1 = ⎟⎟⎠

⎞⎜⎜⎝

⎛+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+−

21

22

21

211 mm

m2u

mmmm

u ;

b) v2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+ 21

122

21

11 mm

mmu

mmm2

u

15. Two bodies of equal masses moving in opposite directions with the same speed collide, if the collision is elastic, each body rebounds with the same speed.

16. A body collides with another body of equal mass at rest. The collision is oblique and perfectly elastic. The two bodies move at right angles after collision.

17. A body of mass m1 collides head on with another body of mass m2 at rest. The collision is perfectly elastic. Then

a) Fraction of kinetic energy lost by the first body is 2

21

21

)mm(

mm4

+.

b) Fraction of kinetic energy retained by first body is 2

21

21

mmmm

⎟⎟⎠

⎞⎜⎜⎝

⎛+− .

18. The loss of energy of the first body is maximum 100% when m1 = m2. 19. Inelastic collision :

1. Linear momentum is conserved. 2. Kinetic energy is not conserved. 3. Total energy is conserved. 4. Temperature changes. 5. The bodies may be deformed. 6. The bodies may stick together and move with a common velocity after collision 7. If the bodies collide and move together after collision; the collision is perfectly inelastic. 8. Two bodies collide in one dimension. The collision is perfectly inelastic, then m1u1 + m2u2 = (m1 + m2) V

9. Common velocity after collision)mm(umum

v21

2211++

=

10. Total loss of kinetic energy in perfect inelastic collision

=21

22121

mm)uu(mm

21

+−. = ( )2

2121

21 uumm

mm21

−+

20. Ballistic Pendulum: A block of mass M is suspended by a light string. A bullet of mass m moving horizontally with a velocity 'v' strikes the block and gets embedded in it. The block and the bullet rise to a height h. Then

a) mv = (M+m)V b) V = gh2

c) mv = (M+m) gh2

d) v = gh2m

)mM( +

21. If the string of the ballistic pendulum makes an angle θ with vertical after impact and the length of the string is l (when θ≤ 90°)

)cos1(gl2m

mMv θ−+

=

22. If the ballistic pendulum just completes a circle in the plane, velocity of the bullet

gl5m

mMv +=




Collisions

81

23. A ballistic pendulum can be used to determine the velocity of projectiles.

24. Fraction of the kinetic energy lost in the impact of a ballistic pendulum is )Mm(

M+

25. Coefficient of restitution (e) : The coefficient of restitution between two bodies in a collision is defined as the ratio of the relative velocity of separation after collision to the relative velocity of their approach before their collision. (i) e =

approach ofvelocity relativeseparation ofvelocity relative

(ii) e = − 12

12uuvv

−−

(iii) e =21

12uuvv

−−

Eg : The value of e is 0.94 for two glass balls, 0.2 for two lead balls. (iv) for a perfectly elastic collision, e = 1 (v) for a perfectly inelastic collision, e = 0 (vi) If a body falls from a height h1 on to a hard floor and rebounds to a height h2, then

e = 1

2hh .

26. The value of coefficient of restitution is independent of the masses and the velocities of the colliding bodies. It depends on their materials.

27. If a body dropped from a certain height takes a time t1 to strike the ground and time t2 to rise e=t2/t1.

28. If a body dropped from a certain height hits the ground. With a velocity v1 and rebounds with a velocity v2.

1

2vv

e =

29. A small metal sphere falls freely from a height h upon a fixed horizontal floor. If e is the coefficient of restitution, i) the height to which it rebounds after n collisions is hn = he2n ii) the total distance travelled by it before it stops rebounding.

d = ⎟⎟⎠

⎞⎜⎜⎝

⎛

−

+2

2

e1e1h

iii) the velocity with which it rebounds from the ground after nth collision is Vn = ne)gh2( iv) the time taken till it comes to rest is

t = ⎟⎠

⎞⎜⎝

⎛−+

e1e1

gh2

30. A ball is projected with velocity ‘u’ at an angle ‘θ’ to the horizontal plane. It will keep rebounding from the plane for a time

)e1(gsinu2−

θ (e is the coefficient of restitution)

31. In the above case, its horizontal range before coming to rest is

R =)e1(g

2sinu2

−θ .

32. Inelastic collision :




Collisions

82

a) There is loss of energy from the system in the form of heat, sound, light etc., b) For the above case m1u1 + m2u2 =m1v1+m2 v2

c) 21 m1 2

22211

222

21 um

21um

21vm

21v +<+

d) Relative velocity of separation v2 - v1 = e (u1 - u2)

e) V1 = ( ) ( )21

22

21

211 mm

me1u

mmemm

u+

++

+−

V2 =( ) ( )

21

122

21

11 mm

emmumm

e1mu+−

++

+

V1 & V2 are velocities of m1 and m2 after the collision f) Loss of Kinetic energy from the system is

g) ΔK = Elost = 221

2

21

21 ]uu[]e1[mm

mm21

−−⎥⎦

⎤⎢⎣

⎡+




94

CENTRE OF MASS

Synopsis : 1. Every particle is attracted towards the centre of the earth by the force of gravity and the centre

of gravity of a body is the point where the resultant force of attraction of the weight of the body

acts.

2. Centre of mass : If a system of parallel forces proportional to the masses of the various

particles of a body are assumed to act on it, their resultant passes through a fixed point,

irrespective of the direction of the parallel forces and that point is called centre of mass.

3. In a uniform gravitational field, the centre of mass of the body and the centre of gravity

coincide with each other.

4. For small bodies (compared to the size of the earth) the centre of gravity and the centre of mass

coincide.

5. If two particles of masses m1 and m2 are at distances x1 and x2 from the origin and xcm is the

position of the centre of mass of the system, then

21

2211cm mm

xmxmx

++

=

6. For a system of two bodies of masses m1 and m2 separated by a distance d.

a) Centre of mass lies on the line joining the centers of mass of two bodies.

b) Distances of center of mass from first body r1 = 21

2mmrm

+ .

c) Distances of center of mass from second body r2 = 21

1

mmrm

+ .

d) Centre of mass lies nearer to heavy mass.

7. If the position coordinates of particles of masses m1, m1, m1... are (x1, y1), (x2, y2), (x3,y3)...The

mass of the system is M. The position coordinates of the centre of mass are (x,y).

....mmm...xmxmxm

x321

332211+++

++=

= M...xmxm 2211 ++

...mmm...ymymym

y321

332211+++

++=

= M...ymym 2211 ++

8. The distance of centre of mass from the origin is 22 yx + .

9. If the two masses are equal, the centre of mass lies midway between the particles.

10. When a large number of particles are distributed in space, the centre of mass of the system is

given by the coordinates.




Centre of mass

95

∑

∑=

∑

∑=

∑

∑=

=

=

=

=

=

=n

1ii

n

1iii

cmn

1ii

n

1iii

cmn

1ii

n

1iii

m

zmz;

m

ymy;

m

xmx

cm

11. a) The position of center of mass depends on the shape of the body and distribution of mass

b) In symmetrical bodies in which distribution of mass is homogeneous. Centre of mass

coincides with geometrical center.

c) Centre of mass of regular bodies.

1) Uniform rod : middle point of the rod

2) Cubical box: Point of intersection of diagonals

3) Circular ring : centre of the ring

4) circular disc : centre of the disc

5) Sphere : centre of the sphere

12. In vector notation, each particle of the system can be described by a position vector r and the

centre of mass can be located by the position vector cmr .

iiiii

iicm zkyjxirwhere

mrmr ++=

∑∑=

cmcmcmcm zkyjxir ++=

13. Velocity of the centre of mass massTotalmomentumTotal

mVm)v( cm =

∑∑=

14. If two particles masses m1 and m2 are moving with velocities 1v and 2v at right angles to each

other, then the velocity of their centre of mass is given by

Vcm = )mm(vmvm

21

22

22

21

21

+

+

15. The velocity of centre of mass of an isolated system remains constant as long as no external

force acts on the system. Its acceleration is zero.

16. The momentum of centre of mass of the system is equal to the sum of individual momenta of n

particles of the system.

17. If an external force is acting on the system of particles, centre of mass behaves as if total force

is acting only at that point.

18. The centre of mass of the system is static under the action of internal forces.

19. When a body rotates, vibrates or moves linearly, the centre of mass moves in the same way as a

single particle subjected to the same force.




Centre of mass

96

20. Acceleration of the centre of mass = ∑∑

mFext

.

21. Characteristics of centre of mass :

i. The position of centre of mass depends upon the shape of the body and the distribution of mass.

ii. If the origin is at the centre of mass, then sum of moments of mass (Σmixi) of the system

about the centre of mass is zero.

iii. Centre of mass may be within the body or on or outside the material of the body.

iv. It is not necessary that mass should be present at centre of mass. In the case of uniform ring,

centre of mass is outside the material where no mass is present.

v. The location of the centre of mass is independent of the reference frame used to locate it.

vi. The centre of mass of a system of particles depends only on the masses of the particles and

their relative positions.

22. Laws of motion of the centre of mass :

i. The centre of mass of a system of particles moves as though total mass of the system is

concentrated at a point and external forces were applied at that point.

ii. The motion of the centre of mass of the body is called the translational motion of the body.

iii. The internal forces will not effect the motion of the centre of mass.

iv. If no external force acts on a system, the acceleration of centre of mass is zero, the velocity

and momentum of the centre of mass remains constant, though velocity and momentum of

individual particles vary.

v. The motion of the centre of mass can be studied using Newton's laws of motion.




Centre of mass

97

23. When two solid spheres of same materials but with radii r1 and r2 are

kept in contact, the center of mass of the system is at a distance of x

from the center of bigger sphere, where

x = 3

23

1

213

2

rr

)rr(r

+

+

.

24. When a portion of m2 is removed from a body of mass m1 then shift in the position of center of

mass =

distance between cm of themass of removed part

body and removed partmass of remaining part

⎡ ⎤×⎢ ⎥

⎣ ⎦

25. When two circular discs of same material and thickness and radii r1 and r2 are kept in contact,

then the center of mass of the system is at a distance of x from the center of first (large) disc of

mass m1 where x=2

22

1

212

2

rr

)rr(r

+

+

.

26. A circular portion of radius r2 is removed from a circular disc of radius r1

from one edge. Then the shift in the center of mass of the disc is x

= 21

22

rrr+ .

27. From a circular disc of radius r, a circle of diameter r is removed. The shift in the center of mass

of the remainder is x=r/6.

28. When as sphere of radius r2 is removed from a solid sphere of radius r1 from its edge, then the

shift in its centre of mass is x = )rr(

)rr(r3

23

1

213

2

−

−

.

29. When a person walks on a boat in still water, centre of mass of person, boat system is not

displaced.

a) If the man walks a distance L on the boat, the boat is displaced in the opposite direction

relative to shore or water by a distance x = mMmL+ .

b) Distance walked by the mass relative to shore or water is (L – x).

30. Two masses starting from rest move under mutual force of attraction towards each other, they

meet at their centre of mass.

a) In the above case Vcm and acm = 0

b) If the two particles are m1 and m2 and their velocities are v1 and v2, then m1v1 = – m1v2

m1

C1

m2

C2

r2r1

C

x

r +r1 2

m1

C1

m2

C2C

x




Centre of mass

98

c) If the two particles have accelerations a1 and a2.

m1a1 = – m2a2

d) If s1 and s2 are the distances traveled before they meet

m1s1 = m2s2




1

FRICTION

Synopsis : 1. When a body is in motion over another surface or when an object moves through a

viscous medium like air or water or when a body rolls over another, there is a resistance to the motion because of the interaction of the object with its surroundings. Such a resistance force is called force of friction.

2. Friction is a result of molecular interaction. According to modern view, the cause of friction is largely due to atomic and molecular forces between the two surfaces at the point of contact.

TYPES OF FRICTIONAL FORCE : 3. STATIC FRICTION : The frictional force, which is effective before motion starts

between two planes in contact with each other, is known as static friction. Note: 1) Static frictional force is a self adjusting one. 2) The maximum frictional force when the body is ready to start is called limiting

frictional force. 4. DYNAMIC FRICTION: The frictional force, which is effective when two surfaces in

contact with each other are in relative motion with respect to each other, is known as dynamic friction.

5. ROLLING FRICTION: The frictional force, which is effective when a body rolls or rotates on a surface, is known as rolling friction.

6. Limiting friction (Fs) is independent of the area of contact of the surfaces. Rolling

friction depends on the area of contact. 7. Limiting friction is directly proportional to the normal reaction between the surfaces in

contact. Fs ∝ R

sss whereRF μμ= is called the coefficient of static friction. It depends upon the nature of the surfaces in contact and their state of roughness.

8. The angle between R and the resultant of R and F (i.e., R and Rl) is called the angle of friction λ . Tan λ=μ

9. Characteristics of static friction:

State of motion

Dyna

mic F

rictio

n

Pulling force

Stati

c Fric

tion

Frict

ional

force

O

Fs A B

D C

State of rest

RRI

λF= sRμ motion




Friction

2

a) μs between two given surfaces is independent of the normal force between the two surfaces.

b) μs > 0, it can also be greater than one, but in most of the cases it is less than one c) If θs is the angle of limiting friction between two surfaces tan θs = μs 10. When one body moves over the other, the force of friction acting between the two

surfaces is called kinetic friction. 11. The force of kinetic friction is independent of the area of the surfaces in contact and is

proportional to the normal reaction Fk ∝ R. Fk = R.kμ Where μk is coefficient of kinetic friction.

12. When one body moves over another body, the coefficient of friction is less than limiting coefficient of friction and is called the coefficient of kinetic friction.

13. Fk is independent of the velocity of sliding provided the velocity is low. 14. When a body rolls over another, the frictional force developed is called rolling frictional

force and the corresponding coefficient of friction is called coefficient of rolling friction (μr).

15. Rolling friction: a) Rolling friction comes into play when a body such as a wheel rolls on a surface. b) Rolling friction arises out of the deformation of the two surfaces in contact with each

other. c) Greater the deformation greater is the rolling frictional force. d) The rolling frictional force is inversely proportional to the radius of the rolling body. e) The rolling frictional force between two given surfaces is lesser than kinetic and

limiting frictional forces. f) If μR is the coefficient of rolling friction μR < μk < μs for a given pair of surfaces. g) Ball bearings are used in machinery parts because rolling friction is least. h) Radial tyres used in cars reduce rolling friction. 16. When lubricants or viscous liquids are introduced between the surfaces of two solids in

contact, they reduce frictional forces because intermolecular forces in liquids are much weaker than those in solids.

17. Pulling a block or roller a) If the pulling force is such that F cosθ<fs, (fs is limiting friction) the block will be at

rest and the force of friction between block and the surface is f = F cosθ b) The normal force is N = mg - F sin θ c) Force needed to just slide the body is

F= )cos(

sinmgsincos

mg

s

sφ−θφ

=θμ+θ

μ Where φ is the angle of friction between the two

surfaces.

F

mg

F cos θ

(pulling

NF sin

θf




Friction

3

d) If the applied force is greater than the above value, block slides with acceleration and the force of friction between the block and the surface is fk.

e) The minimum possible force among all directions required to just move the body is

mg sinφ (or)2s

s

1

mg

μ+

μ where φ is the angle of friction. The force must be applied at

angle θ to the horizontal at an angle equal to angle of friction φ. 18. Pushing a block or Roller: a) If the pushing force is such that, Fcosθ<f s, the block will be rest and the force of friction

between the block and the surface is f = F cos θ. b) The normal force is N = mg + F sinθ c) Force needed to just slid the body is

F= )cos(

sinmgsincos

mg s

s

sφ+θφμ

=θμ−θ

μ

where φ is the angle of friction. d) Pulling is easier than pushing because lower frictional force, in the case of pulling

need to be overcome. e) If the angle made by the pushing force with the vertical is lesser than or equal to angle

of friction, the block cannot be moved, irrespective of the magnitude of the applied force.

19. A uniform chain of length L lies on a table. If the coefficient of friction is μ, then the maximum length of the chain which can overhang from the edge of the table without

sliding down is 1

L+μμ .

20. Block on a rough fixed horizontal surface a) If we continue to apply a force F = f s, the block slides with an acceleration given by

a = (μs – μk) g b) Once the block slides, force of friction on the block is kinetic frictional force (fk)

c) If the block slides with an acceleration under the influence of an external force F, the acceleration of the block is a =

mfF k−

21. Sliding block on a horizontal surface coming to rest:

a) If a block having initial velocity u slides on a rough horizontal surface and comes it rest, the acceleration of the block is a = – μkg

b) Distance traveled by the block before coming rest is S = g2

u

k

2

μ

F sin θF (pushing

θ

N

mg

f F cos θ

v =fx u

m




Friction

4

c) Time taken by the block to come to rest is t =

gu

kμ

22. Motion on a rough horizontal plane :

(a) Pulled with a horizontal force F: (i) body moving with uniform velocity F = kμ mg. (ii) body moving with uniform acceleration F = m( kμ g + a).

(b) Pulled with a force F inclined at an angle θ with the horizontal and the body moving with uniform velocity.

F =θμ+θ

μsincos

mg

k

k

c) Pushed with a force F inclined at an angle θ with the horizontal and the body moving with uniform velocity:

F =θμ−θ

μsincos

mg

k

k .

23. Block on a rough inclined plane a) Angle of repose α: It is the angle of inclination of the

inclined plane with the horizontal for which block just begins to slide down.

b) If α is the angle of repose μs = tanα c) The angle of repose is the angle of static friction d) The angle of inclination is (θ) less than (α), the block does

not slide down, it is at rest. The force of friction f < f s and is equal to

f = mg sinθ [mg sinθ < f s] e) If the angle of inclination is [θ] equal to [α]. Then the block is in limiting equilibrium.

The force of friction is f = fs = μs mg cos α [mg sinθ = fs] f) If the block slides down the inclined plane with uniform velocity μk = tan θ where θ is

the angle of inclination of the inclined plane. Sliding down the inclined plane: g) If the inclination is maintained at α, the block will eventually slide down with an

acceleration equal to a = g 2s

ks

1

][

μ+

μ−μ

h) If θ ≥ α, the block slides down with an acceleration given by a = g [sinθ - μk cosθ] [mg sinθ > f s] i) If θ ≥ α, and the block slides down from the top of the inclined plane. Velocity at the

bottom of the plane is V = )cot1(gh2cos(singl2 kk θμ−=θμ−θ j) In the above case time of descent is

t = )cos(sing

L2

k θμ−θ

θmg mg cos θ

θ

mg sinf

N




Friction

5

k) The time taken by a body to slide down on a rough inclined plane is 'n' times the time taken by it to slide down on a smooth inclined plane of same inclination and length,

then coefficient of friction is μ = tan θ ⎥⎦

⎤⎢⎣

⎡−

2n11

Moving up the inclined plane: l) If a block is projected up a rough inclined plane, the acceleration of the block is a = – g [sinθ + μk cosθ]

m) Force opposing the motion of the block is F = mg sinθ + μk mg cosθ n) The distance traveled by the block up the plane before the velocity becomes zero is

S = )cos(sing2

u

k

2

θμ+θ

o) The time of ascent is t = k

ug(sin cos )θ + μ θ

. In the above case the block will come down

sliding only if θ ≥ α. p) In the above case if time of decent is n times the time of ascent, then

μ = tanθ ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

−

1n1n

2

2

q) Force needed to be applied parallel to the plane to move the block up with constant velocity is F = mg sinθ + μk mg cosθ

r) Force needed to be applied parallel to the plane to move the block up with an acceleration a is

F = mg sinθ + μk mg cosθ + ma s) If block has a tendency to slide, the force to be applied on the block parallel and up

the plane to prevent the block from sliding is F = mg sinθ - μs mg cosθ 24. Block on a smooth inclined plane a) N = mg cos θ

b) Acceleration of sliding block (a = g sin θ)

c) If l is the length of the inclined plane and h is the height. The time taken to slide down starting from rest from the top is

t = gh2

sin1

sing21

θ=

θ

d) Sliding block takes more time to reach the bottom than to fall freely from the top of the incline.

e) Velocity of the block at the bottom of the inclined plane is V = gh2singl2 =θ same as the speed attained if block falls freely from the top of the

inclined plane. f) If a block is projected up the plane with a velocity u, the acceleration of the block is a = - g sinθ g) Distance traveled up the plane before its velocity becomes zero is

S = θsing2

u2

h) Time of ascent is t = θsing

u

θmg mg cos θ

θmg sin

N




Friction

6

25. When the body moves on a rough horizontal surface, the force of friction is μmg. If s is the displacement, the work done against friction = μmgs. This work is converted into heat.




Friction

7

26. Block pressed against a vertical wall : A body of mass 'm' is pressed against a vertical wall with a horizontal force 'F'. The normal force is F. If the coefficient of static friction is μs, then a) Block will be about to slide down if μs F = mg. b) If μs F ≥ mg, block will not slide and the frictional force

acting on the block is mg. c) If μs F ≥ mg, block will slide and the frictional force acting on the block will be μs F.

27. A vehicle is moving on a horizontal surface. A block of mass 'm' is stuck on the front part of the vehicle. The coefficient of friction between the truck and the block is 'μ'. The minimum acceleration with which the truck should travel, so that the body may not slide down is a = g/μ .

28. Block in a lorry: a) When a block is lying on the floor of an accelerating lorry,

the force of friction acting on the block is in the direction of acceleration of the lorry.

b) Relative to lorry, block experiences a pseudo force ma opposite to the acceleration of the lorry (a = acceleration of lorry)

c) The maximum acceleration of the lorry for which block beings to slide on the floor of the lorry is

a = μs g [ ]gamg ma ss μ=∴μ= d) If a < μs g block does not slide and friction force on the block is f = ma e) If a ≥ μs g block slips or slides on the floor. The acceleration (a) of the block relative

to lorry is a1 = a - μk

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

μ−=∴=μ−

=−

gaamakmgma

mafkma

k1

11

f) In the above case, acceleration of the block relative to earth is μk. Block on Block: 29. Case I: (lower block pulled and there is no friction between lower block and the

horizontal surface) a) When the lower block is pulled upper block is accelerated

by the force of friction acting upon it b) The maximum acceleration of the system of two blocks for

them to move together without slipping is a = μs g, where μs is the coefficient of static friction between the two blocks.

c) If a < μs g blocks move together and applied force is F = (mB + mu )a d) If a < μs frictional force between the two blocks f = mu a e) The maximum applied force for which both blocks move together is Fmax = μs g (mu + mB) f) If F > Fmax blocks slip relative to each other and have different accelerations. The

acceleration of the upper block is μk g and lower block is

a = uB mm

F+

pseudoforce

ma

a

observerm f

RF

f

m

m

mu f fF




Friction

8

30. Case - II (Upper block pulled and there is no friction between lower block and the

horizontal surface) a) When the upper block is pulled, lower block is

accelerated by the force of friction acting upon it. b) The maximum acceleration of the system of two blocks

for them to move together without slipping is amax = g

mm

B

usμ (μs = coefficient of static friction between the two blocks)

c) If a < amax frictional force between the two blocks is f = MB a d) If a < amax' then applied force on the upper block is F = (mB + mu) a

e) The maximum force for which both blocks move together is Fmax = μs gmm

B

u (mu + mB)

f) If F>Fmax blocks slide relative to each other and hence have different accelerations. The accele-ration of the lower block is g

mm

B

ukμ and the acceleration of the upper block

is u

ukm

)gmF( μ− .

31. When moisture is present between bodies, friction increases. 32. If the metal surfaces of ball bearing are not hard, friction will be high. 33. friction is reduced by using alloys for making the moving parts (alloys have low

coefficient of friction) 34. Advantages of friction:

i) Safe walking on the floor is possible because of the friction between the floor and the feet.

ii) Nails and screws are driven in the walls or wooden surfaces due to friction. iii) Friction help the fingers to hold a drinking water tumbler or pen. iv) Vehicles move on the roads without sliding due to friction and they can be stopped

due to friction. v) The mechanical power transmission of belt drive is possible due to friction.

35. Disadvantages of friction: i) Friction results in the large amount of power loss in engines. ii) Due to friction, the wear and tear of the machine increases. iii) Due to friction, heat is generated which goes as a waste.

36. Methods of reducing friction: i) Friction between two surfaces of contact can be reduced by polishing the surfaces. ii) A lubricant is a substance which forms a thin layer between two surfaces in contact

and reduces the friction. The process of reducing friction is called lubrication. Soap water, two–in–one oil and grease are the examples of lubricants.

iii) The free wheels of vehicles like cycles, two wheelers, motor cars, shafts of motors, dynamos etc., are provided with ball bearing to reduce the friction.

iv) Automobiles and aeroplanes have special construction i.e. they are stream lined to reduce the friction due to air.

mu Ff

mB

f




1

ROTATORY MOTION

Synopsis : CIRCULAR MOTION :

1. In translatory motion, every particle travels the same distance along parallel paths, which may be straight or curved. Every particle of the body has the same velocity and acceleration.

2. In rotatory motion, the body rotates about a fixed axis. Every particle of the body describes a circular path and centres of concentric circles lie on the axis of rotation. Every particle of the body undergoes the same angular displacement. But linear velocities of rotating particles differ depending upon their radii of rotation.

3. When a particle describes a circular path, the line joining the centre of the circle and the position of the particle at any instant of time is called the radius vector.

4. As the particle moves round the circle, the radius vector rotates (like the hands of a clock). The angle described by the radius vector in a given interval of time is called the angular displacement.

5. Angular displacement is a vector passing through the centre and directed along the perpendicular to the plane of the circle whose direction is determined by right hand screw rule (It is a pseudo vector).

6. Angular displacement is measured in radians or turns. 7. The rate of change of angular displacement is called angular velocity (ω).

11 rads3060

2rpm1;rpmorradst

−− π=

π=

θ=ω

. 8. Angular velocity is a vector lying in the direction of angular displacement.

9. Linear velocity r )V( ×ω= . 10. Rate of change of angular velocity is called angular acceleration (α). Unit is radsθ2.

timevelocityangularinchange

=α.

11. Linear acceleration = radius × angular acceleration. r a ×α= . 12. Resultant acceleration a = ω where ar = radial acceleration and aT = tangential

acceleration. 13. Angular displacement ( )(velocityangular), ωθ and angular acceleration (α) are pseudo

vectors. 14. For a body rotating with uniform angular acceleration, the following equations hold

good. i) atuvtosimilar.......t0 +=α+ω=ω

ii) 22

0 at21utstosimilart

21t +=α+ω=θ

iii) as2uvtosimilar2 2220

2 =−αθ=ω−ω iv) W = FsWtosimilar =τθ v) Power = FvPtosimilar =τω vi) Torque ( maFtosimilar) =α=τ I vii) Angular momentum (L) = Iω similar to P = mv

viii) Rotational kinetic energy = 2

21

ωI similar to

2mv21.E.K =

15. Angular momentum and torque are pseudo vectors.




Rotatory Motion

2

16. When a body is moving in a circular path with uniform speed, then the acceleration experienced by the body, along the radius of the circle and directed towards the centre is

called normal or radial or centripetal acceleration and is equal to rn4orror

rv 222

2πω

where n is the number of revolutions per second.

17. The force which makes a body move round a circular path with uniform speed is called the centripetal force. This is always directed towards the centre of the circle.

18. If the direction of a force of constant magnitude applied on a body is always at right angles to the direction of its motion, then it describes a circular path with a uniform speed and its kinetic energy remains constant.

19. For a body moving round a circular path with uniform speed, the time period of

revolution T = ωπ2

and frequency n =n2.,e.i

2T1

π=ωπ

ω=

where n is the number of revolutions per second.

20. Centripetal force =2

2mr

rmv

ω=.

21. A body moving round a circular path with uniform speed experiences an inertial or pseudo force which tends to make it go away from the centre. This force is called the centrifugal force and this is due to the inertia of the body.

22. Centrifugal force = −centripetal force (but these are not action and reaction). 23. No work is done by centripetal force. 24. The kinetic energy of the body revolving round in a circular path with uniform speed is

‘E’. If ‘F’ is the required centripetal force, then

rE2F =

25. Uses of centrifugal forces and centrifugal machines.

i) Cream is separated from milk (cream separator) ii) Sugar crystals are separated from molasses. iii) Precipitate is separated from solution. iv) Steam is regulated by Watt’s governer. v) Water is pumped from a well (Electrical pump). vi) Hematocentrifuge, Grinder, Washing machine, etc.

26. The angle through which a cyclist should lean while taking sharp turnings is given by the

relation ⎟⎟⎠

⎞⎜⎜⎝

⎛=θ −

rgvTan

21

.

27. Safe speed on an unbanked road when a vehicle takes a turn of radius r is v = rgμ where μ = coefficient of friction.

28. The maximum speed that is possible on curved unbanked track is given by g = v2h/ar where h = height of centre of gravity and a = half the distance between wheels.

29. After banking of a road, the weight W of the vehicle, the normal reaction N and the centripetal force F form a vector triangle (or) centripetal force is the resultant of W and N. Angle of banking θ is given by tan θ = v2/rg and the height of banking is given by h = θsinL where L is the width of the road.

If θ is very small, then Lh

rgv2

==θ

NW

Fθ

hL




Rotatory Motion

3

30. When a mixture of liquids of different densities is centrifuged, the denser liquid moves as far away from the axis as possible.

MOTION IN A VERTICAL CIRCLE : 31. A particle of mass 'm' suspended by a thread is given a horizontal

speed 'u'. When it is at 'A',it moves in a vertical circle of radius 'r' 32. When the displacement of the particle is 'θ', i.e when the particle

is at p a) Speed of the particle

v = )Co1(gr2u2 θ−− where u is the velocity at A, the lowest point b) Centripetal force mv2 / r = T – mg cos θ c) The speed of the particle continuously changes. It increase while coming down and

decreases while going up d) This is an example for non - uniform circular motion. e) Tangential acceleration = g sin θ f) Tangential force = mg sin θ

g) Tension in the string T = mv2/r + mg cos θ =)]hr(gv[

rm 2 −+

h) Velocity, speed, K.E, linear momentum, angular momentum, angular velocity, all are

variables. Only total energy remains constant

33. If u < gr2 the body oscillates about A.

0 < u < gr2 ; 0 < θ < 90°.

34. If gr2 <u< gr5 the body leaves the path without completing the circle. 35. A body is projected with a velocity 'u' at the lowest point a) Height at which velocity u = 0. is h = u2/2g b) Height at which Tension T = 0 is

h = g3rgu2 +

c) Angle with vertical at which velocity v = 0. is Cos θ = 1− gr2u2

d) Angle with vertical at which the tension T = 0 is Cosθ = 2/3 −u2/3 gr e) Tension in the string at an angular displacement θ with vertical is T = mu2/r − mg (2 − 3 cosθ)

36. When the body is projected horizontally with a velocity u = gr5 from the lowest point A, a) It completes the circle

b) Velocity at the top B = gr called the critical speed c) Tension in the string at the top T1 = 0 d) Tension in the string at the lowest point

T2 = 6mg. It is the maximum tension in the string) e) T2 - T1 = 6mg.

f) Velocity at the horizontal position ie, at C Ve = gr3 g) Tension in the string at C = 3mg

O

B

CP

mgcosθmgsinθmg

A

θ

θ




Rotatory Motion

4

h) Ratio of velocities at A, B and C = 1:3:5 i) Ratio of K.E at A, B and C = 5: 3: 1

j) Velocity at an angular displacement θ is given by V = θ+ cosgr2gr3 k) Vmin does not depend on the mass of the body l) Tension at angular displacement 'θ' is given by T = 3mg (1+Cosθ) m) Tmin in the string does not depend on the radius of vertical circle.

37. When the body is rotated at a constant speed, 'v' a) Tension in the string at the lowest point T = mv2/r+mg b) Tension in the string at the highest point T = mv2/r–mg c) Tension in the string at the horizontal position

T = mv2/r

d) Time period of revolution if v = rg is

T = 2 π g/r 38. A sphere of mass m is suspended from fixed point by means of light string. The sphere is

made to move in a vertical circle of radius r whose centre coincides with point of suspension such that the velocities of sphere are minimum or critical at different points. Then

at the lowest point K.Emax = 25

mgr

at the highest point = K.Emin = 21

mgr during one revolution ΔK.E = ΔP.E = 2mgr 39. A ball of mass 'M' is suspended vertically by a string of length 'I'. A bullet of mass 'm' is

fired horizontally with a velocity 'u' on to the ball sticks to it. For the system to complete

the vertical circle, the minimum value of 'u' is given by u = gl5

mmM +

. 40. A body of mass m is sliding along an inclined plane from a vertical height

h as shown in the figure. For the body to describe a vertical circle of

radius R, the minimum height in terms of R is given by h = 2R5

. 41. a) A particle is freely sliding down from the top of a smooth convex hemisphere of radius

r. The particle is ready to leave the surface at a vertical distance h = r/3 from the highest point.

b) If the position vector of the particle with respect to the centre of curvature makes an angle θ with vertical then Cosθ = 2/3

42. During non–uniform acceleration, r aT ×α= and α×ω= wherev aR is angular acceleration, Ta is tangential (linear) acceleration and Ra is radial or centripetal acceleration.

43. Moment of inertia (I) of a body about an axis is defined as the sum of the products of the masses and the squares of their distances of different particles from the axis of rotation.

44. Moment of inertia or Rotational inertia a) It is the property of a body due to which it opposes any change in its state of rest or

uniform rotation. b) It is the rotational analogue of inertia in translatory motion

h R




Rotatory Motion

5

c) For a particle of mass 'm' rotating at a distance r from the axis of rotation. I = mr2 d) For a rigid body I = mk2 K is called radius of gyration e) Radius of gyration is the distance whose square when multiplied by mass of the body

gives moment of inertia of the body about the given axis. K = m/I

f) S.I unit of moment of inertia is Kg.m2. g) Moment of inertia of a body depends on

i) Mass of the body ii) Distribution of mass of the body iii) Position of axis of rotation iv) Temperature of the body

h) It is independent of angular velocity of rotation of the body. 45. The root mean square of the distance of all the particles, from the axis of rotation is

known as radius of gyration (K).

K = nr.....rrr 2n

23

22

21 ++++

46. The radius of gyration of a rigid body about a given fixed axis is that perpendicular

distance from the given axis where the entire mass of the body can be redistributed and concentrated without altering the moment of inertia of the body about the given axis.

I = MK2 or K = M/I 47. Two small spheres of masses m1 and m2 are joined by a rod of length ‘r’ and of negligible

mass. The moment of inertia of the system about an axis passing through the centre of mass and perpendicular to the rod, treating the spheres as particles is

I =2

21

21 r mm

mm⎟⎟⎠

⎞⎜⎜⎝

⎛+

48. Perpendicular axes theorem : The moment of inertia of a plane lamina about an axis perpendicular to its plane is the sum of the moments of inertia of the same lamina about two mutually perpendicular axes, lying in the plane of the lamina and intersecting on the given axis. Iz = Ix + Iy

49. Parallel axes theorem : The moment of inertia of any rigid body about any axis is equal to the moment of inertia of the same body about a parallel axis passing through its centre of mass plus the product of the mass of the body and square of the distance between the parallel axes. I = IG + Md2.

50. Angular momentum (L ) : (i) The moment of linear momentum is called angular momentum of the particle about

the axis of rotation. (ii) ω=ω== I2mrmvrL (iii) It is a vector quantity. (iv) SI unit is kgm2s−1 or Js

(v) )v r(mp rL ×=×= 51. Torgue a) A force F acting on a particle at p whose position vector is r . Then

the torque F about 'O' is defined as Fxr=τ b) It is an axial vector. Its direction is given by right hand thumb rule. c) S.I unit is N.m

r FP

O




Rotatory Motion

6

d) τ = Fr sinθ

52. Angular impulse ( J ) : It is the product of torque and time for which it acts.

Angular impulse = 1212 LLtt J −=ω−ω=α=×τ= III J = change in angular momentum

53. Couple : Two equal forces with opposite directions, not having the same line of action constitute a couple. e.g. : 1. Turning the cock of a water tap. 2. Turning the key in a lock. 3. Winding a wall clock with a key.

54. The moment of couple or torque is the product of either of the forces and the distance of separation between the forces.

55. To balance a couple, another equal but opposite couple is necessary. 56. Resultant of two like parallel forces :

Resultant (R) = P + Q P. AC = Q. BC R lies inside AB, between A and B

57. Resultant of two unlike parallel forces Resultant (R) = P − Q when P > Q P. AC = Q. BC R lies outside AB and is nearer to the greater force

58. Law of parallel forces : If a system of coplanar parallel forces acting on a rigid body keep it in equilibrium, then 1) the sum of like parallel forces is equal to the sum of unlike parallel forces or the algebraic sum of the forces is zero and 2) the sum of clockwise moments is equal to the sum of anticlockwise moments or the algebraic sum of the moments at a point is zero. (This is known as principle of law of moments)

59. When the resultant external torque on a system is zero, the angular momentum of the system remains constant. 2211 ω=ω II =…= constant. Circus acrobats, divers and ballet dancers take advantage of this principle.

60. When polar ice cap melts, the duration of the day increases. (i) If a wheel of radius ‘r’ rolls on the ground without slipping, the linear velocity of its

centre being v, then v = rω. (ii) The instantaneous velocity of the highest point is 2v and (iii) The instantaneous velocity of the lowest point is zero.

61. If V is the velocity of centre of mass of a rolling body the velocity of highest point of rolling body is 2V. Velocity of lowest point of the rolling body is 0. (with respect to an observer out side)

62. The kinetic energy of a rotating body = ⎥⎥⎦

⎤

⎢⎢⎣

⎡=ω

2

222

RKmv

21

21 I

63. The total kinetic energy of a rolling body = ⎥⎥⎦

⎤

⎢⎢⎣

⎡+=ω+

2

2222

RK1mv

21

21mv

21 I

64. Work–energy theorem for a rotating body is given by W = 21

22 2

121

ω−ω II

65. Acceleration of a body rolling down an inclined plane without slipping is

P

A

R

C

Q

B

B

P

A

R

C

Q

B




Rotatory Motion

7

a = 2

2

2RK1

gsina or

MR1

sing

+

θ=

+

θI

66. If a body rolls down an inclined plane of height ‘h’ without slipping, the velocity acquired

is given by

2

2

2RK1

2gh Vor

MR1

gh2V+

=+

=I

67. The time taken by a body to reach the bottom of an inclined plane of length L and height

h is

T =θ

+=

θ

+

sing

)RK1(L2

sing

)RK1(h2

2

2

2

2

2

68. If a solid sphere, a hollow sphere, a circular disc and a ring are allowed to roll down an

inclined plane simultaneously, then the solid sphere reaches the ground first and the ring reaches last.

69. If a1, a2, a3 and a4 are the accelerations of centre of masses of rolling solid sphere, solid cylinder, hollow sphere and hollow cylinder respectively then.

a1 > a2 > a3 > a4 70. If t1, t2, t3 and t4 are the times of travel of rolling solid sphere, solid cylinder, hollow

sphere and hollow cylinder respectively to reach the bottom from the top of an inclined plane then

t1 < t2 < t3 < t4 71. The moment of inertia of a body is the least when the axis of rotation passes through the

centre of gravity of the body. 72. The acceleration of a rolling body is independent of the mass of the body. 73. A body rolls down an inclined plane without slipping only when the coefficient of

friction (μ) bears the relation θ⎟

⎟⎠

⎞⎜⎜⎝

⎛

+≥μ tan

Rkk

22

2

. or μ ≥ ⎟⎟⎠

⎞⎜⎜⎝

⎛+

θ

2

2

Rk1

tan

74. Formulae for moment of inertia for some important cases :

Object Axis of rotation Moment of inertia

1. Disc of radius R 1) through its centre and perpendicular to its plane 2

MR2

2) about the diameter 4

MR2

3) about a tangent to its own plane 4

MR5 2

4) tangent perpendicular to the plane of the disc 2

MR3 2

2. Annular ring or disc of outer and inner radii R and r

1) through its centre and perpendicular to its plane 2

)rR(M 22 +

2) about the diameter 4

)rR(M 22 +




Rotatory Motion

8

3) about a tangent to its own plane 4

)rR5(M 22 +

3. Solid cylinder of length L and radius R 1) axis of cylinder

2MR2

2) through its centre and perpendicular to the axis of cylinder ⎟

⎟⎠

⎞⎜⎜⎝

⎛+

4R

12LM

22

3) diameter of the face ⎟⎟⎠

⎞⎜⎜⎝

⎛+

4R

3LM

22

4. Thin rod of uniform length L

1) through its centre and perpendicular to its length 12

ML2

2) through one end and perpendicular to its length 3

ML2

5. Solid sphere of radius R 1) about a diameter 5

2

MR2

2) about a tangent 2MR57

6. Hollow sphere of radius R about a diameter 2MR

32

7. Thin circular ring of radius R

1) perpendicular to its plane and passing through its centre. MR2

2) about its diameter 2

MR2

8. Hollow cylinder of radius R about axis of the cylinder MR2

9. Rectangular lamina of length l and breadth b

1) through its centre and perpendicular to its plane ⎟

⎟⎠

⎞⎜⎜⎝

⎛+

12b

12lM

22

2) through its centre and parallel to breadth along its own plane 12

Ml2

3) through its centre and parallel to length along its own plane 12

Mb2

4) edge of the length in the plane of the lamina 3

Mb2

5) edge of the breadth in the plane of the lamina 3

Ml2

6) perpendicular to the plane of the lamina and passing through the mid point of the edge of the length

⎟⎟⎠

⎞⎜⎜⎝

⎛+

3b

12lM

22

7) perpendicular to the plane of the lamina and passing through the mid point of the edge of the breadth

⎟⎟⎠

⎞⎜⎜⎝

⎛+

12b

3lM

22

10. Plane elliptical lamina with the values of axes 2a and 2b

Perpendicular to the plane of the lamina and passing through its centre

)ba(4M 22 +




Rotatory Motion

9

75. Motion along an inclined plane :

Object Velocity while rolling down an inclined plane

Acceleration = 2

2

RK1

sing

+

θ

1. Solid sphere 7singl10 θ

7sing5 θ

2. Hollow sphere 5singl6 θ

5sing3 θ

3. Solid cylinder 3singl4 θ

3sing2 θ

4. Disc 3singl4 θ

3sing2 θ

5. Hollow cylinder θsingl 2sing θ

6. Ring θsingl 2sing θ




1

GRAVITATION

Synopsis :

1. All forces in nature can be classified under three categories depending upon their relative strengths. They are

(i) gravitational force,

(ii) electromagnetic force and

(iii) nuclear force.

2. Fundamental forces of the universe: a) Gravitational Force:

i) It is the weakest of all the forces but has the longest range.

ii) It is because of attraction between particles due to the property of mass.

iii) Since it is a weak force, the force effects are considerable only when the interacting objects are massive.

iv) It provides the large scale structure for the universe.

b) Electromagnetic force: i) It is a strong force between two charged particles and has a long range.

ii) It acts through electric and magnetic fields.

iii) It can be attractive as well as repulsive.

iv) According to quantum field theory electromagnetic force between two charges is mediated by exchange of Photons.

c) Nuclear force: i) They are a short range, strong force of attraction between nucleons, which provides stability to the

nucleons.

ii) It is the strongest of all the fundamental forces and has a range of 1 fermi = 10–15 m.

3. a) Order of Range Range of Gravitational force > Range of Electromagnetic force > Range of nuclear force.

b) Order of strength: Nuclear force > Electromagnetic force > Gravitational force

4. The ratio of relative strengths of nuclear, electromagnetic and gravitational forces is 1 : 10−15 : 10−35.

5. Aryabhat in his famous book "Aryabhatiyam" suggested that earth is a solid sphere and it spins around itself.

6. In Rigveda, paths of planets in solar system were suggested to be elliptical.

7. In "Thaithireeya Aruna Patham", existence of several solar systems moving under the influence of a great central force was suggested.

8. The geo–centric theory was proposed by Ptolemy. According to this theory, all the planets revolve round the earth in circular orbits, the earth being at the centre.

9. The helio–centric theory was proposed by Copernicus. According to this theory, all the planets revolve round the Sun in circular orbits, the Sun being at the centre.

10. Kepler’s laws confirm the helio–centric theory.




Gravitation

2

11. Kepler’s first law of motion (Law of orbits) : All the planets revolve round the Sun in elliptical orbits with the Sun at one of the foci.

12. Planets are nine in number which revolve round the Sun and have self rotation. The order of the planets revolving round the Sun as we move away from the Sun is Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.

13. Jupiter is the biggest planet and Mercury is the smallest planet.

14. The planet which is nearest to the Earth is Venus.

15. The moon is the satellite of the earth. Acceleration due to gravity on the surface of the moon is 1.6 ms−2 or 1/6 that of the earth.

16. Mercury and Venus have no satellites, Earth and Pluto have each one satellite, Mars, Jupiter, Saturn, Uranus and Neptune have 2, 16, 22, 12 and 6 satellites respectively.

17. Kepler’s second law : (Law of areas) : The radius vector joining a planet to the Sun sweeps out equal areas in equal intervals of time. (I ω = constant). This law is a direct consequence of the law of conservation of angular momentum.

a) A planet moves fastest when it is nearest to the Sun (perihelion or perigee) and moves slowest when it is farthest from the sun (aphelion or apogee).

b) The line joining the sun and the earth sweeps out equal areas is equal intervals of time i.e. a real velocity is constant.

c) A real velocity is ω= 2r

21

dtdA

m2L

dtdA

= L is the angular momentum of the planet of mass m in the given orbit.

d) Kepler's second law is a consequence of law of conservation of angular momentum

e) According to second law a planet moves faster when it is nearer to sun and moves slower when it is far a way from the sun.

f) According to II law

Vmax rmin = Vmin rmax

III law: Law of Periods: f) Square of the period of any planet (T2) about the sun is proportional to cube of the mean distance (R3) of

the planet from the sun.

T2 α R3 or T2/ R3 = constant. 32

22

31

21

RT

RT

=

g) According to third law, as the distance of the planet increases, duration of the year of the planet increases.

h) Kepler's laws supported heliocentric or Copernicus theory.

18. The period of revolution of the moon round the earth is equal to the period of its self rotation.

19. Newton’s law of universal gravitation : Every two bodies in the universe attract each other with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

20. If m1 and m2 are the masses of two bodies and d is the distance between them, the gravitational force of attraction F which each exerts on the other is given by




Gravitation

3

F = 2

21

d

mmG ⋅

where G is called universal gravitational constant and is equal to 6.67×10−11 Nm2kg−2.

21. G was first accurately determined by Cavendish. It is a scalar quantity.

22. Properties of gravitational force: a) The gravitational force of attraction between two particles from an action and reaction pair, ie equal in

magnitude and opposite in direction.

b) Gravitational force is a central force i.e. it acts along the line joining the two particles.

c) Gravitational force between two particles is independent of the properties of intervening medium.

d) Gravitational force between two particles is independent of the presence of other particles.

e) Principle of superposition: If a no. of particles interact with each other, the net force acting on a given particle is the vector sum of the forces acting upon it, due to its interaction with each of the other particles.

f) They are long range attractive forces.

23. If two identical spheres each of radius r are kept in contact with each other, the gravitational force F between them is proportional to r4.

24. If two identical spheres each of radius r are separated by a certain distance and the distance between the spheres is maintained constant, the gravitational force F between them is proportional to r6.

25. Newton's third law of motion do not apply when (i) velocities of moving bodies are comparable to velocity of light and (ii) gravitational fields are very strong, e.g. gravitational field between objects whose masses are greater than the mass of sun.

26. Universal law of gravitation cannot explain the reason for gravity between objects and force of attraction between two bodies even when they are not in physical contact.

27. The relation between g and G is given by g = 2R

GM

= GR

34

ρπ where M is the mass of the planet, R is its radius

and ρ is the mean density of the planet.

28. The value of g near the equator is 9.78 ms−2 and near the poles it is equal to 9.83 ms−2 and is zero at the centre of the earth.

29. Variations of g are due to i) shape of the earth (Pear shaped, more flattened at the S–pole than at the N–pole), ii) Spin of the earth, iii) Latitude, iv) Altitude and v) Local conditions.

30. Shape : Earth is flat at the poles and some what bulky at the equator. The polar radius is lesser than the equatorial radius by 21 km. Hence g is greater at the polar regions than at the equatorial region.

31. Latitude : i) Because of the spin of the earth, more centrifugal force acts on bodies near the equator. Hence g value is

less at the equator.

ii) Variation of g due to rotation of the earth is given by g1 = g − R λω 22 cos where λ = latitude angle, ω = angular velocity of earth.

iii) The angular velocity of rotation of the earth is 7.27×10−5 rads−1. The linear velocity of a body at the equator is 0.465 kms−1.

iv) Spin of the earth does not affect the value of g at the poles.

v) If the earth stops spinning, g increases slightly near the equator.

vi) If the earth shrinks without change in its mass, g increases.




Gravitation

4

vii) The reduction in value of ‘g’ at the equator is 0.034 ms–2 due to the rotation of earth. (∵ Rω2 = 0.034)

viii) If the earth spins at 17 times the present speed, g becomes zero at the equator.

ix) Isograms are the lines joining the places of equal g on the earth.

x) With the help of isograms, mineral deposits and mineral oils are located.

xi) Etova balance, Gradiometer, Gravimeter are the instruments which are used to measure even the slightest variations in g.

32. Altitude : i) As the height from the surface of the earth increases, the value of g decreases.

ii) If g is the acceleration due to gravity on the surface of earth and gh at a height h above the earth, then gh

=g(1− Rh2

) approximately or gh =2

2

)hR(gR+ exactly.

33. Depth : i) As the depth from the surface of the earth increases, the value of g decreases.

ii) If d is the depth below the surface, then gd = g(1− Rd

).

34. Graph of variation of g with distance from the centre of earth:

35. g above the earth’s surface is inversely proportional to the square of the distance from the centre of the earth.

36. The value of g at the centre of the earth is zero.

37. The necessary centripetal force for a satellite is provided by the gravitational attraction of the earth.

38. According to 'field concept' a mass particle modifies the 'space' around it in someway and sets up a gravitational field.

51. Gravitational field strength is the force experienced by unit mass in the gravitational field. It is a vector. Its magnitude is equal to the value of g.

I =2R

GM

52. Propagation of gravitational fields : a) According to Einstein, gravity is because of distortion of space time due to the presence of matter i.e.

Space time is curved because of the presence of matter

b) According to General theory of relativity whenever mass particles are accelerated, the gravitational field around them changes and they are said to produce gravitational waves, which are ripples in Space time.

c) It is difficult to detect gravitational waves, but there are observational consequences such as near a pulsar, Black hole or when a massive star undergoes a gravitational collapse.

distance fromcentre of earth

Altitude

d = R

g depth




Gravitation

5

d) According to quantum theory all fields are quantum in nature including gravity. According to quantum theory of gravity, gravitational force between two mass particles is mediated by a particle called graviton.

e) A graviton has zero rest mass, travels with the velocity of light, therefore gravitational field propagates with the velocity of light.

53. Evolution of stars, cold stars (white dwarf Neutron stars) and Black holes: Life cycle of a star: a) A star is formed when, a large amount of interstellar gas, mostly H2 and He, starts to collapse on itself due

to the gravitational attraction between the gas atoms or molecules.

b) As the gas contracts it heats up due to atomic collisions.

c) As the gas continues to contract, the collision rate increases to such an extent, that the gas becomes very hot, and the gas atoms are stripped off their electrons, and the matter is in a completely ionized state, containing bare nuclei and electrons. Such a state of matter is called plasma state.

d) Under the conditions specified above, the bare nuclei have enough energy to fuse with each other (Nuclear fusion). Hydrogen nuclei fuse in such a manner to form Helium with a release of a large amount of energy in the form of radiation.

e) The radiation emitted in this process is mostly emitted in the form of visible light, UV light, IR light etc., from its outer surface. This radiation is what causes the star to shine, which makes them visible (Ex. Sun and other visible stars)

f) The star at this stage is halted from gravitational collapse (contraction) since the gravitational attraction of matter in it, towards the centre of the star is balanced by the out ward radiation pressure. A star will remain stable like this for millions of years, until it runs out of nuclear fuel such as H2 and He.

g) The more massive a star is, faster will be the rate at which it will use its fuel because greater energy is required to balance the greater gravitational attraction, owing to greater mass i.e. massive stars burn out quickly.

h) When the nuclear fuel is over, i.e. when the star cools off, the radiation pressure is not sufficient to halt the gravitational collapse. The star then begins to shrink with tremendous increase in the density. The star eventually settles into a white dwarf, Neutron star or a Black hole depending upon its initial mass.

54. Formation of white dwarfs: a) For a star to become a white dwarf, initial mass must be less than ten solar masses. (M < 10 Ms where Ms

is the mass of the sun)

b) As the star collapses after cooling off, the radiation emitted due to fusion of remaining H2 nuclei at the outer edge of the core, causes the lighter outer mantle of the star to expand to several times its original diameter. Such an expanding star is called a "Red Giant"

c) After several millions of years., the H2 fuel is exhausted and the material from the outer mantle of the Red giant is down off and the remaining core left over, which is very dim is called “white dwarf”.

d) A white dwarf is barely visible and has a mass of less than 1.4 Ms.

e) Chandrashekar limit: The maximum mass that a white dwarf can have is called Chandrashekar limit, which is 1.4 Ms. A white dwarf cannot have a mass greater than 1.4 Ms.

f) The volume of a white dwarf is about 10-6 times the volume of the original star.




Gravitation

6

g) In the white dwarf stage further gravitational collapse is halted due to the balance between repulsion of electrons and gravitational attraction. The repulsion between electrons is called degenerate electron pressure or degeneracy pressure.

h) The degeneracy pressure is because all the lower available quantum energy states, is filled up by electrons. The pauli exclusion principle prevents further filling up of these energy states. This causes the remaining electrons to fill up higher energy levels causing the required effect. In the white dwarf there is complete break down of atomic structures.

i) Matter in a white dwarf has a very high density. A white dwarf having the mass of the sun has approximately one sixth of the earth.

j) If the mass is greater than 1.4 Ms, the degenerate electron pressure between electrons will not be able to halt further gravitational collapse. The star than collapses into a neutron star or a black hole.

55. Formation of Neutron Stars: a) For a star to become a neutron star, its initial mass must be greater than ten solar masses. (M>10Ms)

b) As a star with initial mass > 10Ms cools off the large mass of the star causes it to contract abruptly and the temperature of the core rises over 100 billion degrees and when out of fuel it explodes violently. The explosion flings most of the star matter into space and is called a supernova. A supernova explosion is very bright and outshines the entire light from the galaxy.

c) The mass of the matter left behind is greater than 1.4 Ms.

d) If the mass of the left over matter is between 1.4 Ms and 3 Ms Neutron stars evolve.

e) When the mass of the left over matter lies in the range 1.4 Ms and 3 Ms, the repulsion between electrons will not be sufficient to stop gravitational collapse. Under such conditions, the protons and electrons present in the star combine to form neutrons. After the formation of neutrons, the outward degeneracy pressure between neutrons prevents further gravitational collapse, and the matter left over is the Neutron star.

f) Neutron star, has a definitely much larger than a white dwarf and has a radius of about 20 kms.

g) Neutron stars are also called pulsars, because they emit regular pulses of radio waves.

h) Neutron stars are not visible.

56. Formation of black holes: a) Black holes are objects in space, whose gravitational field is so strong that even light cannot escape from

it. We cannot see black holes because light emitted by them, would not reach us, however its gravitational effect, will be felt by other objects.

b) The possible existence of a black hole was first pointed out by John Michell (1783).

c) The phrase black hole was first coined by John wheeler.

d) Karl Schwarzschild predicted the existence of a dense object into which other objects could fall, but out of which no object or even light could ever come out. He predicted a magic sphere around this dense object through which nothing can move outwards. The radius of this sphere is known as Schwarzschild radius (Rs).

Schwarzchild radius is R = 2c

GM2

M = mass of object G =universal gravitational constant and c is the velocity of light.




Gravitation

7

e) Any object would be a black hole if and only all of its mass is inside a sphere with a radius equal to Schwarzchild radius. At the Schwarzchild radius escape velocity is equal to the speed of light. This boundary is called the event horizon.

f) Any event that occurs, within the event horizon cannot be observed from outside it.

g) Black holes have infinitely large density. The matter present inside is called singularity. h) Any object, even light, present within the event horizon will be sucked into the black hole

i) Stars turn into black holes, when the mass of the remaining matter after a supernova explosion is greater than 3Ms. The initial mass being greater than 10Ms.

j) When the mass of the remaining star is greater than 3Ms even the degeneracy pressure between neutrons cannot prevent, the gravitational collapse and Black holes are formed.

k) Evidences for existance of black holes i) Cygnus XI is a binary star, which contains a visible star moving around an unseen compansion and which

emits X rays. It is they believed that the unseen companion is a black hole, which sucks off matter from the visible star. As the matter moves towards the Black hole it gets very hot (about 100 billion degrees) and emits X - rays.

ii) Quasars (Quasi Stellar radio sources): Are very distant objects which emit powerful radio waves. A Quasar is an entire galaxy under going a gravitational collapse, due to the presence of a super massive black hole at the galactic center.

57. Formation of black holes:

58. Gravitational potential is the work done in moving a unit mass from infinity to the point under consideration. V

= RGM

− or Ι = dR

dV−

59. The binding energy of a mass ‘m’ at rest on the surface of earth of mass M and radius R is given by

U = −mgR

RGMm

−=.

Generally it is stated as positive energy i.e., as mgR.

Massive Star

(energy source (hydrogen) decreases)

Red giant stage

(material particles blown off)

White dwarf stage

(Chandrasekhar limit exceeded)

Super nova

(further core collapse)

Neutron star

(further core collapse)

Black hole




Gravitation

8

60. Gravitational potential due to a mass M at a distance r is − rGM

.

61. Inertial frames of reference : a) Frames of reference in which Newton Laws of motion are applicable.

b) Inertial frames of reference move with uniform velocity relative to each other. c) Force acting on an object due to its interaction with another object is called a real force

d) All fundamental forces of nature are real.

e) Real forces form action, reaction pairs.

Ex: Normal force, Tension, weight, spring force, muscular force etc.

f) Equation of motion relative to an observer in an inertial frame is amrealF =Σ

(m is the mass of the body having acceleration a relative to the observer.

g) Observers in all inertial frames, measure the same acceleration for a given object but might measure different velocities

h) Observers in all inertial frames, measure the same net force acting on a given object.

i) Basic laws of physics are identical in all inertial frames of reference.

j) Inertial frames of reference are called Newtonian or Galilean frames of reference 62. Non - Inertial frames: a) Frames of reference in which Newton Laws are not applicable.

b) Pseudo Force: Force acting on an object relative to an observer in a non - inertial frame, without any interaction with any other object of the universe.

c) Pseudo force exist for observers only in non - inertial frames, such forces have no existence relative to an inertial frame.

d) If a is the acceleration of a non - inertial frame. The Pseudo force acting on an object of mass m, relative to an observer in the given non- inertial frame is

F pseudo = – m a

i.e. Pseudo force acts on an object opposite to the direction of acceleration of the non - inertia frame.

e) Centrifugal force: It is a pseudo force experienced radically outward by an object relative to the object,

moving in a circular path relative to an inertial frame. The centrifugal force is given by rmv2

.

(V = speed of object relative to inertial frame)

63. Inertial mass and gravitational mass: a) Inertial mass (m1): The inertial mass of a body is the ratio of the force acting on the body to the

acceleration produced by the force.

b) It is difficult to measure inertial mass.

c) Gravitational Mass (mg) : If is the ratio of the gravitational force acting on a body to the acceleration due to gravity.

d) Gravitational mass can be measured using spring balance and common balance

e) Inertial and gravitational mass of a body are equal.

64. PRINCIPLE OF EQUIVALENCE: When experiments are conducted in inertial and non inertial frames under the same conditions, give the same

results, the frames are to be identical. This is the principle of equivalence.




Gravitation

9

65. Orbital Velocity : The velocity required for a satellite to orbit round the earth very close to it is called orbital velocity (vo) or first cosmic velocity.

Vo = RGMgR =

= 7.9 kms−1 for earth bound satellites. Vo = 1.7 kms−1 for moon bound satellites.

a) The velocity with which a satellite must be projected parallel to the earth surface, after parking it in the given orbit, so that it moves around the earth in a circular orbit.

b) If the orbit radius r, orbital velocity is rGMV0 =

c) Orbital velocity is independent of the mass of the satellite. d) V0 depends on mass of the planet and radius of the orbit.

e) For an orbit close to earth surface orbital velocity is V0 = gR

rGM

=

f) V0 close to earth is 7.92 kms-1. g) As height increases, orbital velocity decreases h) The period of a satellite in a circular orbit close to the surface of the earth

T = gR2π

= 84.6 min i) Launching of a Rocket is near equator in west to east direction. 66. A satellite of mass m orbiting close to the earth has kinetic energy and potential energy.

67. Kinetic energy of the satellite = 2mgR

R2GMm

=

68. Potential energy of the satellite = RGMm

−

69. Total energy = K.E + P.E = R2GMm

− (negative sign signifies that the body is bound to the earth)

70. If kinetic energy is E, then potential energy will be − 2E and total energy will be − E. 71. When the altitude of the satellite increases, the potential energy will increase and kinetic energy will decrease. 72. a) The increase in gravitational potential energy of a body of mass ‘m’ taken to a height ‘h’ from the surface of

the earth = mgh⎟⎠

⎞⎜⎝

⎛+ hRR

= )hR(RGMmh

+ .

b) Period of revolution of a satellite

= g)hR(

R2 3+π

c) For a satellite going round the earth in a circular orbit at height h, the orbiting velocity = hRgR2

+

d) If the satellite is very close to the earth, then T = gR2π

= 84.6 minutes.




Gravitation

10

e) If the satellite is very close to the earth, then Rg

=ω= 1.24x10−3 rads−1.

73. Geo Stationary Satellite : a) An orbit in which the time period of revolution of a satellite is 24 hours is called geostationary orbit or

parking orbit or synchronous orbit. It appears stationary with respect to the earth.

b) Radius of the geo–stationary orbit is approximately 42,400 km. Speed of geo–stationary satellite in it is 3.1 kms−1.

c) The relative velocity of a geostationary satellite with respect to the earth is zero.

d) Height of the parking orbit is 36,000 km approximately from the surface of earth.

e) Geo stationary satellite orbits above the equator in the equatorial plane.

f) Geostationary satellites are used

i) to study the upper layers of atmosphere

ii) to forecast the changes in the atmosphere

iii) to know the shape and size of the earth

iv) to identify the minerals and natural resources present inside and on the surface of the earth

v) to transmit the T.V. programs to distant places

vi) to study the properties of transmission of radio waves in the upper layers of the atmosphere and

vii) to undertake extensive research work on the planets, satellites and comets etc., which are present in space.

e) Moon is natural satellite of earth.

74. Escape velocity: a) The velocity which a body must be projected so that it never returns back or goes out of the earth

gravitational field.

b) The escape velocity of a body on earth or on any planet is

Ve = gR2 or Ve = RGM2

c) It depends upon

Mass M of the earth or planet

Radius R of the earth or planet

d) It is independent of mass of the body and angle of projection.

Its value on earth surface is 11.2Kms-1

e) If r. m. s. velocity of gas molecules is equal or greater than escape velocity, then there will be no atmosphere.

f) Orbital velocity and escape velocity are related as

Ve = 2 v0 or Ve/ V0 = 2

f) When a body is projected with escape velocity its total energy is zero.

g) Ve on earth = 11.2 kms–1,

Ve on moon = 2.5 kms–1




Gravitation

11

75. If the gravitational force varies inversely as the nth power of distance R, then the orbital velocity Vn)/2-(1R α and

the time period T)/21(nR +α .

76. Relation between velocity of projection and shape of the orbit : When a body is projected with a velocity ‘v’ from any height in a horizontal direction, then

a) If v < vo, the body falls and hits the ground in a parabolic path.

b) If v = vo, the body revolves round the earth in a circular orbit.

c) If v > vo and v < ve, the body revolves round the earth in an elliptical orbit.

d) If v = ve, the body escapes into space in a parabolic path.

e) If v > ve, the body escapes into space in a hyperbolic path.

77. Both the escape velocity and the orbital velocity are independent of the mass of the body.

78. Launching speed is about 8.5 kms–1 for a satellite at 300 km above the ground.

79. For a body revolving around the earth to escape from the orbit, the velocity of the body must be increased by 41.4% and the kinetic energy should be increased by 100%.

80. By launching a rocket near the equator in west to east direction, advantage is to the extent of 0.45 kms–1 in the launching speed.

81. If the r.m.s velocity of gas molecules is equal to the escape velocity, then there won’t be any atmosphere.

82. An astronaut in the space–craft feels weightless–ness. (since the reaction force exerted by the artificial satellite on the astronaut is zero.)




1

SIMPLE HARMONIC MOTION Synopsis : 1. The motion that repeats itself after regular intervals of time is called periodic motion.

2. If a particle in the periodic motion moves to and fro over the same path, the motion is said to be vibrating or oscillating.

e.g. : Oscillations of the balance wheel of a watch, stretched violin string, loaded spring etc.,

3. Damped Oscillations: Many oscillating bodies do not move back and forth between precisely fixed limits, because frictional dissipate the energy of motion. Such oscillations are called damped oscillations.

Ex: Stretched violin string soon stops vibratory.

4. Equilibrium Position: the point at which no net force acts on the oscillating body is known as equilibrium position or Mean position.

i) At the Mean position displacement of the body is Minimum.

ii) At the Mean position velocity of the body is Maximum.

iii) At the Mean position acceleration of the body is Minimum.

iv) At the mean position P. E. of the body is Minimum.

v) At the Mean position K.E. of the body is Maximum

5. Extreme Position: the point at which maximum force acts on the oscillating body is known as Extreme Position.

i) At the extreme position displacement of the body is maximum.

ii) At the extreme position velocity of the body is minimum.

iii) At the extreme position acceleration of the body is maximum.

iv) At the extreme position P. E. of the body is maximum

v) At the extreme position K. E. of the body is minimum.

6. If a particle moves along a straight line with its acceleration directed towards a fixed point in its path and the magnitude of the acceleration is directly proportional to the displacement from its equilibrium position, then it is said to be in simple harmonic motion.

7. If the to and fro motion is along a straight line it is called linear SHM. If the displacement is measured in terms of angles then it is called angular SHM.

8. A particle in SHM has (a) variable displacement, (b) variable velocity, (c) variable acceleration and (d) variable force.

9. Examples for linear SHM are 1. Vertical oscillations of a loaded spring. 2. Oscillations of a paper boat on water waves. 3. Vibrations of a tuning fork. 4. Oscillations of a simple pendulum with small amplitude etc.

10. Examples for angular SHM are

i) The oscillations of a freely suspended magnet in the earth’s magnetic field.

ii) Oscillations of a torsional pendulum.

iii) Oscillations of a balancing wheel in a watch.

11. The time taken for one complete vibration or oscillation is called time period (T). 12. The number of oscillations or vibrations made per second is called frequency (n). 13. The maximum displacement of the particle measured from the equilibrium position is called amplitude (r or).




Simple Harmonic Motion

2

14. Phase: Phase at any instant is that which gives the state of the vibrating particle with respect to time in a specified direction with reference to a fixed point (Mean position)

Ex: (1) If the phase is zero, ie., the particle is crossing the mean position.

(2) If the phase is π/2 ie., the particle is at the extreme position.

15. The initial phase at t = 0 of a particle in S. H. M. is called phase constant or epoch (φ).

Ex: If Y = A sin (ωt + π/4) at t = o φ = π/4

16. Representation of S. H. M.: The motion of a particle executing periodic motion along a circular path is not simple harmonic motion. But the foot of the perpendicular dropped from the instantaneous position of a particle, executing periodic motion along any diameter is simple harmonic motion.

In general the simple harmonic motion is represented as Y = A sin (w t + φ)

Y = instantaneous displacement

A = Amplitude

Wt + φ = phase; φ is called initial phase.

i) If the motion starts from Mean Position φ = 0

ii) If the motion starts from the extreme position φ = π /2

17. Simple harmonic motion can be expressed by periodic functions like Asinωt, Acos ωt, or combination of these functions.

Y = A sin ωwt + B cos ωt

18. S.H.M. can also be represented in the following way ie., F = ma

F = m. 2

2

dtyd = –KY or

2

2

dtyd + ⎟

⎠⎞

⎜⎝⎛mk y = 0

CHARACTERISTICS OF S. H. M. 19. Instantaneous displacement: The distance of the particle from mean position in a particular direction at any

instant of time is known as instantaneous displacement.

It is given by Y = A sin (ωt + φ)

If the particle starts from Mean position, φ = 0 then

i) Y = 0, at Mean position

ii) Y = A at extreme position

iii) Y = 2A at θ = ωt = 30° = π/6 or after a time interval t = T/12 sec.

iv) Y =2

A at θ = ωt = 45° = π/4 or t = T/8 sec.

v) Y =23 A at θ = ωt = 60° =π/3 or t =T/6 sec.

20. Velocity: The rate of change of displacement is called velocity.

∴ v = dtdy = Aw cos (wt + φ)

θ θ θφφ

Y

X X

Y

P (t=sec) P (t=sec)PA (t=0)

A (t=0)

Phase constant=Y=Asin t-

φω φ

Phase constant=Y=Asin t+

φω φ

φ=0ωY=Asin t

x (t=0)Y





3

v = ω 22 yA −

If the particle starts from the mean position, φ = 0 then

i) v = Aω, ie., maximum at Mean Position

ii) v = 0, ie., minimum at extreme Position

iii) v = 2

Aω at θ = ωt = 60° = π/3 or after a time interval of t = T/6 sec

iv) v = 2

Aω at θ = ωt = 45° = π/4 or after a time interval of t = T/8 sec

v) v = 23 Aω at θ = ωt = 30° = π/6 or after a time interval of t = T/12 sec

21. Acceleration: the rate of change of velocity of a particle in S H M is called acceleration.

∴ a = dtdv = – Aω2 sin (ωt + φ) = – ω2y

a α – y

If the particle starts from the Mean position, φ = 0, then

i) a = 0, i.e., minimum at mean position

ii) a = ω2 A, i.e., maximum at extreme position.

iii) a = 2A2ω at θ = ωt = 30° = π/6, after a time interval of t = T/12 sec

iv) a = 2A2ω at θ = ωt = 45° = π/4, after a time interval of t = T/8 sec

v) a = 2

A3 2ω at θ = ωt = 60° = 3π , after a time interval of t = T/6 sec

22. The projection of uniform circular motion of a particle over any diameter of the circle is in SHM.

23. If two simple harmonic motions of same amplitude and frequency at right angles to each other are superposed, the resulting motion will be linear if the phase difference is 0 and circular if the phase difference is π/2 radians.

24. Time Period: Time taken by vibrating particle in S.H.M. to complete one vibration is called Time period or period of oscillation.

General formula : T = 2π

onaccelerati ntdisplaceme

i) ∴T = 2πay

ii) Time period of a simple pendulum = T = 2π

gl

l = length of the simple pendulum

g = acceleration due to gravity at a place.

iii) Time period of Torsion pendulum

T = 2π CI





4

I = moment of Inertia about the suspension wire C = couple per unit twist.

iv) Time period of a loaded spring

a) T = 2π km

k = Force constant or spring constant

m = mass attached to the spring.

b) T = 2π gx

x = extension produced in the spring due to attachment of the load 'm'

g = acceleration due to gravity.

v) When a hole is drilled along the diameter of the earth and if a body is dropped in it, it moves to and from about the centre of the earth and is in S.H M. with a time period of

T = 2πgR = 84.6 minutes or T =

GD3π

D = Mean density of the earth

G = Gravitational constant

25. Kinetic energy at any instant

)yr(m21

tcosrm21

222

222

−ω=

ωω=

26. Potential Energy :

0222

022 UsinAm

21Uxm

21.E.P +θω=+ω=

Where m = mass of S.H.M.

x = displacement of S.H.M. from its mean position

A = amplitude of oscillation

θ = phase angle from its mean position

Uo= P.E. of S.H.M. at its mean position.

v) During one complete vibration average potential Energy is given by = 1/4 mω2A2

27. Kinetic Energy: i) The K. E. of a particle in S.H.M is given by K. E = 1/2 mω2 (A2 – y2)

= 1/2 mω2A2 cos2 (ωt ± φ)

ii) At mean position (y = o) K. E is maximum

iii) At extreme position (y = A) K. E. is zero.

iv) During one complete vibration average kinetic Energy = 1/4 mω2A2

28. Total Energy: i) T. E. = P. E. + K. E. = 1/2 mω2A2 +U0

ii) When a particle is in S. H. M. At any position T. total energy is constant.

Energy and displacement curve.





5

29. In SHM, average kinetic energy = average potential energy = half of the total energy, when friction is zero.

30. For a body moving with SHM, velocity is 90° out of phase with the displacement and acceleration is 180° out of phase with the displacement. Velocity and acceleration have a phase difference of 2/π radians.

31. If n is the frequency of SHM, then the frequency with which kinetic energy or potential energy oscillates is 2n.

Simple pendulum: 32. The period of oscillation of a simple pendulum is independent of amplitude (for small values only), length being

constant.

33. At constant length, the period of oscillation of a simple pendulum is independent of size, shape or material of the bob.

34. Time period of a simple pendulum (T) = gL2π .

35. Tension in the string of simple pendulum

Tmin = mg Cos θ (when bob is at extreme position)

T = mg (3 – 2 Cosθ) (When bob is at any position)

where θ is any angular amplitude.

36. I – T2 graph of a simple pendulum is straight line passing through origin.

37. l–T graph of a simple pendulum is parabola. 38. At the point of intersection of l–T graph and l–T2 graph of a simple pendulum

i) T = 1 second ii) n = 1 Hz.

iii) cm254

gl2≅

π= on the surface of the earth

39. APPLICATION :

i) When length changes 2

1

2

1LL

TT

=

ii) For small percentage changes < 5%. 100LL

21100

TT

×Δ

=×Δ

iii) For percentage ≥ 5%.If the percentage change is "n" then ⎟⎟⎠

⎞⎜⎜⎝

⎛+=×

Δ100nn2100

TT 2

iv) When the elevator is going up with an acceleration a, then its time period is given by

T = ag

L2+

π and frequency n is given byn = L

ag21 +π

v) When the elevator is moving down with an acceleration a, then its time period is given by

T = ag

L2−

π and frequency n is given by

n = L

ag21 −π

X = AX = A X = 0

P.E. K.E.

E

l –T graphY

l

l –T2 graph

X

T2





6

vi) When the elevator is at rest or moving up or down with constant velocity the time period is given by T =

2Lg

21n;

gL

π=π

vii) When the elevator is moving down with an acceleration (–a) then its time period is given by T=

.Lag

21n;

agL2 +

π=

+π

viii) In case of downward accelerated motion is

a>g the pendulum turns upside and oscillates about the highest point with T = ga

L2−

π

ix) If a simple pendulum of length 'L' suspended in a car that is travelling with a constant speed around a circle of radius 'r', Then its time period of oscillation is bygiven

T = 22

2r

vg

L2

⎟⎟⎠

⎞⎜⎜⎝

⎛+

π

x) If a simple pendulum of length 'L' suspended in car moving horizontally with an acceleration 'a' is given by

22 )a(g

L2T+

π=

The equilibrium position is inclined to the vertical by an angle 'θ'.

Where θ = tan–1⎟⎟⎠

⎞⎜⎜⎝

⎛ga opposite to the acceleration.

xi) If the bob of a simple pendulum is given a charge 'q' and is arranged in an electric field of intensity 'E' to oscillate.

a) opposite to g, → Electric force Eq will be opposite to the force mg. Hence g1=g–mEq

Then T1=2π

mEqg

l

−. So time period increases.

b) In the direction of g → Electric force Eq will be in the direction of force mg. Hence

g1 = g + mEq then

T1 = 2π

mEqg

l

+

so time period decreases.

c) Perpendicular to g → Electric force Eq will be perpendicular to the force mg. Hence

g1 = 2

2mEqg ⎟

⎠⎞

⎜⎝⎛+ Then

T1=2π2

mEq2g

l

⎟⎠⎞

⎜⎝⎛

+

.

So time period decreases.

Eq

-ve charge

+q

+ve charge





7

xii) If a simple pendulum of length L is suspended from the ceiling of a cart which is sliding without friction on

an inclined plane of inclination 'θ'. Then the time period of oscillations is given by T = θ

πcosgL2 Since

the effective acceleration changes from g to g cosθ.

xiii) Time period of a pendulum of length comparable to the radius of earth is

T = g)RL(

LR2+

π .

xiv) The maximum time period of simple pendulum (pendulum of infinite length) is given by = gR2π = 84.6

minute = 1.4 hour

xv) The time period of a simple pendulum having a length equal to the radius of the earth is T = g2/R2π and is equal to 42.3 2 minutes or 59min and 5 sec.

xvi) When a pendulum clock is taken from the equator to the poles the time period decreases. Hence it makes more oscillation and gains time and moves fast.

xvii) When a pendulum is taken from the earth to moon, the time period increases (as g is less on moon). Hence it makes less number of oscillations and looses time or moves slow.

xviii) When a pendulum clock is taken from the earth to the moon, to keep the time correct its length must be decreased.

xix) If the pendulum of a clock is made of metal, it runs slow during summer and fast during winter due to thermal expansion or contraction.

xx) If a boy sitting in a swing stands up, as centre of Mass rised up, length of the pendulum decreases and hence the period of the swing decreases.

xxi) If the bob of a pendulum is made hollow and filled with water, and the water is drained up as the water goes down, centre of mass shifts down, and then rises to its original position. Hence time period first increase and then decreases and reaches its original value.

40. If a pendulum clock is shifted from earth to moon then it runs 6 times slower.

41. For a simple pendulum at a place L/T2 is a constant and g = 22

TL4π .

42. The tension in the string at any position is equal to T = mg cos θ + l

mv 2.

43. If a simple pendulum is arranged in an artificial satellite its time period becomes infinity.

44. Work done by the string of the simple pendulum during one complete vibration is equal to zero.

45. A simple pendulum fitted with a metallic bob of density ds has a time period T. When it is made to oscillate in a liquid of density d1 then its time period increases.

T = 2π

⎟⎟⎠

⎞⎜⎜⎝

⎛−

s

ldd1g

l

46. When two simple pendulum of lengths l1, and l2, l2 > l1 are into vibration in the same direction at the same instant with same phase. Again they will be in same phase after the shorter pendulum has completed n oscillations. To find the value of n,

nTs = (n –1) Tl and T α l





8

∴ S

LTT

1nn

=−

or s

L

ll

1nn

=−

or n =

L

S1

1

ll

−

S – shorter L – Longer

47. Seconds pendulum: i) The simple pendulum whose time period equal to 2 seconds is called seconds pendulum.

ii) its length at place where g = 9.8 m/s2 is 100 cm.

iii) Since T = 2 sec

L = 2

2

4T.gπ

⇒ L = 4.4

g2π

∴L = 2

gπ

iv) The length of a pendulum at a place where g = g1 is l1 and its length at a place where g = g2 is l2 To keep the time period constant at T = 2 sec. its length has to be decreased or increased corresponding to the value of 'g' at that place

Decrease in length = 2

21 ggπ

− (if g1 > g2)

Increase in length = 2

12 ggπ

−(if g2 > g1)

48. A pendulum clock runs slow when i) L increases and ii) g decreases.

49. A pendulum clock runs fast when i) L decreases and ii) g increases.

50. A pendulum clock stops functioning at any place where the gravity is absent such as in an orbiting satellite, the centre of the earth the time period is infinity.

51. If the length of a simple pendulum is increased by x% (when x is very small), the period increases by x / 2 percent.

52. If the value of g increases by x%, the time period decreases by x / 2 percent.

53. If a wire of length L, area of cross–section A and Young’s modulus Y is loaded by a mass m, the period of

oscillation (T)=YAmL2π or T=

gx2π where x is the elongation produced.

54. If a U–tube contains a liquid up to a vertical height h and the liquid in one limb is slightly pushed and released, the oscillation of liquid column is simple harmonic with a time period g/h2π .

55. GRAPHS: (particle starts from mean position) i) Displacement – time graph ii) Velocity – time graph

T / 2

–A

+A Y t T

T / 2

–

+Aω O T





9

iii) Displacement – velocity graph

iv) Acceleration – time graph

v) Acceleration – displacement graph

vi) Force – time graph vii) Force – displacement viii) Potential Energy – time graph ix) Potential Energy – displacement graph x) Kinetic Energy – time graph

X

a

TT/2 t

U

T

Ek

OT/2

t

V

AAW

T / 2

–Aω2

+Aω2

OtT

T / 2 F t T

X

F

o –x

U

X





10

xi) Kinetic Energy – displacement graph

xii) Total Energy – time graph xiii) Total Energy – displacement graph xiv) L – T graph for a simple pendulum xv) L – T2 graph for a simple pendulum.

SPRINGS : 56. The spring constant of a spring may be defined as the force required to produce an extension of one unit in

the spring. K = F / x.

57. Potential energy of the spring = 22

Kx21

KF

21Fx

21

==

58. If a spring is cut into two pieces (of equal size), each piece will have a force constant double the original. 59. When a spring of force constant k is cut into n equal parts, the spring constant of each part is nk 60. If a uniform spring of spring constant K is cut into two pieces of lengths in the ratio l1 : l2, then the force

constants of the two springs will be

2

212

1

211 l

)ll(KK and

l)ll(K

K+

=+

=

61. The spring constant of a spring is inversely proportional to the number of turns. F / x or Kn = constant or K1n1 = K2n2.

62. For a given spring greater the number of turns, greater will be the work done. n w α

2

1

2

1

nn

ww

=

63. If two springs of force constants k1 and k2 are joined in series, the combined force constant

k =21

21kk

kk+

.

64. If two springs of force constants k1 and k2 are joined in parallel, the combined force constant k = k1 + k2. 65. When a body is just dropped on a spring, the maximum compression is double that of when the body rests on

it in equilibrium.

–x + x

E

O

TE

t

-x xO

E

T

Y

lX

T2

l





11

km2T π=

21 kkm2T+

π=

21 kkm2T+

π=

21

21kk

)kk(m2T +π=

km2T π=

21 kkm2T+

π=

21

21kk

)kk(m2T +π=

21 kkm2T+

π=

)mm(kmm2T

21

21+

π=




1

ELASTICITY Synopsis :

1. Deforming force : The external force acting on a body on account of which its size or shape or both change is defined as the deforming force.

2. Restoring force : The force which restores the size and shape of the body when deformation forces are removed is called restoring force. Deforming force and restoring force are not action reaction pair. Restoring force opposes the change in the size and shape of a body.

3. Rigid body : A body whose shape and size cannot be changed however large the applied force is called rigid body. There is no perfectly rigid body in nature.

4. Elasticity : The property of a body by virtue of which it regains its original size and shape immediately after the deformation forces are removed is called elasticity. Elasticity is a molecular phenomenon. It is because of cohesive forces.

5. Elastic body : A body which shows elastic behaviour is called elastic body. E.g. steel, rubber. Quartz is very nearly perfectly elastic body.

6. Plastic body : A body which does not show elastic behaviour is called plastic body. E.g. putty, clay, mud, wax, lead, dough, chewing gum, butter wax etc.

7. Out of the given materials, a body in which it is more difficult to produce strain is more elastic. OR The body which requires greater deforming force to produce a certain change in dimension is more elastic.

a) steel is more elastic than rubber

b) glass is more elastic than rubber

c) water is more elastic than air

d) springs are made of steel but not of copper because steel is more elastic than copper.

8. By the process of hammering or rolling the body elasticity increases.

9. By the process of annealing, the elastic property of a body is reduced.

10. For invar steel (Fe-64%, Ni-36%) the elastic property is constant irrespective of change in temperature. (used in making pendulum clocks)

11. FACTORS EFFECTING ELASTICITY : a. Effect of temperature : In general as the temperature increases the elastic property of a material

decreases.

b. Effect of impurities : Addition of impurity to metal may increase or decrease the elasticity. If the impurity has more elasticity than the material to which it is added, it increases the elasticity.

If the impurity is less elastic than the material it decreases the elasticity.

12. Stress : The restoring force developed per unit area of cross-section of the deformed body is called stress.

Stress = AF

area tionalsecCrossforce Restoring

=−

Unit= pascal ,cm

dyne ,mN

22

Dimensional formula : M1L−1T−2

13. i) Pressure is always normal to the area, while stress can be either normal or tangential.

ii) Pressure on a body is always compressive, while stress can be compressive or tensile.




Elasticity

2

iii) Pressure is a scalar, while stress is a tensor.

Stress is of three types : i) Longitudinal stress : If the restoring forces are perpendicular to the area of cross-section and are along

the length of the wire, the stress is called longitudinal stress.

During longitudinal stress, the body undergoes change in length but not in shape and volume.

ii) Tangential stress (or shearing stress) : If the restoring forces are parallel to the surface, the stress is called shearing stress.

iii) Bulk stress (or volume stress) : If a body is subjected to equal forces normally on all the faces, the stress involved is called bulk stress.

14. Strain : The deformation produced per unit magnitude is called strain.

a) longitudinal strain = change in lengthoriginal length

=le

b) shearing strain = θ = layers two the between distance larperpendicu

layers two between ntdisplaceme lateral

=llΔ

c) Bulk strain=v

vvolume originalvolume in change Δ−

=

d) Transverse strain or lateral strain

=r

rradius originalradius in change Δ−

=

e) shearing strain = 2 x longitudinal strain

f) bulk strain = 3 x longitudinal strain

g) longitudinal strain : shearing strain : bulk strain

= 1:2:3

Shear strain is equivalent to two equal longitudinal elongation and compressional strains in mutually perpendicular directions.

The maximum value of the stress within which the body regains its original size and shape is called elastic limit.

15. Hooke’s law : Within the elastic limit of a body, stress is directly proportional to strain.

constantEstrainstress

==

Unit of E : CGS : 2cm

dyne , SI : 2m

newton or pascal

i) Within the proportionality limit stress-strain graph is a straight line passing through the origin.

ii) A spring balance works on the principle of Hooke’s law.

iii) Modulus of elasticity does not depend upon the dimensions of the body but is a property of the material of the body.

iv) Within the proportionality limit, the load extension graph is a straight line passing through the origin.

16. Behaviour of a wire under the action of a load :




Elasticity

3

A = Proportionality limit B = Elastic limit C = Yielding point D = Breaking point Sb = Ultimate tensile strength

a) Stress is proportional to strain upto a limit, which is called proportionality limit. A is the limit of proportionality. Upto this limit, Hooke’s law is obeyed.

b) The smallest value of stress which produces a permanent change in the body is called elastic limit. c) If the wire is loaded beyond the elastic limit, a stage is reached where the wire begins to flow with no

increase in the load and this point is called yield point. d) Beyond the yield point, if the load is increased further the extension increases rapidly and the wire

becomes narrower and finally breaks. The point at which the wire breaks is called breaking point. e) Maximum stress required to break the wire is called ultimate tensile strength.

f) The capacity of a material to withstand large stresses without permanent set is called resilience.

g) The wire regains its original length if the elastic limit is not exceeded.

h) The wire does not obey Hooke’s law between the proportionality limit and elastic limit. But wire regains its original length when the load is removed.

i) A permanent set (OP) is produced in the wire beyond elastic limit.

j) The stress required to reach the breaking point is called breaking stress.

k) If the gap between elastic limit and breaking point (BD) of a metal is large, it is called a ductile metal.

l) If the wire breaks soon after exceeding limit, the metal is said to be brittle. (If the gap BD is small).

17. Types of moduli of elasticity : There are three moduli of elasticity.

1) Young modulus ‘Y’

2) Rigidity modulus ‘n’

3) Bulk modulus ‘K’

1) Young’s modulus : Young’s modulus is the ratio of longitudinal stress to longitudinal strain within the elastic limit of a body.

Y=elx

AF

leAF

strain allongitudinstress allongitudin

==

X

Y

O Strain

Stre

ss E

DCB

A

OI

Sb




Elasticity

4

When a mass “M” is attached to the lower end,

Y=AeFl but F=Mg, A=πr2

22 r

l.geM Y,

el.

r

Mg Yπ

⎟⎠

⎞⎜⎝

⎛=π

=∴

i) The stress required to double the length of a wire (or to produce 100% longitudinal strain) is equal to Young’s modulus of the wire.

ii) Y of a perfectly elastic material is infinite and that of a perfectly inelastic material is zero.

2) Rigidity modulus : Rigidity modulus is the ratio between shearing stress and shearing strain within the elastic limit of a body.

η = θ

=θ

=AF

tanAF

strain shearingstress shearing

(for small values of θ, tanθ = θ) (or) n =l

l.AF

Δ.

i) If η is low for a wire, it can be twisted very easily.

ii) Since phosphor-bronze has very low rigidity modulus, it is used as a suspension fiber in moving coil galvanometers.

Rigidity (shear) modulus is used to calculate the strain produced in a rod under twisting stress. It is also used to calculate the restoring torque when a wire or a cylinder is twisted. Torque C produced per unit twist

of a wire of length l and radius r is given by C =4r

2lπη where η is rigidity modulus.

iii) If a rod of length l and radius r is fixed at one end and the other end is twisted by an angle θ, then lφ = rθ. where φ is angle of shear.

3) Bulk modulus : Bulk modulus is the ratio between volume stress and volume strain within the elastic limit of a body.

K = volume stress F / Avvolume strain

v

=Δ⎛ ⎞−⎜ ⎟

⎝ ⎠

.

(–sign indicates the decrease in volume)

K = v

PvΔ

i) If a block of coefficient of cubical expansion γ is heated such that the rise in temperature is θ, the pressure to be applied on it to prevent its expansion=Kγθ where K is its bulk modulus.

ii) When a rubber ball of volume ‘V’ bulk modulus ‘K’ is taken to a depth ‘h’ in water, decrease in its volume is

KhdgVV =Δ (d = density of material).

All modulii of elasticity Y, n, K have same units and dimensions -> [M1L−1T−2] → N.m−2

i) Solids possess Y, n and K.

ii) Liquids and gases possess only K.

iii) Bulk modulus of gases is very low, while that of liquids and solids is very high.

l

Δl

θ




Elasticity

5

iv) Isothermal bulk modulus of a gas=pressure of the gas (P)

v) Adiabatic bulk modulus of a gas = γP where γ = ratio of two specific heats.

4) Compressibility : The reciprocal of bulk modulus is called compressibility

C =K1 For incompressible substances C = 0, K = ∞

18. Poisson’s ratio (σ) =strain elongation allongitudin

strain ncontractio lateral

or rtransverse strain rllongitudinal strainl

Δ−=

Δ

i) Poisson’s ratio has no unit and has no dimensions.

ii) Theoretical limits of σ = –1 to 0.5.

iii) Practical limit of σ = 0 to 0.5

iv) If σ = 0.5 the substance is perfectly incompressible.

19. Relation among elastic constants Y, η, K, σ :

i) 9 1 3Y K

= +η

ii) Y 2 (1 )= η + σ

iii) Y = 3K(1−2σ) iv) 3K 22( 3K)

− ησ =

η +

20. Applications of ‘Y’ : i) A long wire suspended vertically can elongate due to its own weight.

ii) Elongation of a wire due to its own weight e=Y2dgl2 ; l is length of the wire, d is density of the wire,

Y=Young’s modulus of the material of the wire, g=acceleration due to gravity

iii) A very long wire suspended vertically can break due to its own weight.

iv) Maximum length of the wire that can be hung vertically without breaking=s/dg where s is breaking stress.

v) Breaking stress : a) The breaking stress of a wire is the maximum stress the material can withstand.

b) Breaking stress=section-cross of area initial

force breaking

vi) Breaking force =

Breaking stress x area of cross-section

vii) Breaking force : 1) is independent of length of the wire

2) depends on the area of cross-section and nature of material of the wire.

3) breaking force α area of cross-section.

4) If we cut a cable that can support a maximum load of W into two equal parts, then each part can support a maximum load of W.

21. Elastic hysteresis is the result of elastic after effect. There is a lag between stress and strain. The lag is known as elastic hysteresis.




Elasticity

6

22. Elastic Fatigue : a) The state of temporary loss of elastic nature due to continuous strain is called elastic fatigue.

b) Due to elastic fatigue :

i) a wire can be broken within the elastic limit

ii) a wire can be cut into pieces without using instruments

iii) railway tracks and bridges are declared unsafe after long use

iv) spring balances show wrong readings after long use.

23. Strain energy : strain energy of a stretched wire =

E = 2

elongation x Force2

Fxe=

Potential energy stored per unit volume in a strained body is called strain energy density.

Potential energy stored in a wire due to twisting= τθ21 .

25. Strain energy density :

= strain x stress x 21

volumework

=

=y

)stress(x21(strain) y x x

21 2

2 =

If ‘K’ is the force constant, energy stored for extension ‘e’ is given by E = 2Ke21

26. Laws of elongation : i) e α l; elongation is proportional to length of wire

ii) e α F; elongation is proportional to force applied

iii) e α 1/A or 1/r2 elongation is inversely proportional to area of cross-section or square of the radius.

iv) elongation is inversely proportional to Young’s modulus.

v) For two wires made of same material, 1

2

2

1

2

1

2

1AA

.FF

.ll

ee

=

vi) For two wires made of same material, when same force is applied on them 21

22

2

1

2

1

r

r.

ll

ee

= .

vii) For two wires, made of same material, and of same volume when same force is applied, elongations ratio

is given by 41

42

21

22

22

21

2

1

r

r

A

A

l

lee

=== (since, V = A × l =constant A1l1 = A2l2 21

22

1

2

2

1

r

rAA

ll

== )

viii) If l1 and l2 are the length of a wire under tension T1 and T2, the actual length of the wire = 12

1221TT

TlTl−−

27. Springs :

i) For a spring that obeys Hooke’s law, equivalent force constant or spring constant is K = l

YA .




Elasticity

7

ii) K α Y, K α A, K α 1/l

iii) If a spring (or a wire) of force constant K is cut into ‘n’ equal parts, the force constant of each part of the wire is ‘nk’.

iv) If a spring (or a wire) of force constant k is cut in the ratio of m:n, km=m

k)nm( + ; kn= nk)nm( +

v) Potential energy of a stretched spring =K

F21Kx

21Fx

21 2

2 ==

vi) Two springs have force constants K1 and K2

a) When they are stretched by the same force and if their elastic energies are E1 and E2. 1

2

2

1KK

EE

=

b) When they are extended by the same length 1

2

2

1KK

EE

=

c) When they are extended till their energies are same, 2

1

2

1KK

FF

=

d) The potential energy of a spring increases, whether it is stretched or compressed.

e) Springs in series Keff=21

21KK

KK+

f) Springs in parallel Keff=K1+K2

g) The reciprocal of spring constant is called compliance.

vii) When a spiral spring is stretched, strain involved is longitudinal strain. (thickness is small)

viii) When a helical spring is stretched, strain involved is longitudinal and shearing strain. (thickness is large)

ix) When a wire is stretched, modulus of elasticity involved is Young’s modulus

x) When a wire is twisted, modulus of elasticity involved is rigidity modulus.

xi) Inter atomic force constant k = Y. r = Young’s modulus x (inter atomic distance)

28. Thermal force : i) When a metal bar is fixed between two walls and the temperature is raised, the bar tries to expand and

exerts a force on the walls. This force is called thermal force given by F=YAαθ

α=co-efficient of linear expansion of the bar

θ = rise in temperature

Y = Young’s modulus, A=area of cross-section

Thermal force is independent of the length of bar.

ii) Thermal stress :

Thermal stress = αθ=αθ

= YA

YAarea

force thermal

iii) If a load ‘M’ produces an elongation ‘e’ in a wire the rise in temperature required to produce the same

elongation is αYA

Mg (since tle

Δα= )

where A = area of cross-section of the wire and α = coefficient of linear expansion of the material of the wire.




1

SURFACE TENSION

Synopsis : 1. The properties of a surface are quite different from the properties of the bulk material. 2. A molecule well inside a body is surrounded by similar particles from all sides. But a molecule

on the surface has particles of one type on one side and of different type on the other side. 3. The attractive forces between the molecules of a substance are called cohesive forces. 4. The attractive forces between the molecules of a different substances are called adhesive

forces. 5. A molecule of water well inside the bulk experiences cohesive forces but a molecule at the

surface experiences both cohesive and adhesive forces. This asymmetric force distribution is responsible for surface tension.

6. The maximum distance upto which the cohesive force between two molecules exists is called the molecular range and of the order 10−9 m or 1 nm.

7. An imaginary sphere drawn around a molecule with a radius of molecular range is called the sphere of influence of that molecule.

8. Intermolecular force of attraction varies inversely as the eighth power of the intermolecular distance. 9. The surface of the liquid behaves like a stretched rubber sheet. Surface tension is due to cohesion between

the molecules of the liquid. 10. Surface tension is the force per unit length of a line drawn on the liquid surface and acting perpendicular to it.

T = lF S.I unit is N/m.

11. Force required to pull a circular ring of radius r from the surface of water of surface tension T is F=4πrT. 12. Force required to pull a rectangular plate of length ‘l’ and thickness ‘t’ from the surface of water of surface

tension T is F = 2(l + t)T. 13. Force required to pull a circular disc of radius R with hole of radius r from the surface of water of surface

tension T is F=2π (R + r)T. 14. A slide is suspended from one arm of a balance and is counter balanced. Now the slide is lowered into a

beaker of water until it just touches the surface of water. If m is the additional mass to keep the balance beam

horizontal, then T =)bl(2

mg+

. Where l is length and t is thickness of slide.

15. When a cylindrical glass tube is closed at one end and it is made heavy such that it floats in the vertical

position. If the depth of heavy end is h below liquid surface, then 22 rT mgh

r dgπ +

=π

.

16. Drops of liquid of density d1 are floating half immersed in a liquid of density d2. If the surface tension of liquid is

T, then the radius of the drop will be r = 1 2

3T(2d d )g−

.

17. A molecule in the surface has greater potential energy than a molecule well inside the liquid. The extra energy that a surface layer has is called the surface energy.

18. Surface tension of a liquid is also equal to the surface energy per unit surface area. Unit is J/m. 19. Surface energy is defined as the quantity of work done in increasing the surface area of the liquid through unity

(also equal to the surface tension of liquid)

surface energy =area

energyareawork

=

20. Work done = surface tension x increase in surface area. 21. At constant temperature, liquid surface does not obey Hooke’s law, and surface tension is independent of the

surface area. 22. The free liquid surface tries to attain minimum surface area. This is the reason for a free liquid drop (like rain

drop) to attain a spherical shape. 23. The work done (W) in blowing a soap bubble of radius r is W = 8πr2T. 24. Work done in increasing the area of circular soap film from radius r1 to r2; W = ]rr[T2 2

12

2 −π .




Surface tension

2

25. Work done in increasing the radius of a bubble from r1 to r2 is given by W = 8π( 21

22 rr − )T.

26. When a number of small liquid drops coalesce to form a large drop, energy is released (since surface area decreases).

27. Energy is required to break a liquid drop into smaller drops (since surface area increases). 28. The work done by an agent to split a liquid drop of radius R into ‘n’ identical drops is

W = 4πR2T(n1/3–1) where T is surface tension. 29. The amount of energy evolved when ‘n’ droplets of a liquid of radius ‘r’ combine to form a large drop is E =

4πr2T(n–n2/3). 30. ‘n’ drops each of radius ‘r’ combine to form a big drop of radius ‘R’. Then the work done (or) the energy

expended is )Rnr(T4 22 −π . 31. Factors influencing surface tension :

a) Temperature : Usually it decreases with increase of temperature. When temperature of liquid increases K.E. of its molecules also increase. Hence cohesive forces between the molecules become weak.

At critical temperature the interface between liquid and its vapour disappear and so surface tension of a liquid becomes zero.

The surface tension of molten cadmium and molten copper increases with increase of temperature. Surface tension of a liquid is zero at its boiling point.

b) Impurities : If the liquid surface is contaminated by impurities, its surface tension decreases. e.g. when oil or kerosene is sprayed on water surface, its surface tension decreases. Mosquito breed floating on water can be destructed using this technique. Surface tension increases when an inorganic substance (highly soluble) is dissolved in water. e.g. When NaCl, ZnSO4 are dissolved, surface tension of water increases. Surface tension decreases when an organic substance is dissolved (weakly soluble) in water. e.g. When soap is mixed with water surface tension decreases. Surface tension of soap solution is about 32 dy cm–1.

32. Of all liquids, mercury has maximum surface tension. 33. Angle of contact : It is the angle between the tangent drawn at the point of contact of a solid and liquid and the

surface of solid within the liquid. 34. The angle of contact may assume any value between 0° and 180°. 35. The angle of contact depends on solid-liquid pair, temperature and impurities. 36. The angle of contact is not altered by the amount of inclination of solid object in the liquid. 37. θ is independent of manner of contact i.e. glass plate in a liquid or liquid drop on a plate or liquid in a solid

vessel. 38. Angle of contact increases with increase in temperature. 39. For pure water and glass, the angle of contact is zero. 40. Angle of contact decreases on adding soluble impurity, detergent and wetting agent to a liquid. 41. For mercury and glass, angle of contact is about 140° .~− 139°C. 42. (a) For Ag and H2O angle of contact is 90°.

(b) For ordinary water θ lies between 8° and 18°. (c) Angle of contact of chromium with water is as high as 160°.

43. If a liquid wets the solid, then the angle of contact is less than 90° and if the liquid doesn’t wet the solid, then the angle of contact is greater than 90°.

44. Angle of contact in case of solid, liquid and air in contact :

Let T1 is surface tension for air-liquid surface, T2 for air-solid and T3 for liquid-solid surfaces respectively. If θ be the angle of contact of the liquid with the solid, then

Solid SolidT 3 T3T2 T2

T1 T1air

θ θ

liquid Mercury




Surface tension

3

1

32T

TTcos

−=θ

a) If T2 is greater than T3, cosθ will be positive i.e., θ will be less than 90°. b) If T2 is less than T3, cosθ will be negative i.e., θ will be between 90° and 180°. c) If T2>T1+T3, there will be no equilibrium, and the liquid will spread over the solid.

45. Capillarity : The property of rise or depression of the liquid due to surface tension in a tube is known as capillarity.

46. Oil ascends in a wick due to capillarity. 47. Flow of ink through a nib is due to capillarity. 48. A painter’s brush under water has its hair spread but on withdrawal from water they adhere to each other due

to surface tension. 49. Ploughing of land brings moisture to the top by capillary action. 50. The addition of a detergent decreases the surface tension and angle of contact. 51. Wetting agents are used in detergents in order to clean clothes. 52. The addition of a water proofing agent like waxy substance to a liquid increases angle of contact. 53. If the angle of contact (θ) is acute (θ < 90°), there will be capillary rise. e.g. : water in glass capillary. 54. If the angle of contact (θ) is obtuse (θ > 90°), there will be capillary depression. e.g. : mercury in capillary. 55. If the angle of contact is 90°, there will be neither rise nor fall. e.g. : water in silver capillary. 56. Rise of liquid in tubes of insufficient length : If a liquid can rise upto a height ‘h’ in the tube but its total

length outside the water surface is less than ‘h’ the liquid will not overflow out of the tube. Instead of it, the liquid will rise to the top of the tube.

57. Excess pressure in a drop of liquid of radius r is given by P = 2T/r. 58. Excess pressure in a soap bubble of radius r is given by P = 4T/r. 59. Excess pressure inside a soap bubble present in a liquid P = 2T/r, where r is radius and T is surface tension. 60. If the surface be curved in two directions and radii of the two curvatures be r1 and r2

respectively the total difference of pressure on the two sides of the surface will be

given by P= ⎥⎦

⎤⎢⎣

⎡+

21 r1

r1T .

61. Pressure difference across a surface film :

a) When free surface of the liquid is plane (fig a), the surface tension acts horizontally and its normal component is zero, thus no extra pressure is communicated to the inside or outside.

b) When free surface is convex, the forces due to surface tension acting on both sides of a line on the surface have components acting downwards which gives excess pressure inside the liquid.

c) Similarly when free surface is concave, the pressure inside the liquid is decreased. d) Thus there is always an excess pressure on the concave side.

62. In case of concave meniscus the pressure below the meniscus is lesser than above it by

⎟⎠

⎞⎜⎝

⎛rT2 .

63. In case of convex meniscus the pressure below the meniscus more than above it by ⎟⎠

⎞⎜⎝

⎛rT2 .

64. The spherical surface of the liquid in the tube is called meniscus. 65. If the adhesive force is large compared with cohesive force, the liquid has concave meniscus upwards.

e.g. water and glass tube.

r1r2

p0p0

rT2po −

rT2po +




Surface tension

4

66. If the adhesive force is less than cohesive force, the liquid has a convex meniscus. e.g. mercury and glass tube.

67. Shape of liquid meniscus in a capillary tube : For a liquid molecule at P, force of adhesion Fa acts at right

angles to the tube at the point P, force of cohesion Fc acts at an angle of 45° to the vertical. The resultant force on it will be the resultant of these two forces of adhesion and cohesion. a) When Fc= aF2 , i.e., the cohesive force is 2 times the

adhesive force the molecules of the liquid are neither raised nor lowered and the liquid surface remains flat or plane (fig a)

b) When Fc< aF2 , i.e., the cohesive force is less

than 2 times the adhesive force, the molecules of the liquid near the walls of the tube are raised up against the tube making the liquid surface concave upwards as in the case of water. (fig b)

c) When Fc> aF2 , i.e., the cohesive force is greater than 2 times the adhesive force, the liquid molecules near the walls of the tube are depressed making the surface convex upwards as in case of mercury. (fig c)

68. When a charge either positive or negative is given to a soap bubble, it expands due to repulsions among the charges.

69. Surface tension by capillary rise method.

T =θ

≈θ

+cos2rhdg

cos2)3/rh(rdg if h >> r

In the case of pure water, T =2

rhdg .

70. In capillary rise the force due to surface tension in upward direction is equal to the weight of liquid column mgcosrT2 =θπ .

71. If the radii of the two limbs of a U tube are r1 and r2, then the difference between the levels of a liquid poured in

it is 1 2

2T 1 1hdg r r

⎛ ⎞= −⎜ ⎟⎝ ⎠

(here r2 > r1), d is density and T is surface tension of the liquid.

72. If a vessel has a small hole of radius ‘r’ at its bottom then the maximum height of water that can be filled into it

so that it does not leak out through the hole is 2Thrdg

= .

73. Jurin’s law : According to Jurin’s law, inversely the height of the liquid (h) risen in capillary tube is proportional to the radius (r) rh = constant

74. A graph between h and r is a rectangular hyperbola. 75. If a liquid rises to a height ‘h’ in a capillary tube and the tube is inclined at an angle ‘ α ’ to the vertical, the

length of the liquid column inside the tube increases but the vertical height to which the liquid rises remains the same.

α

=cos

hL where L = length of the liquid column inside the tube.

76. If a capillary tube is dipped in water in a satellite, the water level will rise to the full length of the tube. 77. For the liquids of low surface tension wetting property is more. 78. Critical temperature : The temperature at which surface tension of the liquid becomes zero is known as

critical temperature. 79. In case of molten copper and molten cadmium T increases with increase of temperature. 80. Surface tension of liquid metals is very very high. 81. ST of a liquid is zero at its boiling point.

T T

(a) (b)(c)

(a) (b) (c)

PP

PFa

Fa

Fa

Fc

Fc Fc

FF

F




Surface tension

5

82. Over small ranges of temperature, the surface tension of a liquid decreases linearly with the rise of temperature.

83. T=To(1 − αt) where T is surface tension at t°C, To is the surface tension at 0°C and α is the coefficient of surface tension.

84. When two soap bubbles of radii a and b coalesce under isothermal condition, the resultant bubble has a radius

R such that R = 22 ba + . 85. If two soap bubbles of radii a and b coalesce (a > b), then the radius of curvature of the interface between the

two bubbles will be ⎟⎠

⎞⎜⎝

⎛− ba

ab .

86. A spherical soap bubble of radius r1 is formed inside another of radius r2. The radius of the single soap bubble which maintains the same pressure difference as inside the smaller and outside the larger soap bubble

is ⎟⎟⎠

⎞⎜⎜⎝

⎛+ 21

21rr

rr.

87. If a small drop of water is squeezed between two plates so that a thin layer of large area is formed, then the pressure inside the water layer is less than the pressure on the plates. The force pushing the two plates together is given by F = excess pressure × area of the layer.

F =dT2

× area

where d = thickness of layer.

88. If m is mass of water drop, 22TmF

d=

ρ (ρ - density of water).

89. When two soap bubbles of different sizes are in communication with each other, the air passes from the smaller one to the larger one and the larger one grows at the expense of the smaller one. i.e., size of smaller bubble decreases and that of larger bubble increases. This is because excess pressure inside the smaller bubble (smaller radius) is greater than that in the larger bubble (greater radius).

90. Energy required to raise a liquid in a capillary tube: When a capillary tube is depressed vertically into a liquid which wets the walls of the tube, there is a rise of the liquid inside the tube. The energy required to raise the liquid in the capillary tube is obtained from the surface energy of the air glass surface.




1

FLUID MECHANICS Synopsis :

FLUID STATICS 1. A substance which can flow is known as fluid. Hence the term includes both liquids and gases.

2. The study of fluids at rest is called fluid statics.

3. Solids are incapable of flow because the intermolecular forces are very strong. Hence solids possess a definite shape and volume.

4. The intermolecular forces are weak in liquids. Hence liquids do not possess a definite shape but take the shape of the container.

5. In the case of gases, the intermolecular forces are practically non-existent. Therefore gases possess neither a definite shape nor a definite volume.

6. Density of a homogeneous substance is defined as the ratio of its mass to its volume. In other words density is the mass per unit volume.

Mathematically, d=vm , where d=density, m=mass and v=volume

S.I unit of density is kgm–3.

7. Specific gravity of a material is defined as the ratio of its density to that of water at 4oC It is a mere number and has no units. It is also known as relative density.

8. Specific gravity of a substance can be said to be numerically equal to its density in grams/c.c.

9. If equal volumes of two liquids of densities d1 and d2 are mixed together, then the density d of the mixture is

d=2

dd 21 +

10. If equal masses of two liquids of densities d1 and d2 are mixed together, then the density ‘d’ of the resultant mixture is

d=21

21dd

dd2+

11. Pressure is defined as the ratio of the normal force acting on the area on which the force acts.

P =AF where P=pressure, F=normal component of force and A = area on which force acts.

S.I unit of pressure is pascal (Pa).

Pressure is a scalar quantity.

Fluid Pressure : 12. The pressure due to a liquid column of height ‘h’ and density ‘d’ is given by

P = hdg where g = acceleration due to gravity

This is called gauge pressure.

13. The pressure exerted by the atmospheric air at any point is equal to the weight of air contained in a column of unit cross sectional area and extending up to the top of the atmosphere. This is called atmospheric pressure. Often it is expressed interms of the height of an equivalent mercury column (in a barometer).




Fluid Mechanics

2

14. The total pressure P acting at the bottom of an open liquid column of height ‘h’ and density ‘d’ is given by P=Pa+hdg where Pa=atmospheric pressure

15. The total pressure P is called absolute pressure.

16. Absolute pressure=Gauge pressure + Atmospheric pressure

17. Atmospheric pressure=1.013×105 Pa =1.013×106 dynes/cm2

=76 cm of Hg =760 torr = 1.013 bar

18. If ‘h’ is the difference in heights of mercury in the two limbs of a manometer, then

the gauge pressure=hdg and

the total pressure = hdg + atmospheric pressure

19. In a U-tube or Hare’s apparatus, if h1 and h2 are the heights of water and liquid columns (in balancing

method) respectively, then the specific gravity of the liquid=2

1hh

.

20. A sphygmomanometer is a type of blood pressure gauge commonly used by physicians.

21. Pascal’s law : When ever pressure is applied on any part of a fluid contained in a vessel, it is transmitted undiminished and equally in all directions.

22. The Bramah’s press works on the principle of Pascal’s law. It is used to compress cotton bales, extract oil from seeds and drill holes in large metal sheets.

23. If in a hydraulic press (Bramah’s press), the area of the smaller and larger pistons are ‘a’ and ‘A’ and a force ‘f’ is applied on the smaller piston, then the force ‘F’ developed on the larger piston is given by F =

afA .

24. If we take ‘n’ containers each having the same base area and each containing the same liquid to the same height ‘h’, the pressure acting on the bases of each container is equal. This is known as hydrostatic paradox or Masson’s paradox.

25. The pressure acting on the walls of a container having liquid in it is given by hdg21 .

26. The pressure at a point in a liquid is same in all directions.

27. Archimede’s principle: When a body is immersed wholly or partially in a fluid at rest, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body.

28. The loss of weight (ΔW) of a solid when immersed in a liquid is given by ΔW=vdg where v=volume of the displaced liquid, d=density of the liquid and g=acceleration due to gravity.

29. From Archimedes’ principle

i) relative density of solid =

waterin solid of weightof loss

air in solid of weight

ii) relative density of a liquid =

waterin sinker of weightof lossliquid in sinker of weightof loss

30. Some applications of Archimede’s principle are

i) hot air (or helium) balloon




Fluid Mechanics

3

ii) determination of purity of precious metals like gold, silver etc. iii) submarines, etc.

31. The upward force is called the buoyant force or force of buoyancy.

32. Buoyant force depends on the volume of the displaced liquid and not on the volume of the body.

33. Buoyant force depends on the density of the liquid and not on the density of the body.

34. When a body is immersed in a fluid, when

i) if the weight of the body (W) is more than the upthrust (WI) i.e., W>WI, the body will sink.

ii) if the weight of the body (W) is equal to upthrust (WI) i.e., W=WI, the body will float, the whole if its volume being inside the liquid and

iii) if the weight of the body (W) is less than the upthrust (WI) i.e., W<WI, the body will float with a part of it being outside the liquid.

35. The buoyant force acts vertically upwards through the center of gravity of the displaced liquid.

36. The center of gravity of the displaced liquid is called center of buoyancy.

37. When a body floats the vertical line joining the center of buoyancy and the center of gravity of the body is called central line.

38. When a floating body is slightly disturbed, the point where the vertical line from the center of buoyancy intersects the central line is called meta center.

39. If the meta center lies above the center of gravity, the body will remain in stable equilibrium.

40. If the meta center lies below the center of gravity, the body will be in unstable equilibrium.

41. If the meta center coincides with the center of gravity, the body will be in neutral equilibrium.

42. When a solid of density ‘ρ’ floats in a liquid of density ‘d’, then the volume fraction of solid immersed in liquid is given by

Vi= dρ where Vi=volume fraction of the solid inside the liquid.

Vo=1–Vi where Vo=volume fraction of the solid outside the liquid.

43. When an ice block floating on water melts, the level of water remains the same.

44. If a floating piece of ice contains an air bubble, the level of water does not change when the ice metals.

45. If a floating ice block contains a piece of cork embedded inside, there is no change in the level of water when the ice melts.

46. A floating block of ice contains a piece of lead. The level of water decreases when the ice melts.

47. When a block of ice floating on a liquid denser than water melts, there is an increase in the level of the liquid.

48. When a block of ice floating on a liquid whose density is less than that of water melts, there is a decrease in the level of the liquid.

49. A man is sitting in a boat which is floating in a pond. If the man drinks some water from the pond, the level of the water remains the same.

50. A boat carrying a number of stones is floating in a water tank. If the stones are unloaded into the water, the water level in the tank decreases.

51. A hydrometer is used to measure the density of a liquid directly.

52. A lactometer is used to determine the purity of milk.

FLUID DYNAMICS :




Fluid Mechanics

4

53. The study of fluids in motion is called fluid dynamics.

54. Fluids flow from one place to other because of pressure differences.

55. Streamline flow : The flow of a liquid is said to be streamlined or orderly if the particles of the liquid move along fixed paths known as streamlines and velocity of the particles passing one after the other through a given point on a streamline remains unchanged in magnitude as well as direction at that point.

56. Streamline in flow is also called laminar flow or steady flow.

57. A streamline in general follows the shape of the tube through which the liquid flows. Thus it may be straight or curved.

58. Steady flow or streamline flow :

i) Streamline flow is that flow in which every particle flows along the path of its preceding particle.

ii) The path taken by a particle in a flowing fluid is called its line of flow.

iii) The tangent at any point on the line of flow gives the direction of motion of that particle at that instant.

iv) Streamlines may be straight or curved.

v) Two streamlines cannot intersect each other.

vi) There is no radial flow in the tube.

vii) The mass of fluid entering the tube in unit time is equal to the mass of fluid leaving the tube in unit time.

viii) Pressure over any cross-section is constant.

ix) The velocity at any point of the liquid remains the same throughout the time for which the flow is maintained.

x) The energy supplied to the fluid for maintaining its flow is mainly used in overcoming the viscous drag between different layers.

59. Tube of flow : A bundle of streamlines having the same velocity of fluid elements over any cross-section perpendicular to the direction of flow is known as tube of flow.

60. Turbulent flow : If the velocity of a point of a fluid varies in time it is called turbulent flow. In turbulent flow the liquid flows in a disorderly fashion growing eddies and vortices.

61. Rate of flow : The rate of flow of a liquid is the volume of a liquid that flows across any cross-section in unit time and is given by

Q=time

volume m3/s

Q=AV

where A is the area of cross-section of the tube V is velocity of the liquid

62. Principle of continuity : In case of steady flow of incompressible and non viscous fluid through a tube of non-uniform cross-section, the product of the area of cross-section and the velocity of the flow is same at every point in the tube.

A × V =constant

A1V1=A2V2 It is called equation of continuity

63. Equation of continuity represents the law of conservation of mass in case of moving fluids.




Fluid Mechanics

5

64. Total energy of a liquid : The total energy at any point in a flowing liquid is of three kinds. (a) potential energy, (b) kinetic energy, (c) pressure energy.

a) Potential energy : The energy possessed by a liquid by virtue of its height above some arbitrary level is called potential energy.

Potential energy of mass m of the liquid=mgh

Potential energy per unit mass m=gh

b) Kinetic energy : The energy possessed by a liquid by virtue of its motion is called kinetic energy.

Kinetic energy per unit mass= 2v21

c) Pressure energy : The energy possessed by a liquid by virtue of the pressure acting on it is called pressure energy.

Pressure energy in volume dw = PdV

Pressure energy per unit mass =ρP

Where ρ is density of liquid.

65. Bernoulli’s theorem : If an ideal fluid (non viscous, incompressible) is in streamline flow in a tube of non uniform cross-section the sum of the pressure energy, kinetic energy and potential energy at any point per unit mass or per unit volume is constant.

gh2

VP 2++

ρ=constant

hg2

Vg

P 2++

ρ=constant

here g

Pρ

=pressure head

g2V 2

=velocity head

h=gravitational head 2V

21ghP ρ+ρ+ =constant

P → is called static pressure 2V

21

ρ → dynamic pressure

Applications of Bernoulli’s theorem : 66. Torricelli’s theorem : The velocity of efflux of a liquid through an

orifice (small hole) of a vessel is equal to the velocity acquired by a freely falling body from a height which is equal to that of liquid level from the orifice.

gh2V =

67. Time taken by the efflux liquid to reach the ground is given by

g)hH(2t −

=

68. Horizontal range of liquid is given by

h

gh2V =

hH

H-h

x




Fluid Mechanics

6

x= )hH(h2g

h)-2(H x gh2 −=

69. Horizontal range is maximum when orifice is at the middle of liquid level and bottom.

2Hh = ; xmax=H

The horizontal range (x) of liquid coming out of the holes at depths h or (H−h) from its free surface is the same.

70. Time taken for the level to fall from H1 to H2

[ ]21o

HHg2

AAt −=

where Ao is area of orifice, A is area of cross-section of container.

71. If the hole is at the bottom of the tank, time t taken to emptied the tank

gH2

AAt

o=

Dynamic Lift : 72. The upward lift experienced by a body in motion when immersed in a fluid is called dynamic lift. 73. The dynamic lift experienced by a body when it is in motion in air is called aerodynamic lift. 74. Lift on an Aircraft wing : Aeroplane wings are so designed (i.e., streamlined) the velocity of air flow

above the wing is higher than the velocity of air flow under the wing. This difference of air speeds, in accordance with Bernoulli’s principle, creates pressure difference, due to which an upward force called “dynamic lift” acts on the plane.

Dynamic lift=pressure difference x area of the wing

=(P2−P1) ×A

= [ ]xAVV21 2

122 −ρ

75. Magnus effect : When spinning ball is thrown, it deviates from its usual path in flight. This effect is called Magnus effect and plays an important role in tennis, cricket and soccer etc.

76. If the ball is moving from left to right and also spinning about a horizontal axis perpendicular to the direction of motion, then relative to the ball air will be moving from right to left. The resultant velocity of air above the ball will be v+rω while below it v−rω. So in accordance with Bernoulli’s principle pressure above the ball will be less than below it. Due to this difference of pressure an upward force will act on the ball and hence the ball will deviate from its usual path.

77. If ball is thrown with back spin, the pitch will curve less sharply prolonging the flight.

78. If the spin is clockwise, the pitch will curve more sharply shortening the flight.

79. When wind blows over a house with high speed pressure on the roof will be less that of inside the house and so the roof is lifted and blown away by the wind.

80. Atomizer (sprayer); paint gun and Bunsen burner, pilot tube, carburetor, filter pump (aspriator) work n basis of Bernoulli’s principle.

V - largeP - small

V - smallP - large

V

V

Vrω

rω




Fluid Mechanics

7

81. Venturimeter : Venturimeter is used to measure flow speed and rate of flow in a pipe.

Velocity of flow V1=

1AA

gh22

2

1 −⎟⎟⎠

⎞⎜⎜⎝

⎛

Rate of flow Q=

1AA

gh2A2

2

11

−⎟⎟⎠

⎞⎜⎜⎝

⎛

VISCOSITY : 82. When water flows in a uniform horizontal tube there is fall in pressure along the

tube in the direction of flow. The reason for this fall in pressure is that force is required to maintain motion against friction. This friction is nothing but viscous forces of the liquid.

83. When liquid flows, it can be assumed to be composed of different horizontal layers. The upper layer tries to drag forward due to cohesive forces between the molecules of adjacent layers) the lower layer increasing the velocity of lower layer whereas the lower layer tries to drag backward the upper layer decreasing the upper layer’s velocity. So there exists a velocity gradient perpendicular to the plane of liquid.

84. The velocity of the layers goes on decreasing as the depth increases and finally the deepest layer in contact with the horizontal surface is at rest.

85. Viscosity : The property of a fluid which opposes the relative motion between different layers is called viscosity.

86. Viscosity is the internal resistance or friction exhibited between the layers of a fluid.

87. Viscous force : (Newton’s formula) The viscous force (F) acting tangentially on a layer of a fluid is

directly proportional to the (i) surface area A of the layer, (ii) velocity gradient ⎟⎠

⎞⎜⎝

⎛dxdV which is

perpendicular to the direction of flow.

F=dxdVAη−

This law is called Newton’s law of viscous flow in streamline motion. The constant of proportionality η is called the coefficient of viscosity.

88. Coefficient of viscosity “η” : The viscous force acting tangentially on unit area of the liquid when there is a unit velocity gradient in the direction perpendicular to the flow is called the coefficient of viscosity. It is also called coefficient of dynamic viscosity.

v+dv

vdx

h

A1

V1

P1

P2

V2A2

0




Fluid Mechanics

8

The S.I unit of coefficient of viscosity is 2mSN − or Pa-s or decapoise.

The CGS unit of η is 2cm

sdyne − or poise.

1 Pa-s=10 poise

The dimensional formula of η is ML−1T−1

For ideal liquid (non viscous, incompressible) η=0.

89. Coefficient of kinematic viscosity “μ” : The ratio of the coefficient of dynamic viscosity to the density

of liquid ⎟⎟⎠

⎞⎜⎜⎝

⎛ρη is called coefficient of kinematic viscosity. It is often used by the mathematicians and

engineers in fluid dynamics.

a) Dimensional formula L2T−1

b) The S.I unit of ‘μ’ is m2s−1

c) The unit in CGS system is stoke and 1 stoke=10−4 m2s−1

90. Effect of temperature : In case of liquids, coefficient of viscosity decreases with increase of temperature as the cohesive forces decrease with increase of temperature.

91. In the case of gases, coefficient of viscosity increases with increase of temperature because the change in momentum of molecules increases with increase of temperature.

92. Effect of pressure : a) For liquids the value of η increases with increase of pressure.

Viscosity of water decreases with increase of pressure.

b) For gases, the value of η increases with increase of pressure at low pressure. But a high pressure η is independent of pressure.

93. The quality of fountain pen ink depends largely on its viscosity. The normal circulation of blood through our arteries and veins depends on the viscosity of blood. The shape of ship or the car is streamlined to minimize the effects of force of viscosity.

94. The cloud particles come down slowly and appear floating in space due to the viscosity of air.

95. The cause of viscosity in liquids is the cohesive forces among molecules whereas in gases it is due to diffusion.

96. Poiseuille’s equation : a) When a liquid flows through a capillary tube with streamline motion, the

velocity of the liquid layer along the axis of the tube is maximum and gradually decreases as we move towards the walls where it becomes zero.

b) The volume of liquid flowing per second (rate of flow Q) through the tube depends on the following factors

i) the viscosity of liquid (η)

ii) the radius of the pipe

Temperature

visc

osity

l

Tube




Fluid Mechanics

9

iii) the pressure gradient ⎟⎠

⎞⎜⎝

⎛lP where P is pressure difference across the length l of the capillary tube.

Q=l

Pr8

4

ηπ

This formula is known as Poiseuille’s formula.

This is applicable under the following conditions.

1) The flow must be steady and laminar.

2) The liquid in contact with the walls of the capillary tube must be at rest.

3) The pressure at any cross-section of the capillary tube must be same.

97. When a liquid is flowing through a tube, the velocity of the flow of a liquid at a distance x from the axis of

the tube is given by V= [ ]22 xrl4

P−

η.

The velocity distribution curve of the advancing liquid in a tube is a parabola.

98. Ohm’s law in fluid dynamics (Poiseuilli’s equation)

a) The Poiseuilli’s formula Q=l

Pr8

4

ηπ can be written as Q=

RP similar to Ohm’s law i=V/R

where R=4rl8

π

η is called fluid resistance.

b) when two capillaries are joined in series across constant pressure difference P the fluid resistance R=R1+R2.

R = 42

241

1

r

l8

r

l8

π

η+

π

η and Q =

21 RRP+

c) When two capillaries are joined in series, the rate of flow is the same but the pressure difference across the two tubes is different.

The total pressure difference P=P1+P2

d) If two capillaries are joined in parallel, the pressure difference across the two tubes is the same but the volume of fluid flowing through the two tubes is different. The total volume of the fluid flowing through the tubes is one second is Q = Q1 + Q2.

When two capillaries are joined in parallel across a constant pressure difference P, then fluid resistance R is given by

21 R1

R1

R1

+= where R1= 141

8 lrη

π and R2= 2

42

8 lrη

π

R1

R2

P




Fluid Mechanics

10

The volume of liquid flowing through first capillary Q1=1R

P and the volume of liquid flowing through

second capillary Q2=2R

P .

99. Motion of objects through viscous medium :

a) Stoke’s law : According to this law the viscous force acting on a freely falling smooth, spherical body is directly proportional to

i) coefficient of viscosity of fluid (η)

ii) radius of the spherical body (r)

iii) velocity of the body (V)

Fη=6πηrV. This is called Stoke’s formula.

This is true for small values of V in a large expansion of fluid.

b) when a sphere falls vertically through a viscous fluid it is subjected to the following forces

i) Its weight W acts downward

ii) The viscous resistance Fη acts upwards

iii) The Archimedes upthrust of buoyancy W1 acts upwards.

c) Terminal velocity : When a body is dropped in a fluid at one stage the resultant force acting on it will be zero and it travels with uniform velocity and this is called terminal velocity.

If Vt represents the terminal velocity and F the maximum viscous resistance, then F=W−W1

6πηrVt = g)d(r34 3 ρ−π

Vt = η

ρ− )d(gr92 2

Where d=density of the body, ρ=density of the liquid

d) Terminal velocity is directly proportional to square of the radius of the sphere, difference of densities and inversely proportional to the coefficient of viscosity of the fluid.

100. Critical velocity : The minimum velocity at which a liquid flow changes from streamline to turbulent flow is called critical velocity.

a) The critical velocity (Vc) of the fluid depends on i) viscosity (η), ii) the diameter of tube (D) through which the fluid is flowing, iii) density of the fluid ρ.

Vc=ρ

ηD

R

where R is a constant of proportionality and is called Reynold’s number. It has no dimensions.

b) For a laminar flow, the value of R lies between 0 and 1000.

c) For values of R>2000, the flow will be turbulent.

d) For values of R between 1000 and 2000 the flow is unstable and switches from laminar flow to turbulent and vice versa.

r

W

W1

Fη

Time

V

Vt




Fluid Mechanics

11

e) The orderly flow or streamline flow is produced when we use narrow tubes and liquids of low density and high viscosity.

f) Reynold’s number is the ratio of force of inertia to force of viscosity.

101. In laminar flow, viscous force is proportional to the velocity.

102. In turbulent flow, viscous force is proportional to the square of velocity.

103. The bodies of air planes, torpedoes, ships, bombs and automobiles are streamlined to avoid wastage of energy in movement through the fluid. The man riding a race bicycle bends his body forward in order to have a streamlined shape.

104. A small value of Reynold’s number means that the viscous forces predominate whereas the larger values of it indicate that the forces of inertia predominate.

105. In streamline flow, rate of flow is proportional to pressure difference. In turbulent flow; rate of flow is proportional to P approximately.

Pressure difference

Q

P

Rat

e of

flow

Stre

am lin

e flo

w

Turbulent flow




1

THERMAL EXPANSION OF MATERIALS EXPANSION OF SOLIDS

Synopsis : TEMPERATURE 1. The invention of thermometer and development of the concept of temperature mark the beginnings of

the science of thermodynamics. 2. The temperature of a body is a state which determines the direction of flow of heat or the degree of

hotness of a body. 3. Heat is the cause and temperature is the effect. 4. A body at a higher temperature need not necessarily contain more heat. 5. Two bodies at the same temperature may contain different amounts of heat. 6. Two bodies containing the same amount of heat may be at different temperatures. 7. The direction of flow of heat from a body does not depend on its heat content but depends on its

temperature. 8. In principle, any system whose properties change the temperature can be used as a thermometer. 9. There are four scales of temperature. They are Celsius scale, Fahrenheit scale, Reaumer scale and

Kelvin (or Absolute or thermodynamic temperature) scale. 10. The most fundamental scale of temperature called Kelvin scale is based on the laws of thermodynamics. 11. The melting point of ice at standard atmospheric pressure is taken as the lower fixed point. 12. The boiling point of water at standard pressure is taken as the upper fixed point. The upper fixed point is

determined by using Hypsometer. 13. The distance between the lower and upper fixed points is divided into definite equal divisions. 14. Different scales of temperature. 15. The reading on one scale can be readily converted into corresponding one or the other by the relation

80R

18032F

100C

100273K

=−

==−

16. If in a certain arbitrary scale of temperature, p° is the lower fixed point and q° is the upper fixed point, any temperature x in this scale can be converted to Celsius or Fahrenheit scale by using the formula

18032F

pqpx

100C −

=−−

=

17. The differences of temperature on different scales can be converted using the formula

80R

180F

100C

100K Δ

=Δ

=Δ

=Δ

18. Different types of thermometers and their ranges :

Clinical thermometer –95°F to 110°F Mercury t4hermometer −38°C to 350°C Alcohol thermometer −110°C to 78°C Hydrogen gas thermometer −260°C to 1600°C Platinum resistance thermometer −200°C to 1200°C Pyrometer very high temperatures

19. Advantages of mercury as a thermometric fluid.

i) Mercury remains as a liquid over a wide range of temperature

ii) Pure mercury can be readily and easily obtained.

iii) Its vapour pressure at ordinary temperature is negligible.




Thermal Expansion of materials

2

iv) It has high conductivity and low thermal capacity. So it quickly attains the temperature of the body by taking a negligibly small quantity of heat.

v) It does not wet glass and is opaque.

20. Of all the thermometers, gas thermometers are more sensitive because of their high volume expansion. They have the same scale for all gases.

21. Using a constant volume hydrogen thermometer, temperatures ranging from −200°C to 1100°C can be measured. It is generally used to calibrate other thermometers.

22. To have more surface contact with heat, the thermometric bulb will be in the shape of a cylinder.

23. To determine the maximum and minimum temperatures attained during a day at a place, Six’s maximum and minimum thermometer is used.

24. If X is any thermometric property such as pressure or volume or resistance which has values at 0°, 100°

and t° on any scale as X0, X100 and Xt, then t = ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−

0100

0tXX

XX100.

25. Temperatures on the Celsius scale denoted by the symbol °C (read “degrees Celsius”). Temperature changes and temperature differences on the Celsius scale are expressed in C° (read “Celsius degrees”). For eg: 20°C is a temperature and 20 C° is a temperature difference. In general, all substances whether they are in the form of solids, liquids or gases expand on heating except water between 0°C and 4°C and some aqueous solutions. This is known as thermal expansion.

EXPANSION OF SOLIDS : 26. Solids expand on heating due to increased atomic spacing.

27. A solid can be considered as periodic arrangement of atoms in the form of lattice.

28. At any particular temperature, the atoms are in a specific state of vibration about a fixed point called as equilibrium position in the lattice.

29. As the temperature increases, the amplitude of vibration of the atoms increases.

30. If the lattice vibrations are purely harmonic the potential energy curve is a symmetric parabola and there is not thermal expansion.

31. If the lattice vibrations are anharmonic, the potential energy of an oscillator is an asymmetric function of its position and thermal expansion is observed.

32. Coefficient of linear expansion (α) : The ratio of increase in length per one degree rise in temperature to its original length is called coefficient of linear expansion.

α =)tt(l

ll

121

12−

−

Unit of α is 1oC

− or K−1

33. The change in length is calculated using ΔL=L α Δt

OInteratomic distance

Inte

rato

mic

Pot

entia

l Ene

rgy

ro

OInteratomic distance

Inte

rato

mic

Pot

entia

l Ene

rgy

ro

0 Co 32 Fo 0 Ro 273.15 K

C F R K

100 Co 212 Fo 80 Ro 373.15 K

Celsius (C)

Fahrenheit(F)

Reaumer (R)

Kelvin(K)





3

34. Coefficient of area or superficial expansion (β) : The increase in area per unit area per one degree rise in temperature is called coefficient of areal expansion.

β =)tt(a

aa

121

12−

−

Unit of β is 1oC

− or K−1

35. The change in area is calculated using formula Δa=a β Δt.

36. The coefficient of volume or cubical expansion (γ) is the increase in volume per unit volume per degree rise in temperature.

γ =)tt(V

VV

121

12−

−

Unit of γ is 1oC

− or K−1

37. The change in volume is calculated using formula ΔV=V γ Δt.

38. For all isotropic substances (solids which expand in the same ratio in all directions) α : β : γ = 1:2:3 or γ=3α; β=2α; γ=α+β.

39. If αx, αy and αz represent the coefficients of linear expansion for an isotropic solids (solids which expand differently in different directions) in x, y and z directions respectively, then γ=αx+αy+αz and the average

coefficient of linear expansion α=3

zyx α+α+α.

40. The numerical value of coefficient of linear expansion of a solid depends on the nature of the material and the scale of temperature used.

41. The numerical value of coefficient of linear expansion of a solid is independent of physical dimensions of the body and also on the unit of length chosen.

42. The increase in length or linear expansion of a rod depends on nature of material, initial length of rod and rise of temperature.

43. The numerical value of α or β or γ in the units of per °C is 9/5 times its numerical value in the units of per °F.

44. α per °F= α.95 per °C.

45. α per °R= α.45 per °C.

46. Variation of density with temperature : The density of a solid decreases with increase of temperature.

t1d

d ot γ+

= or )t1(dd ot γ−≈ where do is density at 0°C.

47. If R1 and R2 are the radii of a disc or a plate at t1°C and t2°C respectively then R2=R1(1+α(t2−t1)).

48. A metal scale is calibrated at a particular temperature does not give the correct measurement at any other temperature.

a) When scale expands correction to be made Δl=L α Δt, correct reading=L+Δl

b) When scale contracts correction to be made Δl=L α Δt, correct reading=L−Δl. L=measured value.

c) Lmeasured=Ltrue[1−α(Δt)]

49. When a metal rod is heated or cooled and is not allowed to expand or contract thermal stress is developed.





4

Thermal force F=YA α (t2−t1)

Thermal force is independent of length of rod.

Thermal stress σ=Y α (t2−t1)

Y=Young’s modulus

α=coefficient of linear expansion

t2−t1=difference of temperature

A=area of cross-section of the metal rod.

For same thermal stress in two different rods heated through the same rise in temperature, Y1α1=Y2α2.

50. Barometer with brass scale :

Relation between faulty and actual barometric height is given by h2=h1[1+(αs−YHg)(t2−t1)]

h1=height of barometer at t1°C where the scale is marked

h2=height of barometer at t2°C where the measurement is made

γHg=real coefficient of expansion of mercury

αs=coefficient of linear expansion of scale

51. Pendulum clocks lose or gain time as the length increases or decreases respectively.

The fractional change=2

tTT Δα

=Δ .

The loss or gain per day=2

tΔα x86400 seconds.

52. The condition required for two rods of different materials to have the difference between the lengths always constant is L1α1=L2α2.

53. A hole in a metal plate expands on heating just like a solid plate of the same size.

54. A cavity of a solid object expands on heating just like a solid object of the same volume.

55. If a hollow pipe and a solid rod of same dimensions made of same material are heated to the same rise in temperature, both expand equally.

56. If a thin rod and a thick rod of same length and material are heated to same rise in temperature, both expand equally.

57. If a thin rod and a thick rod of same length and material are heated by equal quantities of heat, thin rod expands more than thick rod.

58. A rectangular metal plate contains a circular hole. If it is heated, the size of the hole increases and the shape of the hole remains circular.

59. A metal plate contains two holes at a certain distance apart from each other. If the plate is heated, the distance between the centers of the holes increases.

60. The change in the volume of a body, when its temperature is raised, does not depend on the cavities inside the body.

Applications of linear expansion : 61. Platinum (or monel) is used to seal inside glass because both have nearly equal coefficients of linear

expansion.

62. Iron or steel is used for reinforcement in concrete because both have nearly equal coefficients of expansion.





5

63. Pyrex glass has low α. Hence combustion tubes and test tubes for hating purpose are made out of it.

64. Invar steel (steel+nickel) has very low α. So it is used in making pendulum clocks, balancing wheels and measuring tapes. (Composition of invar steel is 64% steel and 36% nickel).

65. Metal pipes that carry steam are provided with bends to allow for expansion.

66. Telephone wires held tightly between the poles snap in winter due to induced tensile stress as a result of prevented contraction.

67. Thick glass tumbler cracks when hot liquid is poured into it because of unequal expansion.

68. Hot chimney cracks when a drop of water falls on it because of unequal contraction.

69. A brass disc snuggly fits in a hole in a steel plate. To loosen the disc from the hole, the system should be cooled.

70. To remove a tight metal cap of a glass bottle, it should be warmed.

71. While laying railway tracks, small gaps are left between adjacent rails to allow for free expansion without affecting the track during summer. Gap to be left (Δl)=αlΔt=expansion of each rail.

72. Concrete roads are laid in sections and expansion channels are provided between them.

73. Thermostat is a device which maintains a steady temperature.

74. Thermostats are used in refrigerators, automatic irons and incubators.

75. Thermostat is a bimetallic strip made of iron and brass. The principle involved is different materials will have different coefficients of linear expansion.

76. A bimetallic strip is used in dial-type thermometer.

77. If an iron ring with a saw-cut is heated, the width of the gap increases.

78. Barometric scale which expands or contracts measures wrong pressure. On expansion the true pressure is less than measured pressure.

Ptrue=Pmeasured[1−(γ−α)t]

where γ=coefficient of cubical expansion of mercury α=coefficient of linear expansion of the material used in making the scale t=rise of the temperature

79. When a straight bimetallic strip is heated it bends in such a way that the more expansive metal lies on the outer side. If d is the thickness of the each strip in a bimetallic strip, then the radius of the compound

strip is given by R=t)(

d

12 Δα−α.

EXPANSION OF LIQUIDS

20. Liquids expand on heating except water between 0°C and 4°C.

21. The expansion of liquids is greater than that of solids (about 10 times).

22. Liquids do not possess any definite shape and require a container to hold them. Hence only cubical expansion is considered.

23. Since heat effects both the liquid and the container the real expansion of a liquid cannot be detected directly.

24. For liquids there are two types of cubical expansion

i) coefficient of apparent expansion (γa)





6

ii) coefficient of real or absolute expansion (γr)

25. Coefficient of apparent expansion of a liquid is the ratio of the apparent increase in volume per 1°C rise of temperature to its initial volume.

etemperatur in rise x volume originalvolume in increase apparent

a =γ

The unit of γa is °C−1.

26. Coefficient of real expansion is the ratio between real increase in volume per 1°C rise of temperature and the original volume of the liquid.

etemperatur in rise x volume originalvolume in increase real

r =γ

)tt(VVV

121

12r −

−=γ

The unit of γr is °C−1.

27. γr = γa + γvessel = γa+3α.

28. If γv=+ve and γr<γv, γa=−ve, the level decreases continuously when heated.

29. If γv=+ve and γr=γv; γa=0, the level will not change when heated.

30. If γv=+ve and γr>γv; γa=+ve, the level first falls and then rise when heated.

31. If γv=0; γr=γa, the level will increase continuously when heated.

32. If γv=−ve, γa>γr, the level will increase continuously when heated.

33. The real expansion of a liquid does not depend upon the temperature of the container.

34. The apparent expansion of liquid depends on a) initial volume or liquid, b) rise in temperature c) nature of liquid and d) nature of container.

35. γap is determined using specific gravity bottle or pyknometer or weight thermometer.

etemperatur of rise x remaining massexpelled mass

a =γ

etemperatur of rise x liquid remaining of weightexpelled liquid of weight

a =γ

)tt)(W(WWW

1213

32a −−

−=γ

36. Sinker’s method C/tmtm

mm o

1122

21app −

−=γ

m1=loss of weight of body in liquid at t1°C m2=loss of weight of body in liquid at t2°C

37. To keep the volume of empty space in a vessel (volume vg) constant at all temperatures by pouring certain amount of a liquid of volume vl, the condition is vlγl=vgγg where γl=coefficient of cubical expansion of liquid and γg=coefficient of cubical expansion of vessel.

38. The fraction of the volume of a flask that must be filled with mercury so that the volume of the empty space left may be the same at all temperatures is 1/7.

39. The density of a liquid usually decreases when heated. If d1 and d2 are the densities of a liquid at 0°C and t°C respectively, then





7

t1d

dr

ot γ+

= ; dt=do(1−γrt);

Accurate formula Approximate formula

C/tdtd

dd o

1221

21r −

−=γ

ANOMALOUS EXPANSION OF WATER : 40. When water at 0°C is heated, its volume decreases upto 4°C and from 4°C its

volume increases with the increase of temperature. This peculiar behaviour of water is called anomalous expansion of water. Due to the formation of more number of hydrogen bonds, water has anomalous expansion.

41. As the temperature increases from 0°C to 4°C, the density increases and as the temperature further increases the density decreases. Hence water has maximum density at 4°C.

42. Specific volume is the volume occupied by unit mass. It is the reciprocal of density. As the temperature increases from 0°C to 4°C, the specific volume decreases and as the temperature further increases, the specific volume increases.

43. Hope’s apparatus is used to demonstrate that water has maximum density at 4°C.

44. Rubber shows anomalous expansion like water.

45. Dilatometer is used to prove anomalous expansion of water.

46. Aquatic animals are surviving in cold countries due to the anomalous expansion of water.

47. During winter, in cold countries, even if the temperature falls far below 0°C, the water in the frozen lakes or seas at the bottom remains at 4°C.

48. When water freezes, it expands and consequently water pipes burst in winter.

49. When water at 4°C is filled to the brim of a beaker, then it over flows when it is either cooled or heated.

50. A beaker contains water at 4°C and a piece of ice is floating on it. When the ice melts completely, the level of water increases.

51. When a solid is immersed in a liquid (which does not show anomalous expansion) its apparent weight increases with the increase of temperature.

52. If W is the weight of a sinker in water at 0°C and W1 is weight in water at 4°C, then W1<<W.

53. As the temperature of water is increased from 0°C to 4°C, the apparent weight of a body decreases. At 4°C the apparent weight is minimum. On further heating the apparent weight increases.

54. Water has positive coefficient of expansion above 4°C and negative coefficient below 4°C.

55. At 4°C the coefficient of expansion of water is zero.

56. A wooden block is floating in water at 0°C. When the temperature of water is increased, the volume of the block below water surface decreases upto 4°C and beyond 4°C it increases.

57. In a mercury thermometer, the coefficient of apparent expansion of mercury can be determined by

tvlR2

a Δπ

=γ where l=length of the stem, v=initial volume of mercury in the bulb and Δt=rise in

temperature.





8

DETERMINATION OF γreal OF A LIQUID : 58. Specific gravity bottle method vesselappreal γ+γ=γ .

59. Dulong Petit method (U-tube method)

C/thhh o

0

0treal

−=γ (or) γreal =

1221

12

ththhh

−−

.

ht=height of liquid column in limb at t°C h0=height of liquid column in limb at 0°C

60. Regnault’s apparatus

C/]t)hh(H[

hh o

12

12real −−

−=γ

H=height of liquid in wide tubes h2−h1=difference in heights of liquid columns at t°C and 0°C in U-tube

61. The corrected height of a barometer is given by the relation H=ht[1−(γr−α)t] where H=height at 0°C; ht=height at t°C; γr=coefficient of real expansion of the liquid and α=coefficient of linear expansion of the material of the scale.

EXPANSION OF GASES 1. Pressure, volume and temperature are the three measurable properties of a gas. Change in one of

these factors results in a change in the other two factors.

2. Pressure of a gas is measured by manometer Bourden gauge for high pressures and Mcleod gauge for low pressures. These work on Boyle’s law.

3. Volume of a gas is measured by a gas burette or on Eudiometer. 4. A gas has neither unique shape nor unique volume. The gas completely occupies the vessel in which it

is placed.

Coefficients of expansion of gas : 5. When a given mass of gas is heated under constant pressure, its volume increases with increase in

temperature.

6. When a given mass of gas is heated under constant volume its pressure increases with increase in temperature. Hence gases have two types of coefficients of expansions.

i) volume expansion coefficient

ii) pressure expansion of coefficient

7. Volume coefficient of a gas (α) : At constant pressure the ratio of increase of volume per 1°C rise in temperature to its original volume at 0°C is called volume coefficient of a gas.

1221

12

0

0ttVtV

VV or

tVVV

−−

=α−

=α

Vt=V0(1+αt)

The unit of α is °C−1 or K−1.

8. Pressure coefficient of a gas (β) : At constant volume the ratio of increase of pressure per 1°C rise in temperature to its original pressure at 0°C is called pressure coefficient of gas. Unit is °C−1 or K−1.

V

t Co-273.15 Co

P

t Co-273.15 Co





9

1221

12

0

0ttPtP

PP or

tPPP

−−

=β−

=β

Pt=P0(1+βt)

9. Regnault’s apparatus is used to determine the volume coefficient of a gas.

10. Jolly’s bulb apparatus is used to determine the pressure coefficient of a gas.

11. Volume coefficient and pressure coefficient of a gas are equal and each equal to C/273

1° or 0.0036/°C

for all gases.

12. P-t graph or V-t graph is straight line intersecting the temperature axis at −273.15°C. This temperature is called absolute zero. (0 K)

13. Absolute zero is the temperature at which the volume of a given mass of a gas at constant pressure or the pressure of the same gas at constant volume becomes zero.

14. The lowest temperature attainable is −273.15°C or 0 K.

15. The scale of temperature on which the zero corresponds to −273°C and each degree is equal to the Celsius degree is called the absolute scale of temperature or thermodynamic scale of temperature.

T K = t+273.15°C

There is no negative temperature on Kelvin scale.

16. Boyle’s law : At constant temperature, the pressure of a given mass of a gas is inversely proportional to

its volume. P V1 α or PV = K (n, T are constant) or P1V1 = P2V2. In PV = K, the value of K depends on

the mass and temperature of the gas and the system of units.

17. Boyle’s law can also be defined as follows. At constant temperature, the pressure of a given mass of gas is directly proportional to its density.

P2

2

1

1dP

dP or K

dP or d ==α .

18. P-V graph at a constant temperature (isothermal) is a rectangular hyperbola.

19. PV-V graph is a straight line parallel to volume axis.

20. P–V1 graph is a straight line passing through the origin.

21. Many gases obey Boyle’s law only at high temperatures and low pressures.

22. When a Quill tube is kept vertical with the open end upwards, the pressure exerted by gas column is (H+h) where H is atmospheric pressure and h is the length of mercury pellet.

23. When a Quill tube is kept horizontal, the pressure exerted by gas column is equal to atmospheric pressure.

24. When a Quill tube is kept vertical with the open end downwards, the pressure exerted by the gas column is (H–h).





10

25. When a Quill tube is kept inclined to the vertical at an angle θ and open end is upwards, then the pressure exerted by gas column is (H+hcos θ ).

26. Charles’ law : At constant pressure, the volume of a given mass of gas increases by 1/273th of its original volume at 0°C for every 1°C rise in temperature. (or) At constant pressure, the volume of a given

mass of gas is directly proportional to the absolute scale of temperature. V KTV or T =α (at constant P)

27. V-T graph is a straight line passing through the origin.

28. V-t (in °C) graph is a straight line which when produced meets the temperature axis at –273.15°C or 0 K.

29. The pressure of a given mass of gas at constant volume increases by 1/273th of its original pressure at 0°C for every 1°C rise in temperature. (or) At constant volume, the pressure of a given mass of gas is directly proportional to absolute scale of temperature. This is also known as

Gay Lussac’s law. P KTP or T =α (at constant V)

Gas Equation :

30. Combining Boyle’s law and Charle’s law, the resulting expression is an equation of state for ideal gas.

31. For unit mass of a gas (1 gram or 1 kg) PV=rt is called ‘Gas Equation’ PV=mRT (for m grams)

32. “r” is called gas constant (or) specific gas constant.

33. The value of “r” depends on nature and mass of the gas.

34. S.I. unit of “r” is JKg−1K−1. Dimensional formula for “r” is LT−2θ−1.

35. For one mole of a gas PV=RT is called universal (or) ideal (or) perfect gas equation.

36. The value of R is same for all gases irrespective of their nature.

37. If M is gram molecular mass of the gas, then r = R/m.

PV = nRTPVRTMm

=⇒

where n = no. of moles of gas.

38. S.I. unit of R is J mole−1K−1.

Values of R=8.314 Jg mole−1K−1

R = 8314 J kg mole−1K−1

R = 0.0821 litre atmosphere mole−1K−1

R = 8.314x107 ergs mole−1K−1

R = 1.987 cal mole−1K−1

Significance of R:

39. The value of R gives the work done by one mole of any gas when it is heated under constant pressure through one degree Kelvin.

40. The value of R does not depend on the mass of gas or its chemical formula.





11

41. The fact that R is a constant for all gases is consistent with Avagadro’s hypothesis that “equal volumes of all gases under same conditions of temperature and pressure contain equal number of molecules”.

42. The value of universal gas constant per molecule is 1.38x10−23 J mol−1K−1

R = N0K, where K = Boltzmann’s constant,N0 = Avagadro’s number

43. The gas equation in terms of density dTP =constant. Where d=density of ideal gas.

44. When pressure and volume are constant for given ideal gas.

T1mα ,

TKmα ,

1

2

2

1TT

mm

α

45. Two vessels of volumes V1 and V2 contain air pressures P1 and P2 respectively. If they are connected

by a small tube of negligible volume then the common pressure is 21

2211VV

VPVPP

++

= .

46. Dalton’s law of partial pressures : The total pressure of a non-reacting mixture of gases is equal to the sum of the partial pressures.

Partial pressure=mole fraction x total pressure.

48. Vapour is a gas which can be liquified by the application of pressure alone.

49. Critical temperature (Tc), critical pressure (Pc) and critical volume (Vc) are called critical constants of a gas.

50. The temperature above which a gas cannot be liquified by mere application of pressure is called critical temperature.

51. Gases below their critical temperature are called vapours and vapours above their critical temperature are called gases




1

THERMODYNAMICS Synopsis:

1. Thermodynamics deals with the relation between heat energy and other forms of energy.

2. The thermodynamics coordinates of a system which are also called state variables are pressure, volume and temperature which are inter dependent.

3. The temperature of a system can be expressed as a function of pressure and volume is f(P, V)=T.

Calorimetry: 4. Calorimetery is the study of the measurement of quantities of heat.

5. Quantity of heat is the amount of molecular energy stored in a body.

6. Calorie : The quantity of heat required by one gram of water to raise its temperature from 14.5°C to 15.5°C is called one calorie.

7. British Thermal Unit : The amount of heat required by 1 Pound of water to raise its temperature by 1°F is called one British thermal unit.

8. Pound calorie : The amount of heat required by 1 Pound of water to raise its temperature by 1°C is called one pound calorie.

1 pound calorie=453.6 calories 1 calorie=4.186 joule

Heat Capacity : 9. The amount of heat required to produce a specified change of temperature is directly proportional to the

mass of the material.

10. For a given mass of material, the amount of heat absorbed is directly proportional to the temperature increase.

11. The amount of heat required to raise the temperature of the whole body by 1°C is called heat capacity or thermal capacity. Unit is J/K or Cal/°C.

C = dTdQ

12. Specific heat : The quantity of heat required by one gram of a substance to raise its temperature by 1°C is called its specific heat.

or

Heat capacity per unit mass. Unit is J/Kg-K or Cal/g-°C.

s = mdTdQ

dQ = msdT

13. If m is the mass and s is the specific heat of the material of the body, then the thermal capacity = ms cal/°C.

14. Of all solids and liquids, water has the highest specific heat or specific heat capacity. The value is 1 cal/g/°C or 4200 J/kg/K.

15. The specific heat of lead is the least among solids. (i.e., 0.03 cal/g/°C)

16. In liquids, mercury has least specific heat.




2

17. Of all solids, liquids and gases, hydrogen has the highest specific heat. It is equal to 3.5 cal/g/°C.

18. Specific heat depends upon the nature of the substance and does not depend upon mass, volume and heat supplied.

19. Specific heat of copper = 0.1 cal/g/°C. Specific heat of ice = 0.5 cal/g/°C Specific heat of steam = 0.45 cal/g/°C Specific heat of lead = 0.03 cal/g/°C.

20. Specific heat of a solid at its melting point is infinite.

21. Specific heat of a liquid at its boiling point is infinite.

22. The water equivalent of a body is the number of grams of water which require the same amount of heat as the substance for the same rise of temperature. Unit is grams.

Water equivalent=ms grams.

23. Water equivalent is numerically equal to heat capacity.

24. Latent heat (L) is the quantity of heat required by unit mass of a substance to change its state at a constant temperature. Unit of L is cal/g or J/kg.

25. Latent heat of fusion is the quantity of heat required by unit mass of a solid to melt it at its melting point.

26. The latent heat of ice is 80 cal/g or 3.35x105 J/kg.

27. Latent heat of vapourisation is the quantity of heat required by unit mass of a liquid to vapourise it at its boiling point.

28. The latent heat of steam is 540 cal/g or 2.26x106 J/kg.

29. Latent heat of vapourisation of water decreases with the increase of pressure (i.e., increase of boiling point).

30. The latent heat of steam at boiling point t is given by L=600−0.06t.

31. Latent heat of vapourisation decreases with increase in temperature.

32. Latent heat of a substance becomes zero at critical temperature.

33. Latent heat depends on the nature of a substance and pressure.

34. During the change of state, the formula used to calculate the heat lost or heat gained is Q = mL.

35. When one gram of steam at 100°C is mixed with one gram of ice at 0°C, the resultant temperature will be 100°C and mass of steam condensed will be 1/3 gram.

36. When one gram of ice is mixed with one gram of water at 80°C, the resultant temperature will be 0°C and the composition of mixture will be 2 grams of water.

37. Steam causes more burns than water at 100°C. The reason is that steam while condensing to water at 100°C gives out heat at the rate of 540 cal/g.

38. Calorific value of a fuel is the quantity of heat liberated when one gram of the fuel is burnt completely. Unit is cal/g or J/kg. It is determined by using Bomb calorimeter or Bell calorimeter.

39. Calorific value of a food stuff is the quantity of heat liberated when a unit mass of the food stuff is completely utilised by the body. Unit is cal/g or J/kg.

40. Steam is used in heat engines as working substance because of its high latent heat.

41. Heavy water is used as coolant in nuclear reactors because of its high specific heat.




3

42. In extinguishing fire hot water is preferred than cold water since hot water becomes vapour quickly and vapours do not allow fire.

LAW OF MIXTRUES (OR) CALORIMETRY PRINCIPLE : 43. If no heat is lost or gained otherwise, the quantity of heat gained by the cold body is equal to the quantity

of heat lost by the hot body. This is called the principle of the method of mixtures.

44. The principle of method of mixtures is Heat lost = Heat gained.

45. Calorimeter is generally made up of copper because it has low specific heat and high conductivity and hence attains the temperature of contents quickly.

46. To calculate the heat gained or lost when there is no change of state, we use the formula Q=mst.

47. When three substances of different masses m1, m2 and m3 specific heats s1, s2, s3 and at different temperatures t1, t2 and t3 respectively are mixed, then the resultant temperature is

t = 332211

333222111smsmsm

tsmtsmtsm++++

48. When “x” gram of steam is mixed with “y” gram of ice, the resultant temperature is

t = )yx(

)yx8(80+

−

TRIPLE POINT : 49. The temperature and pressure where solid, liquid and vapour states are co-exist is called triple point.

50. The triple point of water is 273.16 K (0.00750°C) and pressure 613.10 Pa (0.459 cm of Hg).

51. A graph drawn between the pressure and temperature representing the different states of matter is called the phase diagram.

52. PA is the steam line and along this line water and steam are in equilibrium state.

53. Above the line water exists and below steam exists.

54. The curve has positive slope showing the boiling point increases with pressure.

55. CP is called Hoar-frost line. Along this line ice and vapour coexist.

56. CP has positive slope.

57. PB is called ice line, along this line water and ice are in equilibrium.

58. Above the ice line water exists. The curve has negative slope showing the melting point decreases with increase of the pressure.

THERMODYNAMICS: 59. Internal energy depends only on temperature and is independent of pressure and volume.

60. Internal energy=P.E+K.E, where potential energy is due to molecular configuration and K.E is due to molecular motion.

61. Internal energy of an ideal gas consists of only the K.E of molecules (P.E is absent because there are no intermolecular forces among the molecules in an ideal or perfect gas).

62. Internal energy of a real gas consists of P.E and K.E.

63. The amount of work performed is directly proportional to the amount of heat produced (W α H).

Temperature

Pre

ssur

e

273.16 K

4.58

mm Solid

Water ABice line

Steam lin

e

Sublimation lineP

CVapour




4

64. W=JH, where J is known as mechanical equivalent of heat or Joule’s constant. J is equal to the amount of work required to produce one calorie of heat. Its value is equal to 4.186 joule/calorie. SI unit is J/kcal. The statement W=JH is also called the non-differential form of the first law of thermodynamics.

65. J=4.186 J/cal=4.186x107 erg/cal=4186 J/Kcal

66. When heat and work are in Joule then J=1.

67. The height form which ice is to be dropped to melt it completely is

h=gJL where L=latent heat of ice.

68. The rise in temperature of water when it falls from a height h to the ground is

Jsgh

=θΔ where ‘s’ is specific heat of water.

69. When a body of mass m moving with a velocity v is stopped and all of its energy is retained by it, then the increase in temperature is

Js2v2

=θΔ

70. When a block of ice of mass M is dragged with constant velocity on a rough horizontal surface of coefficient of friction μ, through a distance d, then the mass of ice melted is

JLMgdm μ

= where m=mass of ice melted.

71. When a block of mass m is dragged on a rough horizontal surface of coefficient of friction μ, then the rise in temperature of block is

Jsgdμ

=θΔ

72. If a bullet just melts when stopped by an obstacle and if all the heat produced is absorbed by the bullet then

Jmv

21mLms

2=+θΔ

where L=latent heat of the material of the bullet s=specific heat

73. A metal ball falls from a height ‘h1’, and bounces to height ‘h2’. The rise in temperature of the ball is

Js)hh(g 21 −

=θΔ

74. Joule’s law or Mayer’s hypothesis : It states that there is no change in internal energy during the free expansion of gas.

SPECIFIC HEAT OF GASES : 75. A gas will have two specific heats.

a) specific heat at constant volume (CV) b) specific heat at constant pressure (CP)

76. Specific heat depends only on the nature of material and temperature.

77. Water has largest specific heat among solids and liquids.

78. Among solids, liquids and gases specific heat is maximum for hydrogen.

79. Specific heat slightly increases with increase of temperature.

80. In liquids specific heat is minimum for mercury.




5

81. The value of specific heat may lie between 0 and ∞ .

82. In isothermal process, the value of specific heat is ∞ but in adiabatic process its value is zero.

83. Specific heat of water is maximum at 15°C and minimum at 37°C.

84. Specific heat of all substances is zero at 0 K.

85. Substances with highest specific heat are bad conductors of heat and with low specific heat are good thermal and electrical conductors.

86. The substance with large specific heat warms up slowly and cools down slowly.

87. CP is greater than CV and

γ=V

PCC

88. CP−CV=R (for 1 mole of gas) where R is universal gas constant

R=8.3 J/mol-K

89. CP−CV=r (for 1 g of gas)

where r is specific gas constant.

CP−CV=R/J (in heat units)

CV, CP and values of different gases :

S.No. Nature of gas CP CV γ=CP/CV

1. Monoatomic 25 R

23 R

35 =1.67

2. Diatomic 27 R

25 R

57 =1.4

3. Tri (or) Polyatomic

4R 3R 34 =1.33

90. γ value is always greater than one. It depends upon the atomicity of a gas. It decreases with increase in atomicity.

1RCP −γ

γ= and 1

RCV −γ=

91. γ of mixture of gases : When n1 moles of a gas with specific heat at constant volume 1VC is mixed with

n2 moles of another gas of specific heat at constant volume2VC then

(CV)mixture=21

V2V1

nn

CnCn21

+

+

(CP)mixture=(CV)mixture+R = 1 21 P 2 P

1 2

n C n Cn n

+

+

)mixture(V

)mixture(Pmixture C

C=γ ; 1 2 1 2

mix 1 2

n n n n1 1 1

+= +

γ − γ − γ −

Fraction of heat absorbed that is converted into internal energy is γ

==1

CC

dQdU

P

V

Fraction of heat absorbed that is converted into workdone=γ

−==11

CR

dQdW

P

Isothermal Process : 92. In this process, the pressure and volume of gas changes but temperature remains constant.




6

93. The system is in thermal equilibrium with the surroundings.

94. It is a slow process.

95. The internal energy of the system remains constant i.e, du = 0.

96. It obeys the Boyle’s law i.e., PV=K.

97. The work done during isothermal expansion at constant temperature is

W=2.303RTlog10 ⎟⎟⎠

⎞⎜⎜⎝

⎛

1

2VV

=2.303RTlog10 ⎟⎟⎠

⎞⎜⎜⎝

⎛

2

1PP

98. The isothermal elasticity =V/dV

dP− =P.

The –ve sign represents, as pressure increases volume decreases.

99. It takes place in a conducting vessel.

Adiabatic Process : 100. The pressure, volume and temperature of a gas change but total heat remains constant. i.e., dQ=0

(Q=constant)

101. It is a quick process.

102. The internal energy changes as temperature changes.

103. The adiabatic process is represented by the equations

PVγ=constant

TVγ−1=constant

P1−γTγ=constant

104. The work done by the system during the adiabatic expansion is

)TT(nC)TT(1

RW 21V21 −=−−γ

=

= )TT(C

n 21P −γ

=1

VPVP 1122−γ−

105. The adiabatic elasticity of gas is γP.

106. The slope of adiabatic curve is γ times greater than the isothermal curve.

107. It takes place in a non conducting vessel.

108. Adiabatic expansion causes cooling and contraction causes heating.

109. If two samples of gases are compressed so that their pressures have the same increase, one sample isothermally and the other adiabatically, final volume is more in adiabatic change. If their pressures decrease by the same factor the final volume is more in isothermal change.

110. Isochoric process : It is a process in which the volume of the system remains constant.

i.e., ΔV=0 for such process ΔW=0.

111. Isobaric process : It is a process in which the pressure of the system remains constant. i.e., ΔP=0.

112. The equation of state for different types of processes :




7

Sl.

No

Name of the

process

Quantity

remains

constant

Quantity

which

becomes

zero

Result

in I law

1. Isothermal temperature dU dQ=dW

2. Isobaric pressure None dQ=dU+dW

3. Isochoric volume dW dQ=dU

4. Adiabatic heat energy dQ dU=−dW

113. Work done is maximum during isobaric process and minimum in adiabatic process.

HEAT ENGINES : 114. Heat engine is a device used to convert heat into mechanical work.

115. Heat engines are of two types namely internal combustion engine and external combustion engine.

a) Internal combustion engine is an engine in which heat is produced in the engine itself.

Ex : Otto engine and Diesel engine.

b) In external combustion engine, heat is produced outside the engine.

Ex : steam engine.

116. Heat engine absorbs a quantity of heat Q1 from a source, performs an amount of work W, and returns to the initial state after rejecting some heat Q2 to a sink. The working substance, which is a gas or liquid undergoes a cyclic thermodynamic process. The source is at a higher temperature than the sink.

117. The efficiency η of a heat engine is given by

η = absorbed heat total

workto converted heat =

1

2

1

21

1 QQ1

QQQ

QW

−=−

= .

From the above expression it is clear that the efficienty of a heat is always less than 1 or 100%.

118. Carnot heat engine essentially consists of four components. They are

i) a cylinder with perfect thermal insulating walls, perfect conducting base and a tight fitting perfect insulating and frictionless piston (It consists of a working substance).

ii) a hot body of infinitely large heat capacity at a constant temperature serving as a source.

iii) a cold body of infinitely large heat capacity at a constant temperature serving as a sink and

iv) a perfectly thermal insulating stand.

119. Carnot Engine : Carnot developed, ideal heat engine that has maximum possible efficiency consistent with the second law. The working substance is imagined to go through a cycle of four processes known as Carnot cycle. The working substance, an ideal gas, undergoes a cycle which consists of two isothermal and two adiabatic processes as follows.

i) Step I (AB) : The gas expands isothermally at source temperature T1 and absorbs heat Q1.

Source at T1

Sink at T2

Working substance Q1

Q2

I(T1) A(P1V1)

P

B(P2V2)

C(P3V3) D(P4V4)

II

III (T2)

IV

V




8

W1 = Q1 = nRT1 . ln ⎟⎟⎠

⎞⎜⎜⎝

⎛

1

2

VV

.

ii) Step II (BC) : The gas expands adiabatically until its temperature decreases to the sink temperature

T2. W2 = 1

)TT(nR 21

−γ−

.

iii) Step III(CD) : The gas is compressed isothermally at T2, rejecting heat Q2.

W3 = Q2 = nRT2 ln ⎟⎟⎠

⎞⎜⎜⎝

⎛

4

3

VV

iv) Step IV(DA) : The gas is compressed adiabatically until it returns to its initial state A.

W4 = 1

)TT(nR 21

−γ−

.

v) All the steps are carried out very slowly so that there are no dissipative effects.

vi) The process is reversible.

vii) W = Q1 – Q2.

viii) 2

1

2

1

TT

QQ

=

It can be shown that the efficiency of Carnot engine is η = 1

2

TT1− .

120. η can be 100% only if T2 = 0. i.e., the sink is maintained at absolute zero which is impossible.

121. A decreases in T2 is more effective than an equal increase in T1 in increasing the efficiency.

122. The efficiency of engine depends upon the temperature of source and sink.

123. The efficiency of the engines working between same source and sink is same.

124. The efficiency of a reversible engine is independent of the nature of the working substance.

125. The net external work done obtained is the area enclosed.

126. Carnot theorem : No engine working between two given temperatures can be more efficient than a reversible engine working between the same two temperatures have the same efficiency, whatever may be the working substance.

127. If many heat engines are operating in series in such a way that the first engine absorbs a quantity of heat Q1 at T1 and the last engine rejects heat Qn at the lowest temperature Tn the efficiency of the combination is given by

η = 1

n

1

n1

1 TT1

QQQ

QW

−=−

=∑ .

128. The Carnot’s cycle working in the opposite direction can perform the following two functions.

i) It can further root the colder body.

ii) It can further heat the hot body.

a) When reversed heat engine is used to further cool the colder system the arrangment is called a refrigerator.

b) When the reversed heat engine is used to further heat the hotter system it is known as Heat pump.

129. Principle of Refrigerator : a) It will absorb an amount of heat Q2 from the sink (contents of refrigerator) at lower temperature T2.




9

b) As heat is to be removed from the sink at lower temperature, an amount of work equal to Q1 – Q2 is performed by the compressor of refrigerator and then it rejects the total heat Q to the same (atmosphere) through the radiator fixed at its back.

130. Coefficient of performance : The ratio of the quantity of heat removed per cycle from the contents of the refrigerator to the work done by the external agency to remove it is called co-efficient of performance of the refrigerator. It is denoted by ‘β’.

β = w

Q2 = 21

2

QQQ−

⇒ β = 1

QQ

1

2

1 −

But 2

1

2

1

TT

QQ

=

β = 1

TT

1

2

1 −

∴ β = 21

2

TTT−

⇒ β = η

η−1 .

131. SECOND LAW OF THERMODYNAMICS : (a) Classius statement : It is impossible for a self acting machine. (The machine which does not

require any external source of energy for working), unaided by any external agency to transfer heat from a body at lower temperature to another at a higher temperature. (or)

Heat cannot of itself flow from a colder body to a hotter body.

(b) Kelvin statement : It is impossible to construct a heat engine operating in a cycle to convert the heat energy completely into work without any change of working substance. (or)

No heat engine can convert whole of the heat energy supplied to it into useful work.

132. Absolute zero is the temperature at which reversible isothermal process takes place without any transfer of heat. i.e., at absolute zero isothermal and adiabatic processes are identical.




1

TRANSMISSION OF HEAT Synopsis :

1. In general heat travels from one point to another whenever there is a difference of temperatures. 2. Heat flows from a body at higher temperature to a lower temperature. 3. Heat is transferred or propagated by three distinct processes, conduction, convection and radiation. 4. Conduction : It is a process in which the heat energy is transferred from a particle to particle, without

the particles leaving the mean positions but vibrating with amplitudes which depend on the temperature. For conduction the medium is actively involved.

5. The substances through which heat is easily conducted or for which the rate of conduction is large are good conductors of heat. All metals are good conductors of heat. In metals thermal conduction is due to vibration of the atoms and free electrons.

6. The substances which do not conduct heat easily are bad conductors. The substances like cork, wood, cotton, wool are bad conductors. Almost all gases and liquids (except mercury) are poor conductors of heat. The bad conductors do not have the free electrons, therefore, they cannot conduct heat. Whatever little heat they can conduct is by vibrations of the molecules.

Coefficient of thermal conductivity : 7. In steady state, the amount of heat Q transmitted through a conductor is directly proportional to

i) the temperature difference between the faces (θ2−θ1)°C. ii) the area of the cross-section of the slab A. iii) the time of the flow of heat t. iv) inversely proportional to the distances between the two faces “d”.

Q α d

t)(A 21 θ−θ

Q = d

t)(KA 21 θ−θ;

)KA/d(tQ 21 θ−θ

=⎟⎠

⎞⎜⎝

⎛

Where K is a constant that depends upon the nature of the material of the rod and is known as coefficient of thermal conductivity.

8. The value of K depends on temperature, increasing slightly with increasing temperature, but K can be taken to be practically constant through out a substance if the temperature difference between its ends is not too great.

9. Coefficient of thermal conductivity is defined as the quantity of heat flowing per second across a cube of unit edge when its opposite faces have a temperature difference of 1°C. Unit of K is cal/cm/s/°C or Wm−1K−1.

10. The dimensional formula of K is M1L1T−3K−1. 11. Temperature gradient : Rate of change of temperature with distance is called temperature gradient. 12. Thermal conductivity of a good conductor is determined by Searle’s apparatus or Forbe’s method. 13. Thermal conductivity of a bad conductor is determined by Lee’s method. 14. When conduction takes place through two layers of a composite wall with different thermal

conductivities, then

2

22

1

11d

)(AKd

)(AKtQ θ−θ

=θ−θ

= ;

2

2

1

1

21

Kd

Kd

A)(tQ

+

θ−θ=

where θ is common temperature or interface temperature.




Transmission of Heat

2

Interface temperature=

2

2

1

1

2

22

1

11

dK

dK

dK

dK

+

θ+

θ

15. If two slabs of thicknesses d1 and d2 and of thermal conductivities K1 and K2 are placed in contact, then the combination behaves as a single material of thermal conductivity K and is given by

K=1221

2121dKdKKK)dd(

++

or K=

2

2

1

1

21

Kd

Kd

dd

+

+.

16. If three slabs have the same length and have cross-sectional areas A1, A2, A3 respectively and when heat flows from the upper surface to the lower without loss of heat, then the equivalent conductivity K is given by

K=321

332211AAA

AKAKAK++

++

17. The thermometric conductivity or diffusivity is defined as the ratio of the coefficient of thermal conductivity to the thermal capacity per unit volume of the material.

Thermal capacity per unit volume= sVm

⎟⎠

⎞⎜⎝

⎛ =ρs where ρ is density of substance.

sKDy Diffusivit ρ

=∴

18. Heat flow through a conducting rod

Current flow through a resistance

Heat current H =

dtdQ

= rate of heat flow

H = R

.D.TRT

=Δ

R = KAl

K=thermal conductivity

Electric current i = dtdq =

rate of charge flow

i = RD.P

RV

=Δ

R = Aσl

σ=electrical conductivity

19. Thermal resistance of metal rod R=A.K

L where L=length of the rod; A=area of cross-section,

K=coefficient of thermal conductivity. 20. Unit of thermal resistance is KW−1. 21. If different rods are connected in series, then heat flowing per second is same, i.e., H1=H2=H3 and the

net thermal resistance R=R1+R2+R3+…

22. If different rods are connected in parallel, then the net resistance R is given by ....3R1

2R1

1R1

R1

+++=

23. Thermal resistivity is the reciprocal of thermal conductivity. 24. Ingen-Haux experiment is used to compare thermal conductivities of different materials. If L1 and L2 are

the lengths of wax melted on rods of thermal conductivities K1 and K2, then

22

21

2

1

L

LKK

=

25. When the temperature falls below 0°C say to −θ°C, the time taken for the thickness of ice growing from

x1 cm to x2 cm on a lake is given by t= )xx(K2L 2

12

2 −θ

ρ where ρ density of ice, K=coefficient of thermal

conductivity of ice, L=latent heat of fusion of ice. 26. Convection : The transmission of heat from one part to another by the actual transfer of particles of

matter is known as convection. 27. Although conduction does occur in liquids and gases also, heat is transported in these media mostly by

convection.





3

28. Convection is the natural way of heat transmission in fluids. The region of a fluid when heated expands, becomes less dense and rises to the other parts of the fluids there by carrying heat.

29. Convection is a quicker process than conduction. For convection molecules must be relatively free. 30. A wind is a convection current in the atmosphere caused by unequal heating. 31. Trade winds and monsoons are convection currents on a global scale. 32. Near sea shore after the sun rise, the direction of wind is from sea towards the land (sea breeze) and

after the sun set the direction of wind is from the land towards the sea (land breeze). 33. Types of convection : Convection is of two types.

a) free convection or natural convection b) forced convection

34. Free or natural convection takes place in the still fluid, in this process the motion of the fluid particles will be due to their getting heated by the hot body. e.g., hot air rises by natural convection. In general natural convection is a consequence of gravity and always takes place vertically carrying the heat upwards.

35. Forced convection takes place when steady stream of air is sent past the hot air. E.g., cool air from open window enters the room and sends the hot air through ventilators.

RADIATION : 36. It is the process of transmission of heat from one place to another without any material medium. 37. It is a quick process than conduction and convection. 38. In this process medium is not heated. 39. Thermal radiation : The heat energy transferred between the objects without the help of any medium is

known as thermal radiation or radiant energy. (or) Heat energy transferred by means of electromagnetic waves is thermal radiation.

40. Nature and properties of radiant energy : i) It consists of long wavelength electromagnetic radiation. ii) The wavelength of these waves is nearly 800 nm to 4,00,000 nm. iii) It occupies the infrared region of the electromagnetic spectrum. iv) It can be transmitted through vacuum. v) These waves propagate in vacuum with a velocity 3x108 ms−1 like light waves. vi) It obeys laws of reflection, refraction, interference, polarization and diffraction. vii) The intensity of radiant energy obeys inverse square law.

41. Detectors of radiant energy : i) To detect radiant energy Bolometer, thermopile, radiomicrometer, pyrometer are used. ii) By using surface Bolometer radiation coming from the surface of a body is measured. iii) By using linear Bolometer, the distribution of energy in a black body spectrum can be explained. iv) Bolometer works on the principle that the resistance changes with temperature.

42. Prevost’s theory of heat exchanges : i) Every object emits and absorbs radiant energy at all temperatures except at absolute zero. ii) The energy emitted by a body does not depend on the temperature of the surroundings. iii) The rate of emission increases with the increase in the temperature of the body. iv) If the body emits more energy than absorbed its temperature decreases. v) If the body absorbs more radiant energy than it emits, its temperature increases. vi) If two bodies continuously emit and absorb same amount of energy, then they are in dynamical

thermal equilibrium. 43. The radiant energy emitted by a body depends on

a) the nature of the surface b) surface area c) temperature of the body

44. Emissive power (eλ) : i) The amount of energy emitted per second per unit surface area of a body at a given temperature for a

given wavelength range (λ and λ+dλ) is called emissive power. ii) At a given temperature if the radiations emitted have a wavelength difference dλ, then the emissive

power is equal to eλdλ. iii) S.I unit of emissive power is Wm−2 and its dimensional formula is MT−3.





4

45. Emissivity (e) : The ratio of radiant energy emitted by a surface to radiant energy emitted by a black body under same conditions is called emissivity. i) for a perfect black body emissivity e=1.

46. Absorptive power (aλ) ; i) At a given temperature, for a given wavelength range, the ratio of energy absorbed to the energy

incident on the body is absorptive power.

incidentenergy radiant of Amountabsorbedenergy radiant of Amounta =λ

ii) For a perfect black body, the absorptive power, aλ=1. iii) A surface can have different absorptive powers for different wavelengths. iv) Whenever radiant energy is incident on a surface, a part of it is absorbed, a part of it is reflected

and the remaining part is transmitted through it. 47. Reflecting power (r) :

incidentenergy radiant of Amountreflectedenergy radiant of Amountr =

48. Transmitting power (t) :

incidentenergy radiant of Amountdtransmitteenergy of Amountt =

49. aλ+r+t=1 Here aλ is absorptive power, r is reflecting power and t is the transmitting power.

50. Perfect black body : i) A body which completely absorbs all the heat radiation incident on it is called a perfect black body. ii) Fery’s black body and wien’s black body are examples of artificial black bodies. iii) A furnace coated with lampblack or platinum black absorbs about 98% of the radiation incident on it. iv) A perfect black body is a good absorber and also a good emitter of heat. v) The reflecting power of a black body is zero.

51. Kirchoff’s law : i) The ratio of emissive power to absorptive power of a substance is constant. ii) This constant is equal to the emissive power of a perfect black body at the given temperature and

wavelength.

i.e., λλ

λ == Econstantae

where Eλ is the emissive power of perfect black body, eλ and aλ are emissive and absorptive powers of a given substance respectively.

iii) Good absorbers are good emitters. iv) Poor absorbers are poor emitter.

52. Applications of Kirchhoff’s law : i) A piece of blue glass absorbs red wavelengths at ordinary temperature. When it is heated strongly

and cooled it appears brighter than a piece of red glass. ii) A piece of yellow glass absorbs blue wavelengths at ordinary temperatures when heated in dark room

it appears blue because it emits blue colour. iii) Fraunhoffer lines in solar spectrum can be explained on the basis of Kirchhoff’s law. They are

absorption lines. iv) Black surfaces are good absorbers and so good emitters but bad reflectors. v) Highly polished surfaces are bad absorbers and so bad emitters but good reflectors.

53. Stefan’s law : i) The amount of heat radiated by a black body per second per unit area is directly proportional to the

fourth power of its absolute temperature. 44 TETE σ=⇒α

where σ =stefan’s constant =5.67×10−8 Wm−2k−4

ii) Dimensional formula of stefan’s constant is MT−3T−4.





5

iii) Radiant energy emitted by a hot body per second=eAσT4 where e is the emissivity of the hot body, A its surface area, T its absolute temperature and σ the Stefan’s constant.

iv) If the surface area of a body is more, it emits more heat energy. Hence it cools quickly. v) A hot copper cube cools in a lesser time compared to a hot copper sphere of same mass because of

least surface area for sphere. vi) Stefan’s law holds good when the surrounding medium of the black body is vacuum.

54. Stefan-Boltzmann’s law : If a black body at absolute temperature T is surrounded by an enclosure at absolute temperature To, then the rate of loss of heat energy by radiation per unit area is given by E= )TT( 4

o4 −σ

55. Newton’s law of cooling : The rate of cooling of a hot body is directly proportional to the mean excess of temperature of the body above the surroundings, provided the difference in temperature of the body and the surroundings is small.

⎟⎟⎠

⎞⎜⎜⎝

⎛θ−

θ+θ=

θs

212

Kdtd where K =

3s4A

msσθ

here dtdθ =Rate of cooling.

θ1, θ2 are the initial and final temperature of the body respectively. θs is temperature of surroundings and K is the cooling constant.

• Newtons law of cooling is applicable when (i) the heat lost by conduction is negligible and heat lost by the body is mainly by convection (ii) the hot body is cooled in uniformly stream lined flow of air or forced convection (iii) the temperature of every part of the body is same.

• Newtons law holds good for small temperature differences upto 30°C. In case of forced convection the law holds good for large difference of temperatures.

i) Rate of loss of heat of a hot body due to cooling dtdms

dtdQ θ

=

Here m = mass of the body s = specific heat of the body ii) Specific heat of a liquid can be determined using Newton’s law of cooling. iii) If m1, m2 and m3 are masses of the calorimeter, water and liquid, s1, s2 and s3 are the specific heats

of the calorimeter, water and liquid and t1 and t2 are the times taken by water and liquid to cool from

θ2 to θ1°C, then 2

1

3311

2211tt

smsmsmsm

=++

.

iv) Newton’s law of cooling is a law connected with the process of convection. v) It can be deduced from Stefan Boltzmann’s law of radiation. vi) A cube, a sphere, a circular plate of same material and same mass are heated to the same high

temperature. Among them the sphere cools at the lower rate because of its least surface area. 56. Diathermanous substances :

The substances which allow the heat radiations to pass through them without getting themselves heated are called “diathermanous” substances. Ex : air, rock salt, fluorspar.

57. Athermanourl substances : The substances which do not allow the heat radiations to pass through them, but absorb heat and get themselves heated are called “athermanous” substances. Ex : glass, moist air, water, wood. i) Out of a rough black surface and polished black surface, the rough black surface emits more radiant

energy than the polished black surface. ii) Cloudy day is cooler than clear day and cloudy night is warmer than clear night because moist air is

athermanous. iii) Green houses are built with glass doors and roofs because glass is “athermanous”. iv) Cooking vessels are coated black outside because black surface is a good absorber and good

emitter.




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