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WW m l2 0 0 6 - 0 71 mI

y^jdhanic"

PHYSICS

EEE

p-

_

a o t

B" e

i

D. B. SINGHVigyan Gurkui, KOTA

Director

ARIHANT PRAKASHANK A L I N D I , T.R NAGAR, M E E R U T - 2 5 0 0 0 2

URGETIIT 2 0 0 6 - 0 7

Other Useful Books"7ext S4 Radian > Steradian

Step III r

F=

kilogram x metre r second x second

(III) Practical units : A larger number of units are used in general life for measurement of different quantities in comfortable manner. But they are neither fundamental units nor derived units. Generally, the length of a road is measured in mile. This is the practical unit of length. Some practical units are given below : (a) 1 fermi = 1 fm = 10~15 m (b) 1 X-ray unit = l x u = 10 m -to,m (c) 1 angstrom = 1 A - 10""" (d) 1 micron = 1 (.im = 10~6 m (e) 1 astronomical unit = 1 Au = 1.49 x 1011 m [Average distance between sun and earth, i.e., radius of earth's orbit] (f) 1 light year = 1 l y = 9.46 x l 0 1 5 m [Distance that light travels in 1 year in vacuum] (g) 1 parsec = 1 pc = 3.08 x 10 16 m = 3.26 light year [The distance at which a star subtends an angle of parallex of 1 s at an arc of 1 Au]. (h) (i) (j) (k) One One One One shake = 10 _s second. slug = 14.59 kg pound =453.6 gram weight metric ton =1000 kg-13

Step IV The unit of force >

2 = kilogram metre per second

4. Abbreviations for multiples and submultiples: Symbol Prefix Factor 10 24 10 21 10 18 10 15 10 12 109 106 103 102 101 10- 1 ur2 10" 3 10" 6 10" 9 10" 12 yotta zetta exa peta tera g'g a mega kilo hecto deka deci centi milli micro nano pico femto Y Z E P T G M k h da d cm m M n P f

io-

15

8 Units and MeasurementsFactor 10 -18 10- 21 10- 24 10 6 109 10 12 Prefix atto zepto yocto million billion Symbol a z 8. Dimensions and Dimensional Formulae: The dimensions of a physical quantity are powers raised to fundamental units to get the derived unit of that physical quantity. The corresponding expression is known as dimensional formula. In the representation of dimensional formulae, fundamental quantities are represented by one letter symbols. Fundamental Quantity Length in metres 2 x 10 26on

y

trillion 5. Some approximate lengths: Measurement Distance to the first galaxies formed Distance to the Andromeda galaxy Distance to the nearest star. (Proxima Centauri) Distance of Pluto Radius of Earth Height of Mount Everest Thickness of this page Length of a typical virus Radius of a hydrogen atom Radius of a proton 6. Some approximate time intervals: Measurement Life time of a proton (predicted) Age of the universe Age of the pyramid of cheops Human life expectancy Length of a day Time between human heart beats Life time of the Muon Shortest lab light pulse Life time of the most unstable particle The Plank time 7. Some approximate masses: Object Known universe Our galaxy Sun Moon Asteroid Eros Small mountain Ocean liner Elephant Grape Speck of dust Penicillin molecule Uranium atom Proton Electron Mass in kilogram 1 x 10 M 2 x 10 41 2 x 10 30 7x 1x 10 22 10 12 5 x 10 15 7 x 107 5xl03 3 x 10~3 7x10 4x10 2x10 9x10-10

Symbol M L T I K mol cd

2 x 10 4 X 10 16 6 X 10 12 6 x 106 9 x 103 1 x 10 _ 4 1 x 10" 8 5 x 107 11 1 X 10~15

Mass Length or Distance Time Electric current Temperature Amount of substance Luminous intensity

Time interval in second1 xlO',39

Method for finding dimensional formulae : Step I : Write the formula of physical quantity. Step I I : Convert the formula in fundamental physical quantity. Step I I I : Write the corresponding symbol for fundamental quantities. Step I V : Make proper algebraic combination and get the result. Example : Find the dimensions of momentum. Momentum = Mass x Velocity Displacement Step II Momentum = Mass x > Time Step III Momentum _ M A > = Solution : Step I

5 x 10 17 1 x 1 0 11 2xl09 9 x 10 4 8 x 1 0 -1 2 x 10" 6 6x10 1x10R

IT]

Dimensional formula of momentum = [Momentum] = [MLT - 1 ] The dimensions of momentum are 1 in mass, 1 in length and - 1 in time. Example: The unit of gravitational constant is Nm /kg . Find dimensions of gravitational constant. Solution : Step I Write physical quantities of corres> ponding units. Here, Nm 2 Force (Length)2 =- = 5 kg 2 (Mass)2 physical quantities in

15

I-23

1 x 1 0, - 4 3

Step II Convert derived > fundamental quantities. Gravitational constant =

Force x (Length) (Mass)2

(Mass x Acceleration) x (Length) (Mass) Mass (Mass)2 (Length)' Mass x Time Change in velocity Time/ Distance

5 x 10 - 1 7-25 -27 -31

(Length)2

^

Time

9Step III Use proper symbols of fundamental quantities. > [L2] Gravitational constant = [MT] [L] [T]

Units and MeasurementsMT T

= [Gravitational constant] =

= [M~ 1 L 3 T~ 2 ]

.. The dimensional formula of gravitational constant 9. Unit and Dimensions of some Physical Quantities s. No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Physical Quantity Displacement or distance or length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensit" J Area Volume Density Relative density or specific gravity Velocity or speed Acceleration or retardation or g Force (F) Formula length

Name of SI Unit metre kilogram second ampere kelvin mole

SI Units m kg s A K mol cd m2 m3 kg/m 3 kg/m 3

Dimensional Formula ML 1 T M W ML T 1 MLT

length x breadth length x breadth x height mass volume density of substance density of water at 4C distance time change in velocity time mass x acceleration

candela square metre cubic metre kilogram per cubic metre

ML 2 T ML 3 T M1L"3T MLT 0 or dimensionless ML1T_1 MVT-2 M1L1T-2 MVT-1 MVT- 1 M 1 L _1 t

kg/m 3 = no unit m/s m/s 2 N or kg m/s kg m/s Ns N/m2 or Pa kg-m /s or J kg m 2 /s 2 or J kg m 2 /s 3 or J/s or watt (W) Nm 2 /kg 2 rad rad/s

12. 13. 14.

metre per second metre per square second newton or kilogram metre square second kilogram metre second newton-sec

per per

15. 16. 17. 18.

Linear momentum (p) Impulse Pressure Work

mass x velocity force x time interval force area force x distance

pascal or newton per square metre kilogram-square metre per square second or joule kilogram square metre per square second or joule watt (W) or joule per second or kilogram square metre per cubic second newton-square metre per square kilogram radian radian per second

-2

MVT-2

19.

Energy

equivalent to work

M1L2T-2

20.

Power (P)

work time

M1L2T-3

21.

Gravitational constant (G) mim2 arc radius angle (9) time

M

- 1

L

3

T

- 2

22. 23.

Angle (8) Angular velocity (co)

MLT or dimensionless M0LT_1

5 Units Measurements andPhysical Quantity Angular acceleration (a) Formula change in angular velocity time taken mass x (distance) 2 distance Name of SI Unit radian second per square rad/s kgm2 m kg m 2 /s MLTM1L2T M ^ T0

SI Units

Dimensional Formula

Moment of inertia (J) Radius of gyration (K) Angular momentum (L) Torque ( ? )

kilogram square metre metre kilogram square metre per second newton metre or kilogramsquare metre per square second newton per metre or kilogram per square second newton per metre joule per metre square newton metre No unit per square

Z?force displacement force length energy area force area change in dimenson original dimension L logitudinal stress logitudinal strain volume s tress or volume strain normal stress volume strain V 7 1_ Bulk modulus shearing s tress shearing strain F 11= /. ^ Au A T~ Ax stress strain prVc or

MVT-1M 1 L 2 T~ 2

N-m or kg m 2 /s 2

(Spring) force constant (k) Surface tension Surface energy Stress Strain

N/m or kg/s N/m J/m 2 N/m 2

M 1 LT~ 2 M1LT-2 M ^ T 2

M1L-1T-2

ALnewton metre newton metre per per square square N/m metre per per square N/m

MLT

Young's modulus (Y) Bulk modulus (B)

M'l^T-2

Compressibility Modulus of rigidity or shear modulus Coefficient of viscosity (r|)

square newton newton metre

N_1 m2 N/m1M

IVTVT21L

- 1

t

- 2

poise or kilogram per metre per second kg m *s newton metre no unit metre per second or hertz per square

or poise

M^^T-1m

Coefficient of elasticity Reynold's number (R) Wavelength (X) Frequency (v)

N/m

il-It-2 MLT

11distance number of vibrations second co = 2 U T> m s" 1 or Hz radian per second second I=ln2n2a2pv or energy watt per square metre transported per unit area per second rad/s s W/M

MVT0MVT-1mVT-

Angular frequency (co) Time period Intensity of wave (I)

MW

1

mVT"3

Gas constant (R) Velocity gradient

PVnT velocity change distance

joule per mole kelvin per second

J mol - 1 K" 1

M1L2T_2K_1 M0L0T_1

6S. No. 48. 49. Physical Quantity Rate of flow Thermal conductivity (K) Formula volume flow time K Name of SI Unit cubic metre per second kilocalory per metre per degree celsius per second joule per kilogram per kelvin joule per kilogram joule-second joule per kelvin watt per square metre per (kelvin) 4 ampere-second or coulomb no unit newton per coulomb or volt per metre volt or joule per coulomb coulomb-metre ohm volt-metre square coulomb per newton per square metre farad ohm-metre SI Units s" m a31

Units and MeasurementsDimensional Formula ML3T_1M

Qc~

kcal m~ lo C_1s_1

l

L

l

T

- 3

e

- l

50. 51. 52. 53. 54. 55. 56. 57. 58. 59.60.

Specific heat (c) Latent heat (L) Planck's constant (h) Boltzmann constant (fc/j) Stefan's constant (a) Charge Dielectric constant Electric field Potential (electric) Electric dipole moment Resistance (R) Electric flux (0 or E) Permittivity of free space (E0) Capacitance Specific resistance electrical resistivity Conductance or Boltzmann

Qrn

?ti At

Q

Jkg^K"1 J/kg

M0L2T-2K-1

ML2T-2 M1L2T_1 mVT^K"14M

energy frequency PV TNA E AtT" q = It K F AV F E = or E = q a W V:

J-sJ/K Wm - 2 K~

!

L

0

T

- 3

K

- 4

A-s or C Unit less N/C or V/m J/C or volt C-m Q V-m C 2 N- 1 m" 2 F Q-m

MLT dimensionlessM

l

L

l

t

-3J-1

m

1l2t-3j-1

1

p = 2qL

mVT1!1 M1L2T_3F2m

r-7