investigating physics€¦ · investigating physics andrew kenny gill & macmillan derivations...

INVESTIGATING PHYSICS

A N D R E W K E N N Y

G I L L & M A C M I L L A N

DERIVATIONS OF FORMULAEAND LINKS

Contents

Derivations of Formulae 1

Links 11

DERIVATIONS OF FORMULAE 1

Derivations of Formulae

Derivation of Formula T � FdWith reference to the illustration, consider the moments of each force acting on thelever about an axis O:

moment of F1 about the axis O

F1 � F ; moment as illustrated is clockwise

moment of F2 about the axis O

F2 � F ; moment as illustrated is clockwise

adding the moments due to both forces

simplifying the right-hand side of the equation

as the forces are equal in magnitude, opposite indirection and whose lines of action do not coincide,they constitute a couple; therefore the sum of theirmoments equals the torque of the couple

T � Fd

M1 � M2 � Fd

M1 � M2 � Fx � F(d � x)

M2 � F(d � x)

M2 � F2(d � x)

M1 � Fx

M1 � F1x

x

d

O

F1 = F

F2 = F

2 INVESTIGATING PHYSICS

Derivation of Formula P � rghBased on the definition of pressure, pressure is force per unit area, we have:

since the force is due to the weight of the fluid

subbing ‘rV ’ for ‘m’

subbing ‘Ah’ for ‘V ’

dividing above and below by ‘A ’p � rgh

p �r(Ah)g

A

p �rVg

A

p �mg

A

p �FA

h

A


Derivation of Formula a � v2rBased on the definition of sine and cosine we get the displacement vector of a pointon a circle:

with reference to the diagram

subbing ‘vt’ for ‘u’, where v � angular velocity, t � time

differentiating displacement to get velocity

subbing ‘ ’ for ‘ ’

differentiating velocity to get acceleration

subbing ‘ ’ for ‘ ’

subbing ‘ ’ for ‘ ’r cos vt Bi � r sin vt

BjrBaB ��v2rB

dvB

dtaBaB � �v2(r cos v t iB � r sin vt jB)

dvB

dt� �v2r cos vt i

B

� v2r sin vt jB

drB

dtvBvB � �vr sin vt i

B

� vr cos vt jB

drB

dt��vr sin vt i

B� vr cos vt j

B

rB � r cos vt iB

� r sin vt jB

rB � r cos u iB

� r sin u jB

i - axis

j - axis

(r cos �, r sin �)

�

r


Derivation of Formula With reference to the illustration, taking the direction of displacement of the pendulumbob from its rest position as positive, based on Newton’s second law we have:

subbing ‘ma’ for ‘F ’

making the assumption that ,we can only make this assumption as

dividing both sides by ‘m’

since where x is the arc length

Since g and l are constant we can say that the pendulum is under-going simple harmonic motion if we assume the arc x is linear.we can only make this assumption as

rearranging the equation so that it is in the standard form forsimple harmonic motion

subbing ‘ ’ for ‘a’

dividing both sides by ‘�x’

square rooting both sides

subbing ‘ ’ for ‘v’

T � 2pAlg

T �2p

2gl

2pT

2pT

� Ag

l

v � Ag

l

v2 �g

l

�v2x�v2x � �g

lx

a ��g

lx

U<5°

u �xl

a ��gxl

a � �gu

U<5°sin u � uma � �mg u

ma � �mg sin u

F � �mg sin u

T � 2pAlg

Simple Pendulum

mg

mg sin �

�

�

l


Derivation of Formula

The relative increase in sound intensity is the ratio of one intensity to another. It ismeasured in bels, B. If the intensity of one sound is 10 times the intensity of another,then the difference between their intensities is one bel.

The relative increase in intensity from I1 to I2 is 1 B if I2 � 10I1

It follows that the relative increase in intensity from I1 to I2 is 2 B if I2 � 10 � (10I1)� 102I1

Continuing this on, the relative increase in intensity from I1 to I2 is 3 B if I2 � 103I1

This can be generalised, the relative increase in intensity from I1 to I2 is n B if I2 � 10nI1

Based on this equation we get:

rewrite the equation, changing it from index notation to logarithmnotation

Number of bels

A decibel, dB, is one-tenth of a bel, as the prefix ‘deci’ means 10�1.

Therefore 1B � 10 dB.

Number of decibels

Sound intensity level

Sound intensity level (I.L.) is defined as the ratio of sound intensity (I) at a point tothe sound intensity at the threshold of hearing (I0).

Subbing ‘I ’ for ‘I2’ and ‘I0’ for ‘I1’ in the equation, Number of decibels weget:

, where I.L. is measured in decibels.

Doubling sound intensity causes an increase of 3 dB in sound intensity level.

Consider the situation where I2 � 2 � I1.

Number of decibels

subbing ‘2I1’ for ‘I2’

� 10 log10 2

� 10(0.30103)

� 3.0

� 10 log102I1

I1

� 10 log10I2

I1

I.L. � 10 log10I

I0

� 10 log10I2

I1

� 10 log10I2

I1

� log10I2

I1

n � log10I2

I1

10n �I2

I1

I.L. � 10 log10

II0


Derivation of the Wheatstone BridgeFormula

where R1, R2, R3 and R4 are the resistances of the four resistors, as illustrated

When the Wheatstone bridge is balanced, no current flows through the galvanometer.

This means that the current flowing through R1 equals the current flowing through R 2;call this current I1.

Similarly, the current flowing through R3 equals the current flowing through R 4; call thiscurrent I2.

As no current is flowing through the galvanometer, the points B and C must be at thesame potential.

This means that the potential difference across R1 equals the potential differenceacross R3; call this voltage V.

V � I1R1 Ohm’s law applied to R1

V � I2R3 Ohm’s law applied to R3

I1R1 � I2R3 call this equation (i)

Similarly, since the potential difference across R2 equals the potential difference acrossR4 we get

I1R2 � I2R 4 call this equation (ii)

dividing equation (i) by equation (ii)

simplifying both fractionsR1

R2�

R3

R4

I1R1

I1R2�

I2R3

I2R4

R2

I1

I2

I2

I1

R1

A D

B

C

R3 R4

R1

R2�

R3

R4


Derivation of Formulae

&

Joule’s law states that W � I2Rt; this means that the heating effect of an electriccurrent is proportional to its current squared.The law holds true for both alternatingand direct currents.

When an a.c. supply is said to have a voltage V or cause a current I, what is beingdescribed is the equivalent voltage or current from a d.c. supply that would have thesame heating effect.

The voltage is called the root mean squared voltage (Vrms); it is equal to the peakvoltage (V0) divided by the square root of two.

Similarly, the current is the root mean squared current (Irms); it is equal to the peakvoltage (I0) divided by the square root of two.

The derivations of these formulae are not on the Leaving Cert. course and are merelydescribed here for illustrative purposes.

The derivation of each one is similar to the other, so I will just outline the derivation

of .

The current–time graph for an a.c. supply is sinusoidal, as illustrated in the diagram below.

The current, I, at any instant is given by I � I0 sin u.

The current changes as a function of time, so it should therefore be clear that ushould be defined in terms of time.

where f is the frequency of the a.c. supply.

u � 2p ft

II0

I = I0 sin �

�0

sin �

�0

Irms �I0

22

Irms �I0

22

Vrms �V0

22

Vrms �V0

22Irms �

I0

22


The derivation of this equation is given below.

where v � angular frequency, u � angle (measured in radians),t � time

where T � periodic time

where f � frequency

combining the equations and

rearranging the above equation

subbing ‘ ’ for ‘v’


Therefore the current, I, can be expressed as a function of time, where I0 � the peakcurrent and f � the frequency of the supply.

I � I0 sin 2pft

The heating effect of this alternating current (I) is equal to the heating effect of adirect current of value Irms . Joule’s law states W � I2Rt, therefore

heat produced by the d.c. equals heat produced by the a.c.

dividing both sides by R

subbing ‘I0 sin 2pft ’ for ‘I ’

In order to equate these we can graph current squared against time for the a.c. andthe d.c. equivalent.

Graph of square of a.c. against time.

t

I02 I2 = (I0 sin 2� ft) 2

I 2

0

Graph of square of d.c. against time

t

Irms2

I 2

0

Irms2 t � (I0 sin 2pft)2t

Irms2 t � I2t

Irms2 Rt � I2Rt

II0

I = I0 sin 2� ft

0 t

u � 2pft

u

tu

t� 2pf

v � 2pf

T �1f

T �2pv

1f

�2pv

T �1f

T �2pv

v �u

t


The area under the curve in both graphs is equal to and respectively.As these are equal, we get:

the area under a curve is equal to its integral

taking the constants outside theintegrals

using the trigonometric identity

taking the constants outside theintegrals and simplifying the cosinefunction

carrying out the integration

subbing in the limits

,

therefore sin 4pf T � sin 4p � 0.Also sin 0 � 0

dividing both sides by T

square rooting both sides

By a similar argument it may be shown that the root mean squared value of alternating

voltage is given by Vrms �V0

22

Irms �I0

22

Irms2 �

I02

2

Irms2(T ) �

I02

2(T )

T �1f

Q fT � 1Irms2(T ) �

I02

2aT �

14pf

(0 � 0)b

�I0

2

2a (T � 0) �

14pf

(sin 4pf T � sin 0)bIrms

2(T � 0)

Irms2[t]T0 �

I02

2a [t]T0 �

14pf

[sin 4pf t]T0b

Irms2

3

T

0

dt �I0

2

2 3

T

0

(1 � cos (4pf t)) dt

sin2A �12 (1 � cos 2A)

Irms2

3

T

0

dt � I02

3

T

0

12(1 � cos 2(2pf t)) dt

Irms2

3

T

0

dt � I02

3

T

0

(sin 2pf t)2dt

Graph of square of a.c. against time,with one period highlighted

tT

I02 I2 = (I0 sin 2� ft ) 2

I 2

0

Graph of square of d.c. against time,with one period highlighted

tT

Irms2

I2

0

equating one cycle by setting the limits as 0 toT, where T is the periodic time3

T

0

Irms2 dt �

3

T

0

(I0 sin 2pf t)2 dt

LIrms

2dt �L

(I0 sin 2pf t)2dt

(I0 sin 2pft)2 tIrms2t


Derivation of Formula

Based on the law of radioactive decay we have:

where N � number of undecayed nuclei, l � decayconstant, t � time


integrating both sides

where C � constant of integration

where N0 � the number of nuclei present when t � 0

subbing ‘ln N0’ for ‘C ’


using the laws of logs,

by definition of half-life, when

T1>2 �ln 2l

ln 2 � lT1>2

ln 12

� �lT1>2

t � T1>2N � 12N0ln

N0>2N0

� �lT1>2

logaa xyb � loga x � logayln

NN0

� �lt

ln N � ln N0 � �lt

ln N � �lt � ln N0

Q C � ln N0

ln N0 � 0 � C

ln N � �lt � C

L1N

dN �L

�ldt

1N

dN � �ldt

dNdt

� �lN

T1>2 � ln 2l

LINKS 11

Links

Links to Websites Related to Topics in Investigating PhysicsChapter 2 – Linear Motion

1. Addition of vectors: p. 16 http://surendranath.tripod.com/Applets/Math/VectorAddition/VectorAdditionApplet.html

2. Finding the resultant of two vectors: p. 17 http://www.walter-fendt.de/ph14e/equilibrium.htm

3. Difference between distance and displacement: p. 18 http://faraday.physics.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/DisplaceDistance/DisplaceDistance.html

4. Distance–time and velocity–time graphs: p. 26 http://www.walter-fendt.de/ph14e/acceleration.htm

Chapter 3 – Force and Momentum

5. Principle of Conservation of Momentum: p. 40http://www.walter-fendt.de/ph14e/collision.htm

6. Demonstrating that F � ma : p. 43 http://www.walter-fendt.de/ph14e/n2law.htm

7. Illustration of the scale of the Earth to the Sun: p. 46 http://www.rense.com/general72/size.htm

8. Acceleration due to gravity: p. 48 http://www.seed.slb.com/uploadedFiles/Science/Laboratory/Air_and_Space/Galileo_Drops_the_Ball/anim/en/index.html?width=740&height=570&popup=true

9. Terminal velocity: p. 51 http://www.waowen.screaming.net/revision/force&motion/skydiver.htm

Chapter 4 – Moments, Density and Pressure

10. Principle of the Lever: p. 67 http://www.walter-fendt.de/ph14e/lever.htm

11. Boyle’s law: p. 77 http://phet.colorado.edu/simulations/sims.php?sim=Gas_Properties

Chapter 5 – Work, Energy and Power

12. Newton’s cradle: p. 91 http://www.physics.org/article-interact.asp?id=32


Chapter 6 – Circular Motion

13. Applet 6.2: illustrating centripetal acceleration: p. 100 http://phet.colorado.edu/simulations/sims.php?sim=Motion_in_2D

Chapter 7 – Simple Harmonic Motion and Hooke’s Law

14. Simple Harmonic Motion: p. 114 http://www.ngsir.netfirms.com/englishhtm/SpringSHM.htm

15. Simple pendulum: p. 116 http://phet.colorado.edu/sims/pendulum-lab/pendulum-lab_en.html

Chapter 8 – Temperature

16. Difference between heat and temperature: p. 122 http://www.yenka.com/freecontent/attachment.action?quick=ad&att=738

Chapter 9 – Heat

17. Explanation of conduction: p. 146 http://www.absorblearning.com/media/attachment.action?quick=s5&att=2016

18. Demonstration of convection currents: p. 148 http://www.absorblearning.com/media/attachment.action?quick=an&att=758

Chapter 10 – Waves

19. Transverse waves: p. 156 http://www.acoustics.salford.ac.uk/feschools/waves/wavetypes.htm

20. Longitudinal waves: p. 156 http://www.ngsir.netfirms.com/englishhtm/Lwave.htm

21. Electromagnetic waves: p. 157 http://micro.magnet.fsu.edu/primer/java/scienceopticsu/electromagnetic/index.html

22. Reflection, refraction and diffraction: p. 160 http://www.lon-capa.org/~mmp/kap13/cd372.htm

23. Explanation of phase: p. 161 http://www.acoustics.salford.ac.uk/feschools/waves/super.htm

24. Polarisation: p. 163 http://www.ngsir.netfirms.com/englishhtm/Polarization.htm

25. Doppler Effect: p. 163 http://www.astro.ubc.ca/~scharein/a311/Sim/doppler/Doppler.html

Chapter 11 – Vibrations and Sound

26. Demonstrating the wave nature of sound: p. 171 http://phet.colorado.edu/simulations/sims.php?sim=Sound

27. Resonance: p. 173 http://www.absorblearning.com/media/attachment.action?quick=yy&att=2505

LINKS 13

28. Production of stationary waves: p. 175 http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/reflect.html

29. Harmonics in strings: p. 176 http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html

30. Stationary waves in pipes: p. 177 http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html

31. Notes from different instruments: p. 180 http://www.absorblearning.com/media/attachment.action?quick=14h&att=2903

32. Addition of waves: p. 180 http://www.eserc.stonybrook.edu/ProjectJava/WaveInt/index.html

33. Vernier callipers: p. 180 http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=52

Chapter 12 – Reflection of Light

34. Applet 12.1 illustrating reflection in a plane mirror : p. 197http://www.freezeray.com/flashFiles/planeMirror.htm

35. Ray diagrams for mirrors: p. 199 http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=48.0

Chapter 13 – Refraction of Light

36. Snell’s law: p. 210 http://www.upscale.utoronto.ca/PVB/Harrison/Flash/Optics/Refraction/Refraction.html

37. Total internal reflection and critical angle: p. 217 (attached to Demonstration 13.1)http://www.freezeray.com/flashFiles/Refraction1.htm

38. Ray-tracing: p. 221 http://www.teachnet.ie/torourke/flashprojects/raydiagrams.html

39. The functioning eye: p. 229 http://www.freezeray.com/flashFiles/eyeDefects.htm

Chapter 14 – Wave Nature of Light

40. Young’s Double Slit Experiment: p. 236 http://micro.magnet.fsu.edu/primer/java/interference/doubleslit/index.html

41. Illustration of path difference: p. 238 http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/youngexpt4.htm

42. Measurement of wavelength of light: p. 240 http://schools.matter.org.uk/Content/Interference/gratings.html

43. Factors affecting distance between fringes: p. 242 http://www.surendranath.org/Applets/Optics/Slits/DoubleSlit/DblSltApplet.html


44. Addition of colours of light: p. 247 http://www.colorado.edu/physics/2000/tv/colortv.html

45. Colours in soap bubbles: p. 252http://www.microscopy.fsu.edu/primer/java/interference/soapbubbles/index.html

Chapter 15 – Static Electricity

46. Illustrating electric field: p. 271 http://phet.colorado.edu/simulations/sims.php?sim=Charges_and_Fields

47. Illustrating electric field lines: p. 271 http://lectureonline.cl.msu.edu/~mmp/kap18/RR447app. htm

Chapter 16 – Voltage and Capacitance

48. Factors affecting the capacitance of a parallel plate capacitor : p. 284 http://micro.magnet.fsu.edu/electromag/java/capacitance/index.html

49. Charging a capacitor : p. 286 http://micro.magnet.fsu.edu/electromag/java/capacitor/index.html

Chapter 17 – Current Electricity

50. Heating effect of an electric current: p. 294 http://micro.magnet.fsu.edu/electromag/java/filamentresistance/index.html

Chapter 18 – Resistance

51. Colour Coded Resistance Calculator : p. 305 http://www.ese.upenn.edu/rca/calcjs.html

52. Rheostat as a variable resistor : p. 305 http://www.absorblearning.com/media/attachment.action?quick=118&att=2669

53. Ohm’s law: p. 306 http://micro.magnet.fsu.edu/electromag/java/ohmslaw/index.html

54. Building a circuit with resistors in series and parallel: p. 308 http://www.walter-fendt.de/ph14e/combres.htm

55. Potentiometer: p. 312 http://www.walter-fendt.de/ph14e/potentiometer_e.htm

56. Resistivity: p. 313 http://phet.colorado.edu/sims/resistance-in-a-wire/resistance-in-a-wire_en.html

57. Micrometer: p. 314 http://www.upscale.utoronto.ca/PVB/Harrison/Micrometer/Flash/MicSimulation.html

58. Metre bridge: p. 325 http://www.walter-fendt.de/ph14e/wheatstone_e.htm

LINKS 15

Chapter 19 – Semiconductors

59. Semiconductor diode: p. 335 http://www-g.eng.cam.ac.uk/mmg/teaching/linearcircuits/diode.html

Chapter 20 – Electromagnetism

60. Magnetic fields around a bar magnet: p. 347 http://www.walter-fendt.de/ph14e/mfbar.htm

61. Magnetic fields due to a straight wire, loop and solenoid: p. 349 http://schools.matter.org.uk/Content/MagneticFields/fields_3.html

62. Force on a current-carrying conductor in a magnetic field: p. 352 http://www.walter-fendt.de/ph14e/lorentzforce.htm

63. d.c. motor: p. 354 http://www.walter-fendt.de/ph14e/electricmotor.htm

64. Force between currents: p. 356 http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/magnetostatics/ParallelWires/Parallel_Wires_640.mpg

65. Faraday’s Electromagnetic Lab: p. 357 http://phet.colorado.edu/simulations/sims.php?sim=Faradays_Electromagnetic_Lab

66. Faraday’s law: p. 358 http://phet.colorado.edu/sims/faraday-mx/faraday-mx.swf

67. Lenz’s law: p. 359 http://micro.magnet.fsu.edu/electromag/java/lenzlaw/index.html

68. a.c. generators: p. 363 http://www.walter-fendt.de/ph14e/generator_e.htm

69. Building a circuit: p. 364 http://phet.colorado.edu/simulations/sims.php?sim=Circuit_Construction_Kit_ACDC

70. Transformers: p. 367 http://micro.magnet.fsu.edu/electromag/java/transformer/index.html

71. Inductance: p. 369 http://www.magnet.fsu.edu/education/tutorials/java/inductivereactance/index.html

Chapter 21 – The Electron

72. Crookes’Tube: p. 381 http://micro.magnet.fsu.edu/electromag/java/crookestube/index.html

73. Deflection of a charged particle in a magnetic field: p. 382 http://www.surendranath.org/Applets/Electricity/MovChgMag/MovChgMagApplet.html

74. Photoelectric effect: p. 389 http://phet.colorado.edu/simulations/photoelectric/photoelectric.jnlp


Chapter 22 – The Nucleus

75. Rutherford’s Gold Foil Experiment: p. 394 http://www.absorblearning.com/media/item.action?quick=bf

76. Emission line spectra: p. 395 http://jersey.uoregon.edu/vlab/elements/Elements.html

77. Bohr model and the production of emission line spectra: p. 396 http://www.upscale.utoronto.ca/PVB/Harrison/BohrModel/Flash/BohrModel.html

78. Ionisation by an alpha particle: p. 400 http://www.absorblearning.com/media/item.action?quick=187

79. G-M tube: p. 403 http://chemistry.binghamton.edu/ilc/labs/radiochem/moviepop/geiger-counter-sim.htm

80. Solid state detector: p. 404 http://micro.magnet.fsu.edu/primer/java/digitalimaging/avalanche/index.html

81. Half-life graph: p. 406 http://lectureonline.cl.msu.edu/~mmp/applist/decay/decay.htm

82. Decay chains: p. 407 http://www.walter-fendt.de/ph14e/decayseries.htm

83. Fission reactions: p. 410 http://phet.colorado.edu/simulations/sims.php?sim=Nuclear_Fission

Chapter 23 – Particle Physics (Option 1)

84. (Cockroft and Walton Experiment: p. 428 attached to heading ‘Cockroft andWalton Experiment’)http://www-outreach.phy.cam.ac.uk/camphy/cockcroftwalton/cockcroftwalton11_1.htm

85. (Quark Model: p. 441 Attached to Table 23.4)http://www.lon-capa.org/~mmp/applist/q/q.htm

86. (The Particle Adventure: p. 443 attached to STS Box ‘We’re all just empty space’)http://www.particleadventure.org/modern_atom.html

87. (Building atoms: p. 443 attached to Table 23.5)http://www.pbs.org/wgbh/aso/tryit/atom/builder.html

88. (Building hadrons: p. 443 attached to Table 23.5)http://www.lon-capa.org/~mmp/applist/q/q.htm

Chapter 24 – Applied Electricity (Option 2)

89. Force on a current-carrying conductor in a magnetic field: p. 448 http://www.walter-fendt.de/ph14e/lorentzforce.htm

90. d.c. motor: p. 449http://www.walter-fendt.de/ph14e/electricmotor.htm

LLIINNKKSS 1177

91. a.c. generators: p. 454 http://www.walter-fendt.de/ph14e/generator_e.htm

92. Transformers: p. 455 http://micro.magnet.fsu.edu/electromag/java/transformer/index.html

Links to Websites with Applets (Interactive Animations) Relevant to the Leaving Cert. SyllabusYou should note that many of these sites provide far more information than is requiredat Leaving Cert. level.They can be a useful support to textbook and classroom learning,but should not be viewed as a means to learning the definitive answers for the LeavingCert. examination. These sites do provide an opportunity to gain a deeper under-standing of various topics and can aid a teacher’s classroom teaching.

http://www.walter-fendt.de/ph14e/

http://phet.colorado.edu/simulations/index.php?cat=Physics

http://www.ngsir.netfirms.com/englishVersion.htm

http://www.surendranath.org/Applets.html

http://www.absorblearning.com/physics/contents.html

http://micro.magnet.fsu.edu/electromag/java/index.html

http://www.freezeray.com/physics.htm

The list of sites below relates to interesting areas of physics. The material on the sitesdoes not relate to the syllabus, but may be of interest to physics students.

http://www.pbs.org/wgbh/nova/elegant/

http://www.pbs.org/wgbh/nova/time/

http://www.particleadventure.org/

http://superstringtheory.com/index.html

http://www.bbc.co.uk/iplayer/categories/factual/science_and_nature/

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