EP015 Note #KMKK

CHAPTER 8
ROTATION OF RIGID BODY

LEARNING OUTCOMES

8.1 Rotational Kinematics

1. Define and use:

i) angular displacement ()

ii) average angular velocity (av)
iii) instantaneous angular velocity ()

iv) average angular acceleration (av)

v) instantaneous angular acceleration ().

2. Relate parameters in rotational motion with their corresponding quantities in linear

motion:

s = rθ, v = rω, at = rα , ac= rω² = v2
r

3. Use equations for rotational motion with constant angular acceleration;

 = ω0 + αt , θ = ω0 t + 1 αt² and ω² = ω0²+2 αθ .
2

8.2 Equilibrium of a Uniform Rigid Body

1. Define torque, τ= r x F
2. Solve problems related to equilibrium of a uniform rigid body

8.3 Rotational Dynamics

1. Define and use the moment of inertia of a uniform rigid body (sphere, cylinder, ring,

disc and rod)

2. Determine the moment of inertia of a flywheel
3. State and use torque, τ =Iα .

8.4 Conservation of Angular Momentum

1. Define and use angular momentum, L=Iω .
2. State and use the principle of conservation of angular momentum.

8.1 ROTATIONAL KINEMATICS
1. A boy and a girl are riding on a merry-go-round which is rotating at a constant rate. The

boy is near the outer edge, and the girl is closer to the center. Who has the greater angular
displacement?
A. The boy
B. The girl
C. Both have the same non-zero angular displacement.
D. Both have zero angular displacement
2. Which of the following statements is CORRECT for a rigid body that is rotating?
A. Its center of rotation is its center of mass.
B. All points on the body are moving with the same angular velocity.
C. All points on the body are moving with the same linear velocity.
D. Its center of rotation is at rest, i.e., not moving.
3. The blades in a blender rotate at a rate of 6.5103 rpm. When the motor is turned off the
blades come to rest in 3.0 s. Calculate the angular acceleration.

(227 rad s2)
4. A pulley of radius 8.0 cm is connected by a string to a rotating motor which is rotating

at 7.0103 rad s1 and reduced to 2.0103 rad s1 in 5.0 s. Calculate
a) the tangential acceleration of the string,

(80 m s2)
b) the time taken to stop the pulley.

(2 s)

101

8.2 EQUILIBRIUM OF A UNIFORM RIGID BODY
5. A uniform ladder AB of length 10m and mass 4.0kg leans against a smooth vertical wall.

The height of the end A of the ladder is 8.0m from the floor. An object P of mass 10 kg is
suspended from point C on the ladder where BC=2.5 m.
(a) What maximum frictional force must act on the bottom of the ladder to keep it

from slipping?
(137.3 N)

(b) What is the necessary coefficient of static friction?
(0.241)

6. A uniform rod AB of length 100 cm and mass 2.0 kg is hinged smoothly at the end A
and is kept in a horizontal position by objects P & Q as shown in figure below. The
mass of Q is 4.0 kg. Determine:
(a) the mass of P
(30 kg)
(b) the reaction force at A
(274.68 N)

102

8.3 ROTATIONAL DYNAMICS

7. The moment of inertia of a rigid body is independent of
A. the mass of the body
B. the speed of rotation
C. the axis of rotation
D. the shape of rotation

8. A flywheel of mass 3.0 kg and radius R = 0.40 m is wrapped with a rope round its rim. A
load of mass 2.0 kg is attached to the other free end of the rope. The load is then released
from rest from a point 20.0 m above the ground. How long does it take the load to fall to
the ground? 1 (Moment of inertia of the wheel = ½ mr2)
(2.7 s)

0.4 m

2.0
kg

8.4 CONSERVATION oF ANGULAR MOMENTUM

9. A man standing on a freely rotating frictionless platform holding two weights with his arms
extended horizontally. If he pulled the weights inward horizontally to his chest, then the

A. angular velocity increase
B. angular velocity decrease
C. angular momentum increase
D. angular momentum decrease

10. A figure skater increases her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to
a final rate of 3.0 rev s1. Her initial moment of inertia was 4.6 kg m2.

a) Calculate her final moment of inertia. (0.77 kg m2)
b) How does she physically accomplish this change?

103

11. A merry-go-round of radius 2.0 m has a moment of inertia 2.5102 kg m2 and is rotating
at 10 rpm. A child of mass 25 kg jumps onto the edge of the merry-go-round. Calculate
the new angular velocity of merry-go-round.
(0.75 rad s1)

12. An object P of mass 300 g is placed on a stationary smooth horizontal disc. A string tied
to P passes through a small hole O at the centre of the disc as shown in FIGURE 7.1. P
revolves about the centre with tangential linear speed of 5.0 m s1 in a circle of radius 50
cm. The string is then pulled downwards slowly until P travels in a circle of radius 30 cm.

a) Sketch the force acting on P as it travels in a circle.
b) Calculate the linear velocity of P in the new circle.

(8.33 ms-1)

104

d) Emphasize the relationship between total SHM energy &
amplitude
e) Apply velocity, acceleration, kinetic energy and potential
energy for SHM

105

Displacement in SHM Acceleration in SHM

Velocity in SHM

106

107

108

CHAPTER 9
SIMPLE HARMONIC MOTION

LEARNING OUTCOMES

9.1 Kinematics of Simple Harmonic Motion

1. Explain SHM
2. Solve problems related to SHM displacement equation, , x = A sin ωt

3. Apply equations:

i) velocity, v  dx   A2  x2
dt

ii) acceleration, a  dv  d2x   2 x
dt dt 2

iii) kinetic energy, K  1 m 2 ( A2  x2) and potential energy, U  1 m2 x2
2 2

4. Emphasize the relationship between total SHM energy & amplitude
5. Apply velocity, acceleration, kinetic energy and potential energy for SHM

9.2 Graphs of Simple Harmonic Motion

6. Discuss the following graphs:

i) Displacement - time
ii) Velocity – time
iii) Acceleration – time
iv) Energy - displacement

9.3 Period of Simple Harmonic Motion

7. Use expression for period of SHM, T for simple pendulum and single spring.

109

9.1 KINEMATICS OF SIMPLE HARMONIC MOTION
1 The amplitude of a system moving with SHM is doubled. The total energy E will then be

A. E
B. 2E
C. 3E
D. 4E

2 When the mass passes through the equilibrium position, its instantaneous velocity
A is maximum.
B is less than maximum, but not zero.
C is zero.
D cannot be determined from the information given

3 A weight of mass m is at rest at O when suspended from a spring as shown in FIGURE
9.1. When it is pulled down and released, it oscillates between positions A and B. Which
of the following statements about the system consisting of the spring and the mass is
CORRECT?

FIGURE 9.1
A. Gravitational potential energy of the system is greatest at A.
B. The elastic potential energy of the system is greatest at O.
C. The rate of change of momentum has its greatest magnitude at A and B.
D. The rate of change of gravitational potential energy is smallest at O.
4 In which group below do all the three quantities remain constant when a particle moves in
simple harmonic motion?
A. Acceleration, force, total energy
B. Force, total energy, amplitude
C. Total energy, amplitude, angular frequency
D. Amplitude, angular frequency, acceleration

110

5 The angle between the string of a pendulum at its equilibrium position and at its maximum
displacement is the pendulum’s

A. period.
B. frequency.
C. vibration.
D. amplitude.

6 For a system in simple harmonic motion, which of the following is the time required to
complete a cycle of motion?

A. amplitude
B. period
C. frequency
D. revolution

7 Measuring the length from the middle (equilibrium) of the pendulum to the highest point
of the swing, it is found to be 15 cm. What is the amplitude?

A. 30 cm
B. 1/15 cm
C. 15 cm
D. 1/30 cm

8 For a system in simple harmonic motion, which of the following is the number of cycles
or vibrations per unit of time?

A. amplitude
B. period
C. frequency
D. revolution

9 A ball moves in a circular path of diameter 0.15 m with constant angular speed of 20 rpm.
Its shadow performs SHM on the wall behind it. Calculate the acceleration and speed of
the shadow at

a) the turning point of the motion,

(v= 0 ms-1 , a = 0.329 ms-2)

b) the equilibrium position and a point 6 cm from the equilibrium position.

(v = 0.15 ms-1 , a = 0 ms-2)
(v = 9.42 x 10-2 ms-1 , a = 0.263 ms-2)

111

10 The position of a particle is given by the expression

x = (4.00 m) cos (3.00 t)

Where x in meters and t in seconds. Calculate

a) frequency, period and amplitude

(1.5Hz, 0.67s, 4.00 m)

b) the position of the particle at t = 0.25 s

(-2.83m)

11 A 50.0 g object connected to a spring with a force constant of 35.0 N m-1 undergoes SHM
on a horizontal, frictionless surface with amplitude of 4.00 cm. Calculate:
a) total energy of the system and speed of the object at x = 1.00 cm

(2.80 x 10-2J, 1.02 m s-1)
b) kinetic energy at x = 3.00 cm and potential energy x = 3.00 cm.

(1.22 x 102J, 1.58x10-2J )

12 A particle moving along the x axis in SHM starts from the origin at t=0 and moves to the
right. The amplitude of its motion is 2.00 cm and the frequency is 1.50 Hz.
a) maximum speed and the earliest time (t > 0) at which the particle has this speed
(18.8 cm s-1, 1/3s)
b) maximum acceleration and the earliest time (t > 0) at which the particle has this
acceleration.
(178cm s-2, 0.5 s)

9.2 GRAPHS oF SIMPLE HARMONIC MOTION

1 A graph of position versus time for an object oscillating at the free end of a horizontal
spring is shown in FIGURE 9.2. The point at which the object has negative velocity and
zero acceleration is

Q
PR

S

A. P FIGURE 9.2 R
B. Q S
C.
D.

112

2 The equation of motion for a particle oscillating in SHM is given as:
x  5sin 3t

where x is the displacement in cm

Determine: (5 cm, 2.094 s)
(a) Its amplitude and period (2.823 cm)
(b) Its displacement at time t = 0.2s
(c) Maximum velocity and acceleration (  0.15 ms-1,  0.45 ms-2)
(d) Velocity when its displacement is 4 cm (  0.09 ms-1)
(e) Sketch the graph of displacement against time

3 The graph below shows the forces acting on a particle of mass, m = 2kg.

What type of motion is the particle undergoing? Give reason for your answer.
(a) What is its amplitude?

(b) Calculate:

(i) angular velocity
(ii) period
(iii) maximum velocity

(0.1m, 10 rads-1, 0.628s, 1ms-1)

113

4

The figure showed the graph of displacement against time for an oscillation in SHM with
angular velocity 3 rads-1.
For this oscillator, sketch and label the graph of

a) Velocity against Time
b) Acceleration against Time

9.3 PERIOD oF SHM
1 How are frequency and period related in simple harmonic motion?

A. They are directly related.
B. They are inversely related.
C. Their sum is constant.
D. Both measure the number of cycles per unit of time.

2 A certain pendulum with a 1.00 kg bob has a period of 3.50 s. What will happen to the
period of the pendulum if the 1.00 kg bob is replaced by a bob with a mass of 2.00 kg?
A. Increase
B. Decrease
C. Stays the same
D. Need more information

3 A pendulum has a frequency of 4 Hertz (Hz). Find its period.
A. 4 sec
B. 2 sec
C. 1/2 sec
D. 1/4 sec

4 What is the period of oscillation of a mass of 40 kg on a spring with constant k = 10 N/m?
(12.57 s)

114

5 Determine the length of the SHM pendulum if period is 0.50s.

(0.06m)

6 A particle of mass 4 kg is vibrating in SHM. The graph for the potential energy U against
displacement x is shown in FIGURE 9.3. Calculate

U(J)
1.0

-0.2 0 0.2 x (m)

a) angular velocity

b) period.

(3.54 rad s-1 , 1.77s)

7 A block of unknown mass is attached to a spring with a spring constant of 6.50 N m-1 and

undergoes SHM with amplitude of 10.0 cm. When the block is halfway between its
equilibrium position and the endpoint, its speed is measured to be 30.0 cm s-1. Calculate:

a) mass of the block,

b) period of the motion and maximum acceleration.

(0.542 kg, 1.81s, 1.20 m s-2)

115

116

117

118

119

120

121

122

123

124

125

126

127

MECHANICAL
WAVES &
SOUND

128

CHAPTER 10
MECHANICAL AND SOUND WAVE

LEARNING OUTCOMES

10.1 Properties of Waves

(a) Define wavelength and wave number.
(b) Solve problem related to progressive wave equation,y (x,t) = A sin (ωt ± kx)

(c) Discuss and use particle vibrational velocity, vy= dy and wave propagation
dt

velocity, v = λf.

(d) Discuss the graphs of

i. Displacement-time, y-t

ii. Displacement-distance, y-x.

10.2 Superposition of Waves
(a) State the principle of superposition of waves for constructive and destructive
interferences.
(b) Use the stationary wave equation: y = A cos kx sin t
(c) Discuss progressive and standing wave.

10.3 Sound Intensity
(a) Define and use sound intensity, I  P .
A
(b) Discuss the dependence of intensity on:
i. amplitude : I  A²

ii. distance from a point source : I  1

r2

10.4 Application of Standing Waves
(a) Solve problems related to the fundamental and overtone frequencies for:
i. stretched string
ii. air columns (open and closed pipe)

(b) Use wave speed in a stretched string, = √

10.5 Doppler Effect

(a) State Doppler Effect for sound waves.

(b) Apply Doppler Effect equation = ( + ) for relative motion between source

+

and observer.

129

10.1 PROPERTIES OF WAVES

1. A waves of a certain frequency travels in a medium at constant speed. Which of
following is TRUE if the frequency is doubled, the speed is constant?

A. The amplitude is halved.
B. The amplitude is doubled.
C. The wavelength is halved.
D. The wavelentgh is doubled.

2. A student attaches one end of fishing line to a lamp post. At the other end he stretches it
outand performs a quick up and down motion to produce waves on the line. To increase
the wavelength he can

A. increase the tension on the hose and shake the end more times per second.
B. decrease the tension on the hose and shake the end more times per second.
C. increase the tension on the hose and shake the end fewer times per second.
D. decrease the tension on the hose and shake the end fewer times per second.

3. A sinusoidal wave travelling from right to left has an amplitude of 0.20 m, a wavelength

of 0.35 m and a frequency of 12.0 Hz. The transverse position of an element of the
medium at t = 0, x = 0 is y = – 0.30 m and the element has a positive velocity.

a) Calculate
i) wave number
[18.0 m-1]
ii) period
[0.083 s]
iii) angular frequency
[75.4 rad s-1]
iv) speed
[4.20 m s-1]

b) Write an expression for the progressive wave function y(x,t).

4. A transverse wave travelling on a taut wire has an amplitude of 0.20 mm and a frequency

of 500 Hz. It travels with a speed of 196 m s-1. The mass per unit length of this wire is

4.1 g m-1. Write an equation of the form y = A sin( t – kx) for this wave, where y
(a)

in meters and t in seconds.

(b) Calculate the tension in the wire.

[158 N]

130

5. A transverse sinusoidal wave on a string has a period T = 25.0 ms and travels in the
negative x direction with a speed of 30.0 m s-1. At t = 0, a particle on the string at x = 0
has a transverse position of 2.00 cm and is travelling downward with a speed of 2 m s-1.

Calculate the

(a) amplitude [2.15 x 10-2m ]
(b) maximum speed and wavelength [5.41 ms-1, 0.750 m]

6. The wave function for a traveling wave on a string (in S.I units) is given by

y(x,t)  0.35sin(10t  3x)

(a) What is the direction of the wave travel? [0.67 m, 5 Hz]
(b) What are the wavelength and frequency of the wave? [3.34 ms-1]

(c) What is the velocity of the wave?

7. Figure 10.1 shows a displacement, y against distance, x graph after time, t for the
progressive wave which propagates to the left with a speed of 20 cm s1.

(a) Determine the wave number and frequency of the wave. [100 πm-1, 10 Hz]
(b) Write the expression of displacement as a function of x and t for the wave

above.

y (cm)

6

01 2 3 x (cm)
4
 6 FIGURE 10.1

FIGURE 10.1

131

8. A sinusoidal wave traveling in the x direction (to the left) has an amplitude of 30 cm, a

wavelength of 20 cm and a frequency of 10 Hz. At t = 0, a particle at x = 0 has a

displacement of 0 cm.

a) Write an expression for the wave function, y(x,t).

b) Determine the speed and acceleration at t = 0.200 s for the particle on the

wave located at x = 10.0 cm.

[-1.89 kms-1, 0 ms-2]

9. If y = 0.02 sin (400t - 30x), calculate the frequency, wavelength, velocity, angular

frequency and wave number.

[ Hz, 2π/30 m, − , rad s-1, 30 m-1]


10. A progressive wave is represented by the equation ( , ) = 2 sin( − ) where y and
x are in centimeters and t in seconds.
a) Determine the angular frequency, the wavelength, the period, the frequency and the
wave speed.
[π rads-1, 2 cm, 2 s, 0.5 Hz, 1x10-2 ms-1]
b) Sketch the displacement against distance graph for progressive wave above in a range
of 0 ≤ ≤ at time, t = 0 s.
c) Repeat question (b) but for time, t = 0.5T.
d) Sketch the displacement against time graph for the particle at x = 0 in a range of
0 ≤ t ≤ T.
e) Sketch the displacement against time graph for the particle at x = 0.5λ in a range of
0≤ t ≤ T.

11.
y (cm)

3

0 x (cm)

1.0 2.0
3

Figure

FIGURE 10.2
Figure 10.2 shows a displacement, y against distance, x graph after time, t for the
progressive wave which propagates to the right with a speed of 50 cm s-1.

a) Determine the wave number and frequency of the wave.

[200π m-1, 50 Hz]

b) Write the expression of displacement as a function of x and t for the wave above.

132

10.2 SUPERPOSITION OF WAVES
1. Two pulses are travelling towards each other at 10 cm s-1 on a long string at t = 0 s, as

shown in FIGURE 10.3.

FIGURE 10.3
Which of the following correctly shows the shape of the string at t = 0.5 s?

2. Two wave pulses with equal positive amplitudes pass each other on a string; one is
travelling toward the right and the other toward the left. At the point that they occupy the
same region of space at the same time
A constructive interference occurs.
B destructive interference occurs.
C a standing wave is produced.
D a traveling wave is produced.

133

3. The principle of superposition states that
A the total energy of the resultant wave is the sum of the energy carried by the
individual wave
B two stationary waves superimpose to produce a progressive wave
C the frequency of the resultant wave is the difference between the frequencies of
the individual waves
D the displacement at a point of the resultant wave is the sum of the displacement of
the individual waves acting at that point.

4. State the differences between progressive wave and stationary wave. Give an example of
each wave.

5. Two sinusoidal waves travelling in opposite directions interfere to produce a standing

wave as

y  (1.50 m) cos(0.400 x)sin(200 t)

where y and x in meters and t in seconds. Calculate the

i) wavelength [15.7 m]
ii) frequency [31.8 Hz]
iii) speed [500 ms-1]

6. Two waves in a long string are travelling in opposite directions interfere to produce a
standing wave as
y =( 0.04m)cos( x ) sin( 40t )
2

where y1, y2 and x are in meters and t in seconds. Calculate: [π, 3π. 5π,...]
(i) positions of the nodes of the resulting standing wave. [0.039m]

(ii) maximum displacement of an element in the string at x = 0.400 m.

7. A stationary wave can be described by the equation

y  2Acoskxsin t

If the amplitude of the two individual waves that produces the stationary wave has
amplitude of 2 cm, wave number k = 1.57 cm-1 and angular frequency of 31.42 rads-1,

calculate

i) the distance between the first two antinodes

ii) the speed of the individual wave [2 cm]
[0.2 ms-1]

134

10.3 SOUND INTENSITY

1. The emission power of a sound source is 100W. Sound is emitted uniformly in all
direction. What is the intensity of sound at a distance 10m from the source?
[0.08 Wm-2]

2. a) Define sound intensity

b)

FIGURE 10.4

In FIGURE 10.4, 120µW of sound power passes perpendicularly through the surface
labeled 1 and 2. These surfaces have areas of A1 = 400 dm2 and A2 = 1200 dm2.
Determine the sound intensity at each surface.

[3x10-5 Wm-2, 1x10-5 Wm-2] ]

3. a) Explain the dependence of sound intensity with distance.
b) A point source produced spherical waves in 3D. The power emitted by the source
is 0.5 W. Calculate the
(i) intensity of the wave at a distance of 3 m from the source.
[4.42x10-3 Wm-2]
(ii) amplitude at a distance of 5 m from the source, if the amplitude of this wave at
a distance 3 m from the source is 2 cm.
[0.012 m]

4. In a KPop concert, a dB meter is registering 10 Wm-2 (130 dB) when placed 2.8 m in
front of a loudspeaker.
(i) Calculate the power output of the speaker.
[ ]
(ii) How far away would the sound level to be at 1x10-3 Wm-2 (90 dB).
[2.8 x 102 m]

5. A rocket in a fireworks display explored high in the air. The sound spreads out uniformly
in all directions. The intensity of the sound is 2.0x10-6 W m-2 at a distance 120 m from
the explosion. Find the distance from the source at which the intensity is
0.80x10-6 W m-2.
[189.74 m]

135

10.4 APPLICATION OF STANDING WAVES

1. A fundamental tone of a sound tone of a sound has a frequency 1MHz. Determine the

frequency of third overtone for odd harmonics exist.

A. 7MHz B. 8MHz

C. 10MHz D. 12MHz

2. When a student slowly blows across the open top end of a test tube, a sound note of the

fundamental frequency, fo is produced. If the student blows harder, the subsequent

frequencies of the sound notes will be

A 2 fo , 3 fo B 2 fo,4 fo

C 3 fo, 4 fo D 3 fo,5 fo

3. The tension in a taut rope is increased 4 times. How does the speed of wave pulses on the
rope change?
A. The speed remains the same.
B. The speed is reduced by half.
C. The speed is doubled.
D. The speed is increased 4 times.

4. An organ pipe of length 33 cm is open at one end and closed at the other. Calculate the
a) frequency of the fundamental note and the first overtone.
[251 Hz, 753 Hz]
b) length of a pipe open at both ends and having fundamental frequency that is equal
to the different between the two frequencies calculated in ( i.).
[0.33 m]

5. A variable length air column is placed just below a vibrating wire that is fixed at both
ends. The length of the air column open at open end is gradually increased until the first
position of resonance is observed at 34.0 cm. The wire is 120 cm long and is vibrating in
its third harmonic. Calculate the speed of transverse waves in the wire.
[194.70 ms-1]

6. Calculate the length of a pipe that has a fundamental frequency of 240 Hz if the pipe is
(i) closed at one end
[0.375 m]
(ii) open at both ends
[0.715 m]

7. The string of a violin has a fundamental frequency of 440 Hz. The length of the vibrating
string is 32 cm and has a mass of 0.35 g. Calculate the tension of the string.
[ . ]

8. Calculate the fundamental frequency of the sound emitted by an open-ends organ pipe
1.7m. (speed of sound in air = 340 m s1)
[100 Hz]

136

9. Calculate the fundamental frequency and the next three frequencies that could cause the
standing wave patterns on a string 30 cm long, has a mass per length of 9 ×
10−3 −1and is stretched to a tension of 20 N.
[78.57 Hz, 157.14 Hz, 235.71 Hz,314.28 Hz]

10.5 DOPPLER EFFECT

1. a) State Doppler Effect.
b) Hearing the siren of an approaching truck, moving with velocity of 80 km h-1 a
man pulls over to the side of the road and stop. As the truck approaching, he hears
a tone of 460 Hz. Calculate the
i) actual frequency sound by the truck.
[429 Hz]
ii) apparent frequency heard by the man when the truck passed him.
[402 Hz]

2. A train station bell gives off a frequency of 500 Hz as the train approaches the station at a
speed of 20 ms-1. Calculate the apparent frequency of the bell to an observer riding the
train.
[529.15 Hz]

3. An ambulance moving at a constant speed is passing a stationary observer at the side of
the road whistle its siren at frequency of 440 Hz. Calculate the speed of the ambulance if
the apparent frequency heard by the observer is 415 Hz.
[20.66 ms-1]

4. The siren of a fire engine emits a sound of 400 hz. What is the frequency observed by an
observer stationary at rest, if the fire engine is moving at 30 ms-1 when it
(a) is approaching observer.
[438.34 Hz]
(b) is moving away from observer.
[368 Hz]

5. The whistle from a stationary policeman at a junction emits sound of frequency 1000 Hz.
If the speed of sound is 330 m s-1, what is the frequency of the sound heard by a
passenger inside a car moving with a speed of 20 ms-1
(a) Towards the junction.
[1060 Hz]
(b) Away from the junction.
[939 Hz]

137

CHAPTER 11: Strain, e
DEFORMATION OF SOLIDS ratio of extension (elongation), e to
original length, l0 . OR
11.1 Stress and strain
Straienl,l0
Learning Outcome
l0 l0
 Distinguish between stress and strain
for tensile and compression force. This type of strain is called tensile strain.
Strain is a scalar quantity.
 Discuss the graph of stress-strain for Strain is unitless
metal under tension.
Graph of Stress-Strain
 Discuss elastic and plastic
deformation.

Stress, s and Strain, e Stress, Elastic Plastic deformation

Consider a rod that initially has uniform deformation D
cross-sectional area, A and length,
l0.Stretch the rod by applying the forces of E
equal magnitude F^ but opposite
directions at the both ends and the rod will ABC
extent by amount e as shown in Figure A.

l0 A Figure B Strain,
F F
OT
e
A : limit of proportionality
Figure A B : elastic limit
C : yield point
D : point of maximum force (stress)
E : breaking point

Stress, s OA
 The force (stress) increases linearly
ratio of the perpendicular force, F┴ to
the cross-sectional area, A. OR with the extension (strain) until
point A.
stress,   F
 Obeys Hooke’s law which states
A “Below the proportionality limit, the
restoring force, Fs is directly
where proportional to the extension, e.”

F : the forceact perpendicular to the crosssection Fs  ke

A: cross- sectional area Where, k is force(Hooke) constant

This type of stress is called tensile stress.  The negative sign indicates that the
Stress is a scalar quantity. restoring force is the opposite
The unit for stress is N m-2 or pascal (Pa). direction to increasing extension.

138

B The stress-strain graphs for various
 This is the elastic limit of the materials

material. Stress,
 Beyond this point, the material is
Steel
permanently stretched and will
never regain its original shape and Glass
length.
 If the force (stress) is removed, the Copper
material has a permanent extension Aluminium
of OT.
 The area between the two parallel line O Strain,ε
(AO and CT) represents the work Figure C
done to produce the permanent
extension OT Explanation for Figure C
 OB region is known as elastic  Rubber undergoes elastic
deformation.
deformation.
C  It is able to regain its original shape
 The yield point marked a change in
and length when the stress is
the internal structure of the material. removed but does not obey Hooke’s
 The plane (layer) of the atoms slide law.
 The strain produced when decreasing
across each other resulting in a the stress is greater than the strain
sudden increase in extension and produced when increasing the stress.
the material thins uniformly.  The shaded area is called the
hysteresis loop and it represents the
CDE energy loss per unit volume.
 This region is known as plastic  This energy lost in the form of heat
dissipation.
deformation.
 When the force (stress) increases, the

extension (strain) increases rapidly.

D
 The force (stress) on the material is

maximum and is known as the
breaking force (stress). This is
sometimes called the Ultimate
Tensile Strength (UTS).

E
 This is the point where the material

breaks or fractures.

139

Distinguish between elastic & plastic
deformation

Elastic Plastic
Deformation Deformation

1. Extension 1. Extension

-does not exceed the -exceeds the elastic
elastic limit of the limit of the material.
material.

2. When stretching 2. When stretching

force is removed force is removed

-wire returns to its -wire does not
original length. return to its original
length.
-wire is permanently Figure E
lengthened. Graph for rubber (elastic material)

3. Hooke’s law 3. Hooke’s law Figure F
-obey. -does not obey.

4. Internal 4. Internal
structure of a structure of a
wire wire
- does not change - change

5. When Hooke’s 5. Energy is
law is obeyed, converted into
the energy heat during
stored during plastic
elastic deformation.
deformation is
fully recovered
when the force is
removed.

Force-extension and stress-strain graphs
Graphs for metal (ductile material)

Figure D

140

11.2 Young’s modulus Relationship between force constant, k
and Young 's modulus, Y for a wire
Learning Outcome
From the statement of Hooke’s law and
 Define Young’s modulus. definition of Young modulus, thus
 Discuss strain energy from force-
F  ke and F  YAe
elongation graph. l0
 Discuss strain energy per unit volume
ke YAe
from stress-strain graph. l0
 Solve problems related to the Young’s

modulus.

Young's modulus, Y k  YA
l0
defined as the ratio of the tensile stress to
the tensile strain if the proportionality Strain energy
limit has not been exceeded. OR When a wire is stretched by a load(force),
work is done on the wire and strain (elastic
Y  Tensile stress, potential) energy is stored within.
Tensile strain,
F
 F  
Y A Proportiona
 e  lity limit
l0
Strain
Y  Fl0 energy
Ae
extension
 It is a scalar quantity.
0 Figure G
 Unit of Young modulus is N m-2 or
Pa. Consider the force-extension graph of
this wire until the proportionality limit
 Does not depend to the length of the (Hooke’s law) as shown in Figure G.
wire but it depend to the material
made the wire. The total work done, W in stretching a
wire from 0 to e is given by
 Does not change if the length of the
wire is increase or decrease.

Table show the value of Young’s
modulus for various material.

Material Young’s modulus,Y
(N m-2 @ Pa)
Aluminium 69.0
Copper 110.0
Steel 200.0
Nylon 3.7
Glass 70.0

141

Bulk modulus

From the definitions of tensile stress and A force applied uniformly over the surface
tensile strain, thus of an object will compress it uniformly.
This changes the volume of the object
This strain energy per unit volume is the without changing its shape.
area under the stress-strain graph until
the proportionality limit (straight line The stress in this case is simply described
graph) as shown in Figure H.
as a pressure (P = F/A). The resulting

volume strain is measured by the fractional
change in volume (θ = ∆V/V0).

The coefficient that relates stress to strain
under uniform compression is known as
the bulk modulus or compression modulus.

Its traditional symbol is K from the

German word kompression (compression)

but some like to use B from the English
word bulk — which is another word for

volume.

The Bulk Modulus Elasticity is a material

property characterizing the compressibility

of a fluid - how easy a unit volume of a

fluid can be changed when changing the

pressure working upon it.

Figure H B   p
V
( )
V

11.3 Bulk modulus The bulk modulus is a property of
materials in any phase but it is more
Learning Outcome common to discuss the bulk modulus for
solids than other materials.
 Define and use bulk modulus,
Compressibility
B   p .
V
( )
V
The reciprocal of bulk modulus is called
 Define and use compressibility,   1 compressibility. Its symbol is κ (kappa).
B
1
. B

A material with a high compressibility
experiences a large volume change when
pressure is applied.

The SI unit of compressibility is the
[Pa−1].
inverse pascal

142

CHAPTER 11

DEFORMATION OF SOLIDS

LEARNING OUTCOMES

11.1 Stress and Strain
a) Distinguish between stress and strain for tensile and compression force.
b) Discuss the graph of stress-strain for metal under tension.
c) Discuss elastic and plastic deformation.

11.2 Young’s Modulus
a) Define Young’s modulus.
b) Discuss strain energy from force-elongation graph.
c) Discuss strain energy per unit volume from stress-strain graph.
d) Solve problems related to the Young’s modulus.

11.3 Bulk Modulus

a) Define and use bulk modulus, B  p .
V
( )
V

b) Define and use compressibility,   1 .
B

143

11.1 Stress and Strain
11.2 Young’s Modulus
11.3 Bulk Modulus

1. A vertical wire of uniform cross-sectional area is stretched by adding masses at its
free end. Which of the following quantities does not affect the strain produced in the
wire?

A. Stress
B. Original length
C. Young’s Modulus
D. Density

2. Which of the following correctly gives examples of ductile and brittle materials?

Ductile Brittle
A. Rubber Steel
B. Steel Glass
C. Glass Copper
D. Copper Rubber

3. A wire is stretch until it undergoes plastic deformation. Which statement is true of
plastic deformation

A. The extension is proportional to force.
B. The wire returns to its original length when the force is removed.
C. The atomic planes in the wire slide over each other.
D. The bonds between atoms are all broken.

4. Shape of true stress-strain curve for a material depends on

A. Strain
B. Strain rate
C. Temperature
D. All

5. When a wire extended by a force, F the extension produced is x. If F=kx, the value of
the force constant k is effected by

A. the force F and extension x only
B. original length and the type of material only
C. force F, original length, and the extension x
D. original length, cross-sectional area, and the type of material

144

6. Which one of the following is true about Bulk Modulus of elasticity?

A. it is the ratio of compressive stress to volumetric strain.
B. it is the ratio of compressive stress to linear strain.
C. it is the ratio of tensile stress to volumetric strain.
D. it is the ratio of tensile stress to linear strain.

7. Which one of the following is the unit of compressibility?

A. m/N
B. m2/N
C. m3/N
D. it is unitless

8. The Young's modulus is 1.50 x 1010 N m-2 for a bone. The bone will fracture if its
stress is greater than 1.50 x 108 Nm-2 is imposed on it.
a) What is the maximum force that can be exerted on the femur bone in the leg if
it has a minimum effective diameter of 2.50 cm?
(73.6 kN)
b) If a 25.0 cm long bone is compressed, by how much will it be shortened?
(2.5 mm)

9. A 30.0 kg hammer with speed 20.0 m s-1strikes a steel spike 2.30 cm in diameter. The
hammer rebounds with speed 10.0 m s-1 after 0.110 s. What is the average strain in
the spike during the impact?
[ Ysteel = 2.00 x 1011 N m-2 ]
(9.85 x 105)

10.

5.0 kg 3.0 kg

FIGURE 11.1

A 2.00 m long steel wire with a diameter of 4.00 mm is placed over a light frictionless

pulley, with one end of the wire connected to a 5.00 kg object and the other end
connected to a 3.00 kg object as shown in FIGURE 11.1. By how much does the

wire stretch when the objects are in motion?
[ Ysteel = 2.00 x 1011 N m-2 ]

(29.3 µm)

145

11. A rubber cord of a catapult has a cross – sectional area of 2.0 mm2 and a length of
10.0 cm. It is stretched to 12.0 cm and then released to vertically launch a stone of
mass 30 g. Determine the velocity of the stone if the Young’s Modulus of rubber is
5.0 x 108 Nm-2.
(16.33 m s-1)

12. a) Explain what is elastic deformation and plastic deformation for a stretched
b) wire.
A piece of bone under tension and compression as in FIGURE 11.2

FIGURE 11.2
i) Which portion of the graph shows that the bone is brittle?
ii) Use the graph to calculate the Young’s modulus for bone under

conditions of compression and tension respectively.
(1.0 x 109 N m-2, 1.75 x 109 N m-2)

c) Given that the average cross sectional area of the bone is 6.0 × 10-4 m2 and its
length is 0.45 m.
i) By using the graph in (b), calculate the compressive force at the instant
the bone breaks
(12 kN)
ii) What is the reduction in length in the bone when the bone is just about
to break?
(9 mm)
iii) Calculate the energy stored in the bone when the bone is just about to
break.
(54 J)

146

13. a) What is meant by stress and strain in relation to a stretched wire? State the
relation between stress and strain and the condition for its validity.

b) The force F against elongation e graphs for wire X and wire Y as shown in
FIGURE 11.3. The wires have the same original length and same cross
sectional area of 4.0 mm2.

F(N) F(N)

300 750
200 500
100 250

0.5 1.0 1.5 e (mm) 2.0 4.0 6.0 e (mm)
Wire Y
Wire X

FIGURE 11.3

i) Calculate the work done to extend wire X and wire Y by 1.0 mm. State
which wire is more rigid
(0.1 J, 0.625 mJ)

ii) From the graphs above, calculate the ratio of the Young’ modulus of
wire X to the Young’ modulus of wire Y
(1.6)

14. The stress-strain graph of a type of rubber is as shown in the Figure 11.4 below.
When a piece of this type of rubber is placed between a vibrating machine and the
floor, the rubber becomes hot. Explain why the observation is consistent with the
graph below.

FIGURE 11.4
147

15. A metal bar that has a cross-sectional area of 4.0 cm2 and length of 50.0 cm is

stretched by a force of 1.0×104 N.

a) Find the stress on the wire.

(2.5x107 Nm-2)

b) If the bar elongates by 0.5 cm, what is the strain in the wire?

(0.01)

c) Determine the Young’s Modulus of the wire.

(2.5 x 109 Nm-1)

d) Calculate the strain energy per unit volume stored in the wire.

(125x103Jm-3)

148

149