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HKAL Extra Exercise : Part 1 Mechanics
Chapter 5 Simple Harmonic Motion
5.1 The Kinematics of SHM (Mathematical Approach)
Displacement
1.
Diagram NOT to scale
mid-point of OB is
A. T / 2.
B. T / 3.
C. T / 6.
20 cm
D. T / 8.
An object performs simple harmonic
3. An object moves vertically with simple
motion. A ticker-tape stuck to it records its
harmonic motion just behind a wall. From
positions at 0.010 s intervals for the first
the other side of the wall the object is
2.
half cycle. The maximum speed of the
object is
A. 3.14 m/s
B. 24.67 m/s
C. 49.35 m/s
D. 98.70 m/s
A particle performs S.H.M. between two
points A and B with period T. If O is the
center of oscillation, the shortest time for
the particle to move from point B to the
visible in each cycle for 3.0 s and hidden
behind the wall for 13.0 s. The maximum
height reached by the object relative to the
top of the wall is 0.60 m. The amplitude of
the motion is
A. 1.20 m.
B. 2.40 m.
C. 3.46 m.
D. 3.60 m.
Phase
4. Which of the following statements is true
of the acceleration of a particle oscillating
with S.H.M.?
A. It is always in the opposite sense to
the velocity of the particle.
B. Its magnitude is a minimum when the
displacement of the particle is a
maximum.
C.
D.
motion, which of the following is/are
correct ?
(1) Displacement from the equilibrium
position is π out of phase with the
acceleration.
(2) Acceleration is π out of phase with
the velocity.
(3) Displacement from the equilibrium
position is π/ 2 out of phase with
It increases as the potential energy
increases.
It varies linearly with the frequency
of oscillation.
6.
5.
When a body performs a simple harmonic
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
the velocity.
A. (2) only
B. (3) only
C. (1) and (3) only
D. (1), (2) and (3)
A body is moving with simple harmonic
motion about point O. Which of the
1/12
following graphs represents the variation
of its velocity v from O, with its
acceleration a?
A.
Q time
B.
v
0
v
a
C.
0
a
D.
v
0
A.
P leads Q by

.
2
B.
Q leads P by

.
2
a
0
C.
P leads Q by

.
4
D.
Q leads P by

.
4
a
An object is placed on a horizontal platform vibrating vertically in S.H.M. with an amplitude
of 4.0 cm. The maximum period of oscillation which will allow the object to remain in contact
with the platform throughout the motion is
A.
B.
C.
D.
9.
The waveforms in the figure show the
time variation of two physical quantities P
and Q. What is the phase relationship
between P and Q ?
v
7.
Acceleration
8.
P
0.4 s
0.5 s
4s
indeterminate
Both the velocity and the acceleration of an object changes with time. The motion of the object
can be
(1) uniform circular motion.
(2) simple harmonic motion.
(3) free fall motion.
A.
B.
C.
D.
(1) only
(3) only
(2) and (3) only
(1), (2) and (3)
5.3 The Dynamics of SHM
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
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Horizontal Block-spring system
10.
Load
m
S2
S1
11.
S2
X
S1
A block of mass m is attached to two
identical springs S1 and S2 as shown. The
force constant of the springs is k. If the
block is made to executes simple
harmonic motion, the period will be
A.
2
m
2k
B.
2
2m
k
C.
2
m
4k
D.
2
4m
k
P
A trolley attached to two fixed supports S1
and S2 by identical springs is displaced
from the equilibrium position along the
direction X and set into oscillation. A load
is dropped onto and is retained by the
trolley when it passes through its
equilibrium position P. Which of the
following statements is/are correct ?
(1) The amplitude of oscillation increases
after the landing of the load.
(2) Linear momentum in the horizontal
direction is conserved just before and after
the load lands on the trolley.
(3) The period decreases after the landing of
the load.
A.
B.
C.
D.
Vertical Block-spring system
12. A body is suspended by a spring and
allowed to swing as a simple pendulum.
When it is moved from the equator to the
north pole, its period will
A. decrease.
B.
C.
D.
increase.
remain constant.
increase and then decrease.
C
13. A small mass is hung vertically from a
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
(1) only
(2) only
(2) and (3) only
(1), (2) and (3)
light spring fixed at its upper end. When
the mass is pulled down 4 cm from its
equilibrium position and released from
rest, it takes 0.6 s to rise back to its
equilibrium position. If the mass is pulled
down 5 cm from its equilibrium position
and released from rest, how long does it
take for the mass to rise 4 cm ? (Assume
that the spring obeys Hooke’s law.)
A. 0.52 s
B. 0.35 s
C. 0.25 s
D. 0.08 s
14. A small block of mass 0.3 kg is suspended
from the ceiling by a light spring of force
3/14
constant 8 N m-1. If the block is projected
vertically downwards with a speed of 1.5
string
performs
simple
harmonic
oscillations in a vertical plane. The tension
m s-1 from its equilibrium position, what is
the maximum acceleration of the block in
its subsequent motion ?
A. 7.8 m s-2
B. 8.0 m s-2
C. 8.7 m s-2
D. 9.0 m s-2
in the string
(1) is independent of the amplitude of the
oscillations.
(2) is independent of m.
(3) has its maximum value when the bob
is at its lowest point.
A. (1) only
B. (3) only
C. (1) and (3) only
D. (1), (2) and (3)
15. A bob of mass m supported by an elastic
16. Two identical pans, each of mass 100 g, are connected by a light string which passes over a
light pulley suspended from the ceiling. The pulley can rotate smoothly about a horizontal axis
through its centre. Two identical weights m1 and m2, each of mass 20 g, are placed on the pans
as shown in Figure 2.1. (Neglect air resistance.)
m1
pan
m2
Figure 2.1
(a) In the space provided below, sketch and label all the forces acting on both m1 and the pan
supporting it. Identify an action and reaction pair.
(3 marks)
pan
m1
(b) Write down all the possible state(s) of motion that the system can take.
(1 mark)
Initially the separation between the pans is 0.94 m. Now m2 is removed from the lower pan and
the system accelerates from rest.
m1
O
x
0.36 m
Figure 2.2
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
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(c) Show that the acceleration of m1 is given by a 
mg
where m and M are the masses
m  2M
of the weight and the pan respectively. Calculate the tension in the string.
(d) Find the speed of the pans when they are at the same level.
(3 marks)
(2 marks)
(e) A light spring of force constant 8 N m-1 is fixed vertically below the descending pan as
shown in Figure 2.2. A light plate is attached to the upper end of the spring. The
descending pan comes into contact with the plate when the two pans are at the same level.
The motion of the system becomes simple harmonic until it comes to rest momentarily.
With the contact point taken as the origin, find the equilibrium position and the angular
frequency of the motion. (Assume that the pan moves together with the plate once they
are in contact and the string does not slack throughout.)
(4 marks)
Simple Pendulum
17.
18.

A simple pendulum is swinging in a
vertical plane. When it is at the position
shown, which of the following diagrams
best represents the forces acting on the
bob ? Neglect air friction.
A.
B.
T
mg
A simple pendulum is displaced an angle
θ and is released from rest. If T is the
tension in the string and m is the mass of
the bob, which of the following statements
is/are correct ?
(1) The restoring force of the harmonic
motion is T tanθ.
(2) At the moment when the bob is released, T
cosθ= mg.
C.
D.
(3) The period of oscillation is independent of
θ when θ is small.
A.
C.
D.
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
(1) only
(1) and (3) only
(1), (2) and (3)
B.
(3) only
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19. A student used a simple pendulum to
measure acceleration due to gravity at the
21. The period of a simple pendulum
undergoing simple harmonic motion may
earth’s surface. The experimental value
was found lower than the standard value.
Which of the following is a possible
reason for this ?
A. The effect due to air resistance
is not negligible.
B. The experiment has been
performed at a place above
sea-level.
C. The experiment has been
be decreased by
A. using a heavier pendulum bob.
B. increasing the amplitude of
oscillation.
C. placing the pendulum at the
North pole.
D. lengthening the string attached
to the bob.
D.
22.
performed at a place below
sea-level.
The stop watch used for the
experiment runs too fast.
O
P
20.
A
P
The figure shows a small heavy bob P
attached to a fixed point A on the ceiling
by a light inextensible string. The bob is
pulled aside with the string taut and then
released from rest. Which of the following
descriptions is/are true ?
(1) When moving towards the lowest point of
its path, The centripetal acceleration of the
bob is increasing.
(2) When the bob is at the lowest point, the
tension in the string equals the centripetal
force
(3) The angular speed of the bob is constant.
A. (1) only
B. (3) only
C. (1) and (2) only
D. (1), (2) and (3)
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
The figure shows a swinging simple
pendulum, starting from a point higher
than P. Which of the following forces
is/are acting on the pendulum bob when it
is at P ? (Neglect air resistance)
(1) a force exerted by the string pointing
towards O
(2) a centrifugal force pointing away
from O
(3) a third force along the direction of
motion of the bob
A. (1) only
B. (3) only
C. (1) and (2) only
E.
(1), (2) and (3)
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5.4 Energy in SHM
Energy versus displacement
23. The tip of each prong of a tuning fork,
emitting a sound wave of frequency 350
Hz, has an amplitude of 0.7 mm. What is
the speed of each tip when its
displacement is 0.6 mm?
A. 0.79 m/s
B. 1.54 m/s
C. 3.39 m/s
D. 15.39 m/s
26. A particle of mass 0.5 kg moves with
S.H.M. of amplitude 0.2 m. If the total
energy of the particle is 0.01 J, then its
period of motion is
A. 2π s.
B. π s.
C. π/2 s.
D. π/4 s.
27. A point mass is attached to the lower end
24. An object undergoes a simple harmonic
motion with an amplitude A, and its total
energy is E. What is the displacement of
the object from the equilibrium position
when its kinetic energy is
A.
A
4
B.
3A
4
C.
A
2
D.
3A
2
E
?
2
25. A particle oscillates with simple harmonic
motion along a straight line with
amplitude A. W1hen the displacement of
the particle from its equilibrium position is
A
, its speed is u. Thke speed of the
2
particle when passing the equilibrium
position is
A.
2u / 3 .
B.
C.
D.
u/ 2.
2 u.
3 u.
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
of a light spring fixed at the upper end.
The mass is made to oscillate vertically. If
the potential energy of the system is taken
to be zero when the mass is at its
equilibrium position, the speed of the mass
at the equilibrium position is directly
proportional to the square root of
(1) the amplitude of oscillation.
(2) the maximum potential energy of the
system.
(3) its mass
A. (2) only
B. (1) and (2) only
C. (2) and (3) only
D. (1), (2) and (3)
28. A metre rule is clamped horizontally to the
edge of a bench so that most of its length
overhangs and it is free to vibrate with
vertical simple harmonic motion. The tip
of the rule vibrates with an amplitude of
3.0 cm and a maximum speed of 1.5 m s-1.
What is the frequency of vibration of the
rule ?
A. 6.5 Hz
B. 7.0 Hz
C. 7.5 Hz
D. 8.0 Hz
7/14
29. A 80-g mass suspended from a light
helical spring oscillates with vertical
On a smooth horizontal surface, a block
connected to the wall with a light spring
simple harmonic motion of amplitude 3.5
cm. If the maximum kinetic energy of the
mass is 4.5 × 10-3 J, its frequency of
performs simple harmonic motion of
amplitude A as shown. If the amplitude is
oscillation is
A. 0.9 Hz.
B. 1.5 Hz.
C. 1.8 Hz.
D. 2.1 Hz.
30. A body hanging on a light spring oscillates
vertically between levels X and Z as
shown below. Its
position is at level Y.
static
equilibrium
X
Y
Z
Which of the following statements is/are
correct ?
(1) The acceleration of the body is zero when
it is at level Z.
(2) The strain energy of the spring is
maximum when the body is at level Z.
(3) The net force acting on the body is at its
maximum when it is at level Z.
A. (1) only
B. (3) only
C. (2) and (3) only
D.
(1), (2) and (3)
31.
A
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
reduced to A/ 2 , which of the following
quantities would be halved ?
(1) the maximum velocity of the block
(2) the period of oscillation of the block
(3) the maximum elastic potential energy
stored in the spring
A. (1) only
B. (2) only
C.
D.
(1) and (3) only
(1), (2) and (3)
32. For an object oscillating with simple
harmonic motion, which of the following
quantities will reach the maximum value
when the object has its acceleration
maximum ?
(1) the speed of the object
(2) the restoring force acting on the
object
(3) the total potential energy of the
system
A. (1) only
B. (3) only
C. (1) and (2) only
D. (2) and (3) only
33. The maximum speed of a simple harmonic
oscillator is 0.5 m s-1 and its amplitude is
1.0 m. What is the displacement of the
oscillator from the equilibrium position
when its speed is 0.4 m s-1?
A. 0.3 m
B. 0.4 m
C. 0.6 m
D. 0.8 m
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34.
A.
K
B. Displacement
C. Force
D. Speed
Displacement
E
a
0
b
x
Acceleration
Acceleration
Speed
x
An object is attached to a light spring
which does not obey Hooke’s law. The
mass is set oscillating so that the system
has a constant total mechanical energy E.
The graph above shows the variation of
the kinetic energy K of the mass with the
extension x of the spring. The object will
experience a force of maximum magnitude
at
A. x = (a + b)/2 only.
B. both x = a and x = b.
C. x = a only.
D. x = b only.
36.
a
0
y
x
The above graph shows the variation of
the acceleration a of a particle with its
displacement x from a fixed point. Which
of the following graphs shows the
variation of its kinetic energy K.E. with
x?
A.
B.
K.E.
0
35.
C.
K.E.
0
y
Kinetic energy
K.E.
x 0
x
D.
K.E.
x
0
x 0
x
The graph describes the motion of a
simple harmonic oscillator. Which pair of
physical quantities is most likely
represented by variables y and x ?
5.5 Damped Oscillation
37. Which of the following physical quantities will decrease with time in damped harmonic
motion ?
(1) Amplitude
(2) Maximum speed (3) Mechanical energy
A. (1) only
B. (3) only
C. (2) and (3) only
D. (1), (2) and (3)
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
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38. A light spring is suspended from a fixed point and a vertical metre rule is placed nearby as
shown in Figure 1.1. A pointer adhered to the lower end of the spring indicates a reading of 5.0
cm. A pan is now attached to the spring. The reading x indicated by the pointer is recorded for
different masses m added to the pan. The result is tabulated as follows :
5.0 cm
x
m
metre
rule
pan
Figure 1.1
m/g
0
30
60
90
120
x / cm
10.1
19.9
30.1
39.9
50.0
(a) (i) Plot a graph of m against x on the graph paper printed on page 2.
(3 marks)
(ii) What is the physical significance of the slope of the graph ? Use the graph, or
otherwise, to find the force constant of the spring, in N m-1, and the mass of the pan.
(4 marks)
(b) All the masses are removed from the pan and a piece of plasticine of mass 35 g is dropped
from rest at a small distance above the pan. The plasticine sticks to the pan and the system
then oscillates with simple harmonic motion.
(i) Find the motion’s period and its equilibrium position indicated by the pointer.
(Assume that the air resistance is negligible.)
(4 marks)
(ii) If, instead, the air resistance is significant, sketch the variation of the displacement of
the system from the equilibrium position with time for a few oscillation cycles. (No
need to draw the scales.)
(2 marks)
displacement/m
0
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
time/s
10/14
5.6 Forced Oscillations
Resonance
39. A mass is suspended by a light spring fixed to the ceiling. When the mass-spring system
undergoes vertical oscillation freely, its frequency is fo . If the system is removed from the
ceiling and forced to oscillate vertically by a periodic driving force of variable frequencies,
which of the following graphs best represents the relationship between its frequency of
oscillation, f, and the applied frequency, fa ?
A.
B.
C.
f
f
0
fo
fa
D.
f
0
fo
fa
f
0
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
fo
fa
0
fo
fa
11/14
40. A system oscillates under the influence of
an external periodic driving force. Which
of
the
following
statements
is
INCORRECT?
A. At resonance the power transferred from
the driving force to the system is a minimum.
B. In steady state the system vibrates at the
frequency of the driving force.
C. The amplitude of vibration becomes very
large when the frequency of the driving force is
close to the natural frequency of vibration of
the system.
41. A system A oscillating at its natural
frequency fA is coupled to a system B of
natural frequency fB, and causes system B
to oscillate. When the steady state is
reached, which of the following
statements is/are correct ?
(1) The rate of transfer of energy from system
A to B is high when fA is much higher than fB.
(2) System B oscillates at frequency fA.
(3) The amplitude of system B depends on the
difference of fA and fB.
A. (1) only
D. At resonance the displacement of the
system is π/2 out of phase with the driving
B. (1) and (2) only
C. (2) and (3) only
force.
D. (1), (2) and (3)
42. A piece of iron is suspended from a vertical spring. The iron (but not the spring) is immersed in
a jar of water, and oscillates with period T0. A vertical sinusoidal force of variable period T is
now applied to the iron using an electromagnet. Which one of the following statements is
correct ?
A. For any value of T, the water temperature rises, due to energy transferred from the
electromagnet.
B. When the electromagnetic is switched off, the period of the oscillations remains T.
C. For T not close to T0, the forced oscillations decrease slowly in amplitude due to damping.
D. The amplitude of the oscillations increases greatly when T is increased far away from T0.
Barton’s Pendulum
43.
paper cone
pendulums
(loaded with
metal rings)
A
string
B
C
heavy
bob
The figure shows a variation of Barton’s pendulum. A, B and C are three paper cone
pendulums of different lengths suspended from the string. The heavy bob is pulled well aside
and released so that it oscillates on a plane perpendicular to the paper. The paper cone
pendulums are forced into oscillations. When the motion settles down after a short time, which
of the following statements is/are correct ?
(1) All the paper cone pendulums are oscillating with the same frequency.
(2) B and C are approximately in phase.
(3) C has the largest amplitude among the three paper cone pendulums.
A. (1) only
B. (3) only
C. (1) and (2) only
D. (1), (2) and (3)
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
12/14
HKAL Extra Exercise : Part 1 Mechanics
Chapter 5 Simple Harmonic Motion (KEY)
1 – 10 : DCCCC, CAADB,
11 – 15 : BCAAC,
16. (a)
33
tension
reaction on m1
pan
m1
action on pan
weight of pan
(b) (Rest or) uniform motion
(c)
(m + M)g – T = (m + M)a
&
T – Mg = Ma
On sloving, a =
weight of m1
11
(or mg = (m + 2M)a)
mg
m  2M
= 0.91 m s-2
& T = Mg + Ma
= 1.09 N
2
2
v = u + 2as
v2 = 2(0.91)(0.18)
(d) By
1
13
1
-1
v = 0.57 m s
(e) At equilibrium,
mg = kx0
0.03 × 10 = 8x0
x0 = 0.0125 below O
For s.h.m., ω =
8
0.02  2  0.1
= 6.0 rad s-1
38. (a) (i)
12
1
1
k
m  2M
=
17 – 20 : DCDA,
1
1
14
21 – 30 : CAACC, AADCC, 31 – 37 : BDCDB AD,
Axes labeled with appropriate scales
Points correctly plotted
Correct graph
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
1
1
13
13/14
m/g
100
80
60
40
20
0
10
20
30
40
x/cm
-20
(ii) Slope × g gives the force constant k
1
3
Slope =
(120  0)  10
(50  10)  10 2
1
 k = 3 (N m-1)
(b) (i)
mo = 15 g (from graph or by calculation)
(m + mo) g = ke
(0.035 + 0.015)(10) = 3e
i.e. x = 0.17 m or 17 cm (or read from the graph)
T = 2
= 2
1
14
1
1
m  mo
k
1
0.05
= 0.81 s
3
14
(ii)
22
displacement
0
time
39 – 40 : DA, 41 – 43 : CBC.
HKAL Extra Exercise_Chapter 5 Simple Harmonic Motion
14/14