Topic 4 Oscillations and Waves Review Questions 2
Solutions
1.
1.
(a)
This question is about simple harmonic motion and waves.
A particle of mass m that is attached to a light spring is executing simple
harmonic motion in a horizontal direction.
State the condition relating to the net force acting on the particle that is
necessary for it to execute simple harmonic motion.
Answer : is proportional to the displacement/distance (of the particle) from
its equilibrium position;
it is directed towards the equilibrium position and opposite to the direction of
displacement;
(b)
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The graph shows how the kinetic energy EK of the particle in (a) varies with
the displacement x of the particle from equilibrium.
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(i)
Answer :
Using the axes above, sketch a graph to show how the potential energy
of the particle varies with the displacement x.
overall correct shape;
with max of 0.06 J at x = ± 0.05 and zero at x = 0;
2
2
(ii)
The mass of the particle is 0.30 kg. Use data from the graph and the
formula below to show that the frequency f of oscillation of the particle
is 2.0 Hz.
E K max 
1 2 2 2
4 π mf x 0
2
Answer :
1 2 2 2
4 π mf x 0 ;
2
from the graph E K max = 0.06 (J);
E K max 
and x0 = 0.050(m);
f=
2 E K max
4 π 2 mx 02
to give f = 2.0Hz
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or
k=
=
2 E K max
;
x 02
2  0.06
0.05 2
= 48;
use of f =
;
1
2π
k
;
m
= 2.0 Hz
(c)
The particles of a medium M1 through which a transverse wave is travelling,
oscillate with the same frequency and amplitude as that of the particle in (b).
(i)
Describe, with reference to the propagation of energy through the
medium, what is meant by a transverse wave.
Answer :
the energy of the wave is propagated in a direction at right angles
to the direction of oscillation of the particles;
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(ii)
The speed of the wave is 0.80 m s–1. Calculate the wavelength of the
wave.
Answer : v = λ f 
(d)
λ = 0.40m
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The diagram shows wavefronts of the waves in (c) incident on a boundary XY
between medium M1 and another medium M2.
The angle between the normal, and the direction of travel of the wavefronts is
30°.
(i)
The speed of the wave in M1 is 0.80 m s–1. The speed of the waves in
M2 is 1.2 m s–1.
Calculate the angle between the direction of travel of the wavefronts in
M2 and the normal.
Answer :
from data booklet: Snell’s law
use of sin θ2 =
v2
sin θ1;
v1

v2
 1.5this marking point is not necessary to award full credit.
v1

θ2 = [sin–1 0.75] = 49°;
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(ii)
On the diagram, sketch the wavefronts in M2.
Answer : Wave speeds up ! Be careful and compare with wave slowing
down
below.
Wave slowing down
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2.
This question is about the simple pendulum.
(a)
A pendulum consists of a bob suspended by a light inextensible string from a
rigid support. The pendulum bob is moved to one side and then released. The
sketch graph shows how the displacement of the pendulum bob undergoing
simple harmonic motion varies with time over one time period.
On the sketch graph above,
Answers:
(i)
label with the letter A a point at which the acceleration of the pendulum
bob is a maximum.
(ii)
label with the letter V a point at which the speed of the pendulum bob is
a maximum.
(i)
one A correctly shown;
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(ii)
one V correctly shown;
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remember:
Energy and the Pendulum
-X max
X=0
+ Xmax
v=0
vmax
+amax
a=0
-amax
+ Fmax
F=0
- Fmax
Ep max
Ep = 0
Ep max
Ek = 0
Ek max
Ek = 0
v=0
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(b)
The point of suspension of the pendulum bob is moved from side to side with
a small amplitude and at a variable driving frequency f.
For each value of the driving frequency a steady constant amplitude A is
reached. The oscillations of the pendulum bob are lightly damped.
(i)
On the axes below, sketch a graph to show the variation of A with f.
(2)
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Answer : (i)
one maximum shown and curve broadly similar to example above;
amplitude falls on each side as shown;
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Damping pp. 207 – 209
SHM is an important motion to study when considering oscillations but it does
not consider friction and other resistance forces. The effect of these forces on an
oscillating system is that the oscillations will eventually stop and the energy of the
system will be dissipated mainly as thermal energy to the environment and the
system itself.
Oscillations taking place in the presence of resistance forces are called damped
oscillations. The behavior of the system depends on the degree of damping.
There are three different types of damping:
1. Under-damping
2. Critical damping
3. Over –damping
Under damping
Whenever the resistance forces are small, the system will continue to oscillate
but with a frequency that is somewhat smaller than that in the absence of
damping. The amplitude gradually decreases until it approaches zero and the
oscillations stop. The amplitude decrease exponentially . Typical examples of
under-damped SHM are shown below.
Fig. 1.17 a) Light damping
Fig. 1.17 b) Heavy damping
Note that the period of oscillation in the heavy damping is SLIGHTLY larger than
the light damping.
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Note that heavy damping oscillations die out faster than light damping.
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(ii)
Explain, with reference to the graph in (d)(i), what is meant by
resonance.
Answer :
resonance is where driving frequency equals/close to natural
frequency; the frequency at the maximum amplitude of the graph;
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Forced Oscillations and Resonance p. 208 – 209 tsokos
Resonance - frequency of external force is equal to natural frequency of the
system. This results in oscillations with large amplitudes. Fig 1.20 p.208
Bridge video
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