A LEVEL

17 Oscillations
17 Oscillations

Learning outcomes
By the end of this topic, you will be able to: 17.2 Energy in simple harmonic motion
17.1 Simple harmonic oscillations 1 describe the interchange between kinetic
and potential energy during simple
1 understand and use the terms displacement,
harmonic motion
amplitude, period, frequency, angular 1
frequency and phase difference in the 2 recall and use E = 2mω 2x02 for the total
context of oscillations, and express the energy of a system undergoing simple
period in terms of both frequency and harmonic motion
angular frequency 17.3 Damped and forced oscillations, resonance
2 understand that simple harmonic motion 1 understand that a resistive force acting on
occurs when acceleration is proportional to an oscillating system causes damping
displacement from a fixed point and in the 2 understand and use the terms light,
opposite direction critical and heavy damping and sketch
3 use a = – ω 2x and recall and use, as a displacement–time graphs illustrating these
solution to this equation, x = x0 sin ω t types of damping
4 use the equations v = v0 cos ω t and 3 understand that resonance involves a
v = ±ω√(x02 – x2) maximum amplitude of oscillations and that
5 analyse and interpret graphical illustrations this occurs when an oscillating system is
of the variations of displacement, velocity forced to oscillate at its natural frequency
and acceleration for simple harmonic motion

Starting points
★ An object that moves to-and-fro continuously is said to be oscillating or
vibrating.
★ Oscillations occur in many different systems from the very small (e.g. atoms)
to the very large (e.g. buildings).
★ Waves can be described by the quantities period, frequency, displacement and
amplitude.
★ Angles may be measured in radians (rad): 2π rad = 360°.

17.1 Simple harmonic oscillations

Some movements involve repetitive to-and-fro motion, such as a pendulum, the beating
of a heart, the motion of a child on a swing and the vibrations of a guitar string.
amplitude Another example would be a mass bouncing up and down on a spring, as illustrated in
equilibrium Figure 17.1. One complete movement from the starting or rest position, move up, then
position down and finally back up to the rest position, is known as an oscillation.
amplitude
The time taken for one complete oscillation or vibration is referred to as the period T
of the oscillation.
one
complete
oscillation The oscillations repeat themselves.
▲ Figure 17.1 Oscillation The number of oscillations or vibrations per unit time is the frequency f.
of a mass on a spring
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 Frequency may be measured in hertz (Hz), where one hertz is one oscillation per
second (1Hz = 1s−1). However, frequency may also be measured in min−1, hour−1, etc.
For example, it would be appropriate to measure the frequency of the tides in h−1. 17
Since period T is the time for one oscillation then
frequency f = 1/T

As the mass oscillates, it moves from its rest or equilibrium position.

The distance from the equilibrium position is known as the displacement.

17.1 Simple harmonic oscillations

This is a vector quantity and therefore has magnitude and direction relative to the
equilibrium position. Displacement may be on either side of the equilibrium position.
The amplitude (a scalar quantity) is the maximum displacement.

Some oscillations maintain a constant period even when the amplitude of the oscillation
changes. Galileo discovered this fact for a pendulum. He timed the swings of an oil lamp
in Pisa Cathedral, using his pulse as a measure of time. Oscillators that have a constant
time period are called isochronous, and may be made use of in timing devices.
For example, in quartz watches the oscillations of a small quartz crystal provide
constant time intervals. Galileo’s experiment was not precise, and we now know that a
pendulum swinging with a large amplitude is not isochronous.
The quantities period, frequency, displacement and amplitude should be familiar from
our study of waves in Topic 7. It should not be a surprise to meet them again, as the idea
of oscillations is vital to the understanding of waves.

Displacement–time graphs
It is possible to plot displacement–time graphs (as we did for waves in Topic 7.1) for
oscillators. One experimental method is illustrated in Figure 17.2. A mass on a spring
oscillates above a position sensor that is connected to a computer through a datalogging
interface, causing a trace to appear on the monitor.

position sensor interface

▲ Figure 17.2 Apparatus for plotting displacement–time graphs for a mass on a spring

The graph describing the variation of displacement with time may have different shapes,
depending on the oscillating system. For many oscillators the graph is approximately
a sine (or cosine) curve. A sinusoidal displacement–time graph is a characteristic of an
important type of oscillation called simple harmonic motion (s.h.m.). Oscillators which
move in s.h.m. are called harmonic oscillators. We shall analyse simple harmonic
motion in some detail, because it successfully describes many oscillating systems, both
in real life and in theory. Fortunately, the mathematics of s.h.m. can be approached
through a simple defining equation. The properties of the motion can be deduced from
the relations between graphs of displacement against time, and velocity against time,
which we met in Topic 2.

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 Simple harmonic motion (s.h.m.)
17 Simple harmonic motion is defined as the motion of a particle about a fixed point
such that its acceleration a is proportional to its displacement x from the fixed
point, and is in the opposite direction.

Note that, since the acceleration and the displacement are in opposite directions then
acceleration is always directed towards the fixed point from which displacement is measured.
Mathematically, we write this definition as
17 Oscillations

a = – ω 2x

where ω 2 is a constant. We take the constant as a squared quantity, because this will
ensure that the constant is always positive (the square of a positive number, or of a negative
number, will always be positive). Why worry about keeping the constant positive? This is
because the minus sign in the equation must be preserved. It has a special significance,
because it tells us that the acceleration a is always in the opposite direction to the
displacement x. Remember that both acceleration and displacement are vector quantities, so
the minus sign is shorthand for the idea that the acceleration is always directed towards the
fixed point from which the displacement is measured. This is illustrated in Figure 17.3.

acceleration
displacement

point from
which displacement
is measured
acceleration
displacement

▲ Figure 17.3 Directions of displacement and acceleration are always opposite

The defining equation is represented in a graph of a against x as a straight line, of

negative gradient, through the origin, as shown in Figure 17.4. The gradient is negative
a
because of the minus sign in the equation. Note that both positive and negative values for
the displacement should be considered.
The square root of the constant ω 2 (that is, ω) is known as the angular frequency of the
oscillation. This angular frequency ω is related to the frequency f of the oscillation by
0 x the expression
ω = 2πf
where one complete oscillation is described as 2π radians.
Since period T is related to frequency f by the expression
▲ Figure 17.4 Graph of the
defining equation for
frequency f = 1/T
simple harmonic motion

then
2π
angular frequency ω =
T

By Newton’s second law, the force acting on an object is proportional to the acceleration
of the object. The defining equation for simple harmonic motion can thus be related
to the force acting on the particle. If the acceleration of the particle is proportional to

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 its displacement from a fixed point, the resultant force acting on the particle is also
proportional to the displacement. We can bring in the idea of the direction of the
acceleration by specifying that the force is always acting towards the fixed point, or by
calling it a restoring force.
17
Solution of equation for simple harmonic motion
In order to find the displacement–time relation for a particle moving in a simple
harmonic motion, we need to solve the equation a = − ω 2x. To derive the solution
requires mathematics which is beyond the requirements of Cambridge International

17.1 Simple harmonic oscillations

AS & A Level Physics. However, you need to know the form of the solution. This is

x = x0 sin ω t

where x0 is the amplitude of the oscillation and at time t = 0, the particle is at its
equilibrium position defined as displacement x = 0. The variation with time t of the
displacement x for this solution is shown in Figure 17.5.

MATHS NOTE
There are actually two solutions to the defining equation of simple harmonic motion,
a = − ω 2 x, depending on whether the timing of the oscillation starts when the
particle has zero displacement or is at its maximum displacement. If at time t = 0
the particle is at its maximum displacement, x = x0, the solution is x = x0 cos ω t (not
shown in Figure 17.5). The two solutions are identical apart from the fact that they
are out of phase with each other by one quarter of a cycle or π/2 radians.
The variation of velocity with time is sinusoidal if the cosinusoidal displacement
solution is taken:
v = − v0 sin ω t when x = x0 cos ω t

x
x = x0sin ω t
x0

0
t

– x0

▲ Figure 17.5 Displacement–time curve for simple harmonic motion

In Topic 2.1 it was shown that the gradient of a displacement–time graph may be
used to determine velocity at any point (the instantaneous velocity) by taking a tangent
to the curve. Referring to Figure 17.5, it can be seen that, at each time at which
x = x0, the gradient of the graph is zero (a tangent to the curve would be horizontal).
Thus, the velocity is zero whenever the particle has its maximum displacement. If we
think about a mass vibrating up and down on a spring, this means that when the spring
is fully stretched and the mass has its maximum displacement, the mass stops moving
downwards and has zero velocity. Also from Figure 17.5, we can see that the gradient of
the graph is at a maximum whenever x = 0. This means that when the spring is neither
under- or over-stretched the speed of the mass is at a maximum. After passing this
point, the spring forces the mass to slow down until it changes direction.

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 If a full analysis is carried out, it is found that the variation of velocity with time is
cosinusoidal when the displacement is sinusoidal. This is illustrated in Figure 17.6.
17 v
v = v0 cos ω t

v0

0
t

–v0
17 Oscillations

a0

0
t
–a0
a = –a0 sin ω t

▲ Figure 17.6 Velocity–time and acceleration–time graph for simple harmonic motion

The velocity v of the particle is given by the expression

v = v0 cos ω t when x = x0 sin ω t

There is a phase difference between velocity and displacement. The velocity curve is
π/2 rad ahead of the displacement curve. The maximum speed v0 is given by

v0 = x0ω

There is an alternative expression for the velocity:

v0 = ±ω ( x02 − x 2 )

which is derived and used in Topic 17.2.

For completeness, Figure 17.6 also shows the variation with time of the acceleration a of
the particle. This could be derived from the velocity–time graph by taking the gradient.
The equation for the acceleration is

a = −a0 sin ω t when x = x0 sin ω t

WORKED EXAMPLE 17A

The displacement x at time t of a particle moving in simple harmonic motion is
given by x = 0.36 sin 10.7t, where x is in metres and t is in seconds.
a Use the equation to find the amplitude, frequency and period for the motion.
b Find the displacement when t = 0.35 s.

Answers
a Compare the equation with x = x0 sin ω t. The amplitude x0 = 0.36 m.
The angular frequency ω = 10.7 rad s−1. Remember that ω = 2πf, so the
frequency f = ω /2π = 10.7/2π = 1.7 Hz. The period T = 1/f = 1/1.7 = 0.59 s.
b Substitute t = 0.35 s in the equation, remembering that the angle ω t is
in radians and not degrees. ω t = 10.7 × 0.35 = 3.75 rad = 215°.
So x = 0.36 sin 215° = −0.21 m.
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 Question 1 A mass oscillating on a spring has an amplitude of 0.20 m and a period of 1.5 s.
a Deduce the equation for the displacement x if timing starts at the instant when 17
the mass has its zero displacement.
b Calculate the time interval from t = 0 before the displacement is 0.17 m.

17.2 Energy in simple harmonic motion

Kinetic energy

17.2 Energy in simple harmonic motion

In Topic 17.1, we saw that the velocity of a particle vibrating with simple harmonic
motion varies with time and, consequently, with the displacement of the particle.
For the case where displacement x is zero at time t = 0, displacement and velocity are
given by
x = x0 sin ωt
and
v = x0ω cos ωt or v = v0 cos ωt
There is a trigonometrical relation between the sine and the cosine of an angle θ, which
is sin2 θ + cos2 θ = 1. Applying this relation, we have
x 2/x02 + v2/x02ω2 = 1
which leads to
v2 = x02ω2 − x 2ω2
and so

v = ±ω ( x02 − x 2 )

The kinetic energy of the particle (of mass m) oscillating with simple harmonic motion is
1
2
mv2. Thus, the kinetic energy Ek at displacement x is given by
1
Ek = 2mω 2(x02 − x 2)
The variation with displacement of the kinetic energy is shown in Figure 17.7.
Ek

x
–x0 0 x0

▲ Figure 17.7 Variation of kinetic energy in s.h.m.

Potential energy
The defining equation for simple harmonic motion can be expressed in terms of
the restoring force Fres acting on the particle. Since F = ma and a = − ω 2 x then at
displacement x, this force is
Fres = −mω2x
where m is the mass of the particle. To find the change in potential energy of the particle
when the displacement increases by Δx, we need to find the work done against the
restoring force.

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 The work done in moving the point of application of a force F by a distance Δx is FΔx.
In the case of the particle undergoing simple harmonic motion, we know that the
17 restoring force is directly proportional to displacement. To calculate the work done
against the restoring force in giving the particle a displacement x, we take account
of the fact that Fres depends on x by taking the average value of Fres during this
1
displacement. The average value of force is just 2mω 2 x, since the value of Fres is zero at
x = 0 and increases linearly to mω 2 x at displacement x. Thus, the potential energy Ep at
displacement x is given by average restoring force × displacement, or
1
Ep = 2mω 2x 2
17 Oscillations

The variation with displacement of the potential energy is shown in Figure 17.8.
Ep

x
–x0 0 x0

▲ Figure 17.8 Variation of potential energy in s.h.m.

Total energy
The total energy Etot of the oscillating particle is given by
Etot = Ek + Ep
1 1
= 2mω 2(x02 − x 2) + 2mω 2x 2

1
Etot = 2mω 2x02

This total energy is constant since m, ω and x0 are all constant. We might have expected
this result, as it merely expresses the law of conservation of energy.
The variations with displacement x of the total energy Etot, the kinetic energy Ek and the
potential energy Ep are shown in Figure 17.9.
E
Etot

Ep

Ek

–x0 x0 x
0
▲ Figure 17.9 Energy variations in s.h.m.

WORKED EXAMPLE 17B

A particle of mass 95 g oscillates in simple harmonic motion with angular frequency
12.5 rad s−1 and amplitude 16 mm. Calculate:
a the total energy
b the kinetic and potential energies at half-amplitude (at displacement
x = 8.0 mm).

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 Answers
a Using Etot = 2mω 2x02,
1

1
17
  Etot = 2
× 0.095 × 12.52 × (16 × 10 −3)2
= 1.90 × 10−3 J
(Don’t forget to convert g to kg and mm to m.)
1
b Using Ek = 2mω 2(x02 − x2)
1
Ek = 2 × 0.095 × 12.52 × [(16 × 10 −3)2 − (8 × 10 −3)2]
   = 1.43 × 10−3 J

17.3 Damped and forced oscillations, resonance

Using Etot = Ek + Ep
1.90 × 10 −3  = 1.43 × 10 −3 + Ep
Ep = 0.47 × 10−3 J

Question 2 A particle of mass 0.35 kg oscillates in simple harmonic motion with frequency
4.0 Hz and amplitude 8.0 cm. Calculate, for the particle at displacement 7.0 cm:
a the kinetic energy
b the potential energy
c the total energy.

17.3 Damped and forced oscillations, resonance

A particle is said to be undergoing free oscillations when the only external force
acting on it is the restoring force.
There are no forces to dissipate energy and so the oscillations have constant amplitude.
Total energy remains constant. This is the situation we have been considering so far.
Simple harmonic oscillations are free oscillations.

In real situations, however, frictional and other resistive forces cause the
oscillator’s energy to be dissipated, and this energy is converted eventually into
thermal energy. The oscillations are said to be damped.

The total energy of the oscillator decreases with time. The damping is said to be light
when the amplitude of the oscillations decreases gradually with time. This is illustrated
in Figure 17.10. The decrease in amplitude is, in fact, exponential with time. The period
of the oscillation is slightly greater than that of the corresponding free oscillation.
displacement

0
time

▲ Figure 17.10 Lightly damped oscillations

Heavier damping causes the oscillations to die away more quickly. If the damping is
increased further, then the system reaches critical damping point. Here the displacement
decreases to zero in the shortest time, without any oscillation (Figure 17.11, overleaf).

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 displacement
17 overdamped

0
time
critically
damped
17 Oscillations

▲ Figure 17.11 Critical damping and ▲ Figure 17.12 Vehicle suspension system
overdamping showing springs and dampers
Any further increase in damping produces overdamping or heavy damping. The
displacement decreases to zero in a longer time than for critical damping (Figure 17.11).
Damping is often useful in an oscillating system. For example, vehicles have springs between
the wheels and the frame to give a smoother and more comfortable ride (Figure 17.12).
If there was no damping, a vehicle would move up and down for some time after hitting a
bump in the road. Dampers (shock absorbers) are connected in parallel with the springs so
that the suspension has critical damping and comes to rest in the shortest time possible.
Dampers often work through hydraulic action. When the spring is compressed, a piston
connected to the vehicle frame forces oil through a small hole in the piston, so that the
energy of the oscillation is dissipated as thermal energy in the oil.
Many swing doors have a damping mechanism fitted to them. The purpose of the
damper is so that the open door, when released, does not overshoot the closed position
with the possibility of injuring someone approaching the door. Most door dampers
operate in the overdamped or heavily damped mode.

Forced oscillations and resonance

When a vibrating object undergoes free (undamped) oscillations, it vibrates at its natural
frequency. We met the idea of a natural frequency in Topic 8, when talking about stationary
waves on strings. The natural frequency of such a system is the frequency of the first mode
of vibration; that is, the fundamental frequency. A practical example is a guitar string,
plucked at its centre, which oscillates at a particular frequency that depends on the speed of
progressive waves on the string and the length of the string. The speed of progressive waves
on the string depends on the mass per unit length of the string and the tension in the string.
Vibrating objects may have periodic forces acting on them. These periodic forces will
make the object vibrate at the frequency of the applied force, rather than at the natural
frequency of the system. The object is then said to be undergoing forced vibrations.
Figure 17.13 illustrates apparatus which may be used to demonstrate the forced vibrations
of a mass on a helical spring. The vibrator provides the forcing (driving) frequency and has
a constant amplitude of vibration.
thread to signal
pulley operator

vibrator

helical
spring

mass

▲ Figure 17.13 Demonstration of forced oscillations

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 As the frequency of the vibrator is gradually increased from zero, the mass begins
to oscillate. At first the amplitude of the oscillations is small, but it increases with
increasing frequency. When the driving frequency equals the natural frequency of
oscillation of the mass–spring system, the amplitude of the oscillations reaches a
17
maximum. The frequency at which this occurs is called the resonant frequency,
and resonance is said to occur.
Resonance occurs when the natural frequency of vibration of an object is equal to
the driving frequency, giving a maximum amplitude of vibration.

17.3 Damped and forced oscillations, resonance

If the driving frequency is increased further, the amplitude of oscillation of the mass
decreases. The variation with driving frequency of the amplitude of vibration of the
mass is illustrated in Figure 17.14. This graph is often called a resonance curve.

amplitude
amplitude

light damping
heavy damping

0 resonant driving
0 resonant driving frequency frequency
frequency frequency
▲ Figure 17.14 Resonance curve ▲ Figure 17.15 Effect of damping on the resonance
curve

The effect of damping on the amplitude of forced oscillations can be investigated by

attaching a light but stiff card to the mass in Figure 17.13. Movement of the card gives
rise to air resistance and thus damping of the oscillations. The degree of damping may
be varied by changing the area of the card. The effects of damping are illustrated in
Figure 17.15. It can be seen that, as the degree of damping increases:
» the amplitude of oscillation at all frequencies is reduced
» the frequency at maximum amplitude shifts gradually towards lower frequencies
» the peak becomes flatter.

Barton’s pendulums may be used to demonstrate resonance and the effects of damping.
The apparatus consists of a set of light pendulums, made (for example) from paper
cones, and a more massive pendulum (the driver), all supported on a taut string.
The arrangement is illustrated in Figure 17.16. The lighter pendulums have different
lengths, but one has the same length as the driver. This has the same natural frequency
as the driver and will, therefore, vibrate with the largest amplitude of all the pendulums
(Figure 17.17).

l
l

▲ Figure 17.17 Time-

exposure photographs
of Barton’s pendulums driver
with light damping,
taken end-on. The
longest arc, in the
middle, is the driver. ▲ Figure 17.16 Barton’s pendulums
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 Adding weights to the paper cones reduces the effect of damping. With less damping,
the amplitude of the resonant pendulum is much larger.
17 There are many examples in everyday life of forced oscillations causing resonance.
One of the simplest is that of pushing a child on a swing. We push at the same
frequency as the natural frequency of oscillation of the swing and child, so that the
amplitude of the motion increases.
The operation of the engine of a vehicle causes a periodic force on the parts of the
vehicle, which can cause them to resonate. For example, at particular frequencies of
rotation of the engine, the mirrors may resonate. To prevent excessive vibration, the
17 Oscillations

mountings of the mirrors provide damping.

A spectacular example of resonance that is often quoted is the failure in 1940 of the first
suspension bridge over the Tacoma Narrows in Washington State, USA. Wind caused
the bridge to oscillate. It was used for months even though the roadway was oscillating
with transverse vibrations. Approaching vehicles would appear, and then disappear, as
the bridge deck vibrated up and down. One day, strong winds set up twisting vibrations
(Figure 17.18) and the amplitude of vibration increased due to resonance, until eventually
the bridge collapsed. The driver of a car that was on the bridge managed to walk to safety
before the collapse, although his dog could not be persuaded to leave the car.

▲ Figure 17.18 The Tacoma Narrows bridge disaster

EXTENSION
Musical instruments rely on resonance to amplify the sound produced. The sound
from a tuning fork is louder when it is held over a tube of just the right length,
so that the column of air resonates. We met this phenomenon in Topic 8.1, in
connection with the resonance tube method of measuring the speed of sound in air.
Stringed instruments have a hollow wooden box with a hole under the strings which
acts in a similar way. To amplify all notes from all of the strings, the sounding-box
has to be a complex shape so that it resonates at many different frequencies.

SUMMARY
» The period of an oscillation is the time taken to » Simple harmonic motion (s.h.m.) is defined
complete one oscillation. as the motion of a particle about a fixed point
» Frequency is the number of oscillations per unit such that its acceleration a is proportional to
time. its displacement x from the fixed point, and
» Frequency f is related to period T by the is directed towards the fixed point, a ∝ −x or
expression f = 1/T. a = − ω 2x.
» The displacement of a particle is its distance (in a » The constant ω in the defining equation for
stated direction) from the equilibrium position. simple harmonic motion is known as the angular
» Amplitude is the maximum displacement. frequency.

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 » For a particle oscillating in s.h.m. with frequency f, » The total energy Etot of a particle of mass m

»
then ω = 2πf and T = 2π/ω .
Simple harmonic motion is described in terms
oscillating in simple harmonic motion with
angular frequency ω and amplitude x0 is 17
of displacement x, amplitude x0, frequency f, and 1
Etot = 2mω 2x02 .
angular frequency ω by the following relations. » For a particle oscillating in simple harmonic motion
displacement: x = x0sin ω t or x = x0cos ω t Etot = Ek + Ep
velocity: v = x0ω cos ω t or v = −x0ω sin ω t or and this expresses the law of conservation of
v = ±ω √(x02 − x2) energy.
acceleration: a = −x0ω 2 sin ω t or a = −x0ω 2 cos ω t. » Free oscillations are oscillations where there

End of topic questions

» Remember that ω = 2πf, and the equations above are no resistive forces acting on the oscillating
may appear in either form. system.
» For a particle oscillating in s.h.m., graphs of the » Damping is produced by resistive forces which
displacement, velocity and acceleration are all dissipate the energy of the vibrating system.
sinusoidal but have a phase difference. » Light damping causes the amplitude of vibration
» Velocity is out of phase with displacement by of the oscillation to decrease gradually. Critical
π/2 radians, meaning velocity is zero at maximum damping causes the displacement to be reduced
displacement and maximum when displacement to zero in the shortest time possible, without any
is zero. oscillation of the object. Overdamping or heavy
» Acceleration is out of phase with displacement damping also causes an exponential reduction
by π radians, meaning acceleration is maximum in displacement, but over a greater time than for
at maximum displacement but in the opposite critical damping.
direction. » The natural frequency of vibration of an object is
» The kinetic energy Ek of a particle of mass m the frequency at which the object will vibrate when
oscillating in simple harmonic motion with angular allowed to do so freely.
1
frequency ω and amplitude x0 is Ek = 2mω 2(x02 − x2) » Forced oscillations occur when a periodic driving
where x is the displacement. force is applied to a system which is capable of
» The potential energy Ep of a particle of mass vibration.
m oscillating in simple harmonic motion with » Resonance occurs when the driving frequency
1
angular frequency ω is Ep = 2mω 2x2 where x is the on the system is equal to its natural frequency of
displacement. vibration. The amplitude of vibration is a maximum
at the resonant frequency.

END OF TOPIC QUESTIONS

1 A particle is oscillating in simple harmonic motion with period 2.5 ms and
amplitude 4.0 mm.
At time t = 0, the particle is at the equilibrium position. Calculate, for this particle:
a the frequency,
b the angular frequency,
c the maximum speed,
d the magnitude of the maximum acceleration,
e the displacement at time t = 0.8 ms,
f the speed at time t = 1.0 ms.
2 A spring stretches by 69 mm when a mass of 45 g is hung from it. The spring is
then stretched a further distance of 15 mm from the equilibrium position, and the
mass is released at time t = 0.
When the spring is released, the mass oscillates with simple harmonic motion of
period T.
The period T is given by the expression
T = 2π √(m/k)
where k is the spring constant and m is the mass on the spring.
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 Calculate:
a the spring constant,
17 b the amplitude of the oscillations,
c the period,
d the displacement at time t = 0.20 s.
3 One particle oscillating in simple harmonic motion has ten times the total energy
of another particle, but the frequencies and masses are the same. Calculate the
ratio of the amplitudes of the two motions.
4 A ball is held between two fixed points A and B by means of two stretched springs,
17 Oscillations

A B
ball as shown in Fig. 17.19.
The ball is free to oscillate along the straight line AB. The springs remain
▲ Figure 17.19
stretched and the motion of the ball is simple harmonic. The variation with time t of
the displacement x of the ball from its equilibrium position is shown in Fig. 17.20.

2
x / cm

0
0.2 0.4 0.6 0.8 1.0 1.2
t/s
–1

–2

▲ Figure 17.20

a i Use Fig. 17.20 to determine, for the oscillations of the ball:

1 the amplitude, [1]
2 the frequency. [2]
ii Show that the maximum acceleration of the ball is 5.2 m s−2. [2]
b Use your answers in a to plot, on a copy of Fig. 17.21, the variation with
displacement x of the acceleration a of the ball. [2]
a /m s–2

0 x /10–2 m

▲ Figure 17.21

c Calculate the displacement of the ball at which its kinetic energy is equal to one
half of the maximum kinetic energy. [3]
Cambridge International AS and A Level Physics (9702) Paper 43 Q3 May/June 2013

284

482807_17_CI_AS_Phy_SB_3e_272-285.indd 284 30/05/20 6:54 PM

 5 A metal plate is made to vibrate vertically by means of an oscillator, as shown in
Fig. 17.22.

sand
17
direction of
plate
oscillations

oscillator

End of topic questions

▲ Figure 17.22

Some sand is sprinkled on to the plate.

The variation with displacement y of the acceleration a of the sand on the plate is
shown in Fig. 17.23.

−2
a/m s
4

–10 –8 –6 –4 –2 0 2 4 6 8 10
y/mm
–1

–2

–3

–4

–5
▲ Figure 17.23

a i Use Fig. 17.23 to show how it can be deduced that the sand is undergoing
simple harmonic motion. [2]
ii Calculate the frequency of oscillation of the sand. [2]
b The amplitude of oscillation of the plate is gradually increased beyond 8 mm.
The frequency is constant.
At one amplitude, the sand is seen to lose contact with the plate.
For the plate when the sand first loses contact with the plate:
i state the position of the plate, [1]
ii calculate the amplitude of oscillation. [3]
Cambridge International AS and A Level Physics (9702) Paper 41 Q2 May/June 2018

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