Section IV: Oscillations and waves
13. Oscillations
13.1
(a)
(b)
(c)
(d)
(e)
Simple harmonic oscillations
describe simple examples of free oscillations
investigate the motion of an oscillator using experimental and graphical methods
understand and use the terms amplitude, period, frequency, angular frequency and phase
difference and express the period in terms of both frequency and angular frequency
recognise and use the equation a = –ω2x as the defining equation of simple harmonic motion
recall and use x = x0 sin ωt as a solution to the equation a = –ω2x
recognise and use v = v0 cos ωt, 𝑣 = ±𝜔√(𝑥0 2 − 𝑥 2 )
describe, with graphical illustrations, the changes in displacement, velocity and acceleration
during simple harmonic motion
(f)
(g)
13.2 Energy in simple harmonic motion
(a) describe the interchange between kinetic and potential energy during simple harmonic
motion
13.3 Damped and forced oscillations, resonance
(a) describe practical examples of damped oscillations with particular reference to the effects of
the degree of damping and the importance of critical damping
(b) describe practical examples of forced oscillations and resonance
(c)
describe graphically how the amplitude of a forced oscillation changes with frequency near to
the natural frequency of the system, and understand qualitatively the factors that determine
the frequency response and sharpness of the resonance
(d) Appreciate that there are some circumstances in which resonance is useful and other
circumstances in which resonance should be avoided
Oscillations
Oscillations: A repetitive back-and-forth or up-and-down motion on both sides of an equilibrium
position.
Types of oscillations:

Free

Forced
Free oscillations: An oscillation whose frequency is the natural frequency of the oscillator.
Natural frequency:
13. Oscillations
The unforced frequency of oscillation of a freely oscillating object.
Physics – Patrick Ho
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Section IV: Oscillations and waves
Forced oscillations:
An oscillation caused by an external driving force and the frequency is equal to
that of the driving force.
Which of these oscillations are free or forced?

Wing beat of a mosquito

Vibrations of a cymbal after it has been struck

Shaking of a building during an earthquake
 Movement of the pendulum in an antique clock
Give two more examples each of free and forced oscillations.
Oscillations can be observed in experiments such as:

Mass-spring system

Pendulum

Loudspeaker cone
The displacement-time graph of oscillating systems is a sine curve, so the motion is described as
sinusoidal.
Revision: what is the amplitude, period and frequency of the sinusoidal graph below?
Phase describes the point that an oscillating mass has reached in a complete cycle.
Phase difference is the fraction of the wave cycle which has elapsed (occurred) relative to the origin
(or another wave). Phase difference can be measured as a fraction, in degrees, or in radians.
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Section IV: Oscillations and waves
Simple harmonic motion
Simple harmonic motion (shm):
Motion of a particle about a fixed equilibrium point, where its
acceleration is directly proportional to its displacement from its
equilibrium position, and is directed towards that position.
Requirements for s.h.m.:

A mass that oscillates

An equilibrium position

A restoring force that acts to return the mass to its equilibrium position
The magnitude of force is directly proportional to the displacement x of the mass from its
equilibrium position and is directed towards the equilibrium position. Thus, this causes the
acceleration to also be directly proportional to displacement x.
Examples of s.h.m.:
• Vibration of musical strings
• A single music note travels through the air, air molecules vibrate with shm
• Alternating current in antenna/aerial in the form of electrons
• Atoms in a molecule or solids
What are not s.h.m. oscillations?
Are these s.h.m. oscillations?
• A guitar string vibrating
• Rotation of the blades of a fan
• A single music note played on a flute
• A conducting sphere vibrating between two parallel, oppositely charged metal plates
• A basketball being bounced repeatedly on the ground
• A spinning top
• A hypnotist pendulum
Displacement, velocity and acceleration changes in s.h.m.
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Section IV: Oscillations and waves
𝑣=
𝛥𝑥
𝛥𝑡
𝑎=
𝛥𝑣
𝛥𝑡
Figure shows the displacement-time graph for an oscillating mass. Use the graph to find:
a) amplitude, period, and frequency.
b) the velocity when t = 0s
c) the maximum velocity
d) the acceleration when t = 3s
Angular frequency
To complete an oscillation, a full cycle of s.h.m. can be represented as 2𝜋 radians.
This means that the phase of the oscillation changes by 2𝜋 per oscillation.
If the frequency of the oscillation is f, this means that there are 2𝜋𝑓 radians per unit time.
Angular frequency of s.h.m.:
1
(𝑓 = 𝑇)
𝜔 = 2𝜋𝑓
2𝜋
𝜔= 𝑇
1.
An object with s.h.m. goes through 5 complete cycles in 2.0s. Calculate:
a)
period T
b)
Frequency f
c)
Angular frequency 𝜔
2.
An atom in a crystal vibrates with s.h.m. with a frequency of 10 12 Hz. The amplitude of its
motion is 4.0 x 10-12 m. Sketch a graph to show how the displacement of the atom varies
during one cycle. What is its angular frequency?
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Section IV: Oscillations and waves
Equations of s.h.m.
Since the displacement-time graph of an s.h.m. is sinusoidal, we compare it against the sine function.
After comparing, the displacement function of an s.h.m. can be represented by:
𝑥 = 𝑥0 sin 𝜔𝑡 OR 𝑥 = 𝑥0 cos 𝜔𝑡
1.
A pendulum oscillates with frequency 2.5 Hz and amplitude 0.3 m. It is at its maximum
displacement at t=0s.
a)
Write an equation to represent its displacement x in terms of amplitude, angular
frequency, and time.
b)
2.
Determine its displacement when t=0.5s.
The vibration of a component in a machine is represented by the equation:
𝑥 = 3.5𝑥10−2 sin 240𝜋𝑡
where the displacement 𝑥 is in metres. Determine the amplitude, frequency, and period of the
vibration.
Velocity is defined as the rate of change of displacement with time. Therefore,
𝑣=
Using trigonometry laws, this velocity equation can also be represented by: 𝑣 = ± 𝜔√(𝑥0 2 − 𝑥 2 )
Since maximum speed happens at x = 0, 𝑣 = ± 𝜔𝑥0
This shows that maximum speed of oscillation is proportional to the amplitude and frequency of
s.h.m.
Differentiation of velocity equation to obtain acceleration equation:
𝑎 = −𝜔2 𝑥
Conclusion from acceleration equation and graph:
• Acceleration does not depend on amplitude of oscillation
• Acceleration is always zero at equilibrium position
• Magnitude of acceleration is directly proportional to distance from equilibrium
• Acceleration depends on frequency of oscillation
• Acceleration against displacement is a linear graph
Therefore, the acceleration equation of s.h.m. can also be used to define s.h.m.
However, you must explain the symbols and the significance of the negative sign.
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Section IV: Oscillations and waves
1.
A mass secured at the end of a spring moves with s.h.m. The frequency of its motion is 1.4 Hz.
a)
Write an equation of the form 𝑎=−𝜔2𝑥.
b)
2.
A short pendulum oscillates with s.h.m. such that its acceleration 𝑎 is related to its
displacement 𝑥 by the equation 𝑎=−300𝑥. Determine:
a)
the frequency of the oscillations.
b)
3.
4.
Calculate the acceleration of the mass when it is displaced 0.05 m from its equilibrium
position.
the maximum speed of the oscillations.
The pendulum of a grandfather clock swings from one side to other in 1s. The amplitude of
the oscillation is 12cm. Calculate:
a)
The period of its motion, and its frequency.
b)
Write an equation of the form 𝑎=−𝜔2𝑥 to show how the acceleration of the pendulum
weight depends on its displacement.
c)
Calculate the maximum speed of the pendulum bob.
d)
Calculate the speed of the bob when its displacement is 6cm.
A trolley of mass m is fixed to the end of a spring. The spring can be compressed and
extended. The spring has a force constant k. The other end of the spring is attached to a
vertical wall. The trolley lies on a smooth horizontal table. The trolley oscillates when it is
displaced from its equilibrium position.
a)
Show that the motion of the oscillating trolley is s.h.m.
b)
𝑚
Show that the period T of the trolley is given by the equation: 𝑇 = 2𝜋√ 𝑘
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Physics – Patrick Ho
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Section IV: Oscillations and waves
Energy changes in s.h.m.
What happens to the energy-displacement graph
if the s.h.m. is given only half of the total energy?
What happens to the energy-displacement graph
if there is some potential energy in s.h.m. when it
is at its equilibrium position?
Figure shows how the velocity v of a 2kg mass was found to vary with time t during an investigation
of the s.h.m. of a pendulum. Use the graph to estimate the following for the mass:
a)
its maximum velocity
b)
its maximum kinetic energy
c)
Its maximum potential energy
d)
Its maximum acceleration
e)
The maximum restoring force that acted on it
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Section IV: Oscillations and waves
Damped Oscillations
In a damped oscillation, the frequency of oscillation
remains constant, but amplitude decreases due to loss
of energy.
What happens to its speed?
Resonance
Resonance: Oscillation that absorbs maximum energy from its driver and forced to oscillate at the
driver frequency, which is similar to its natural frequency, with maximum amplitude.
What happens to the resonance amplitude when there is damping?
List THREE example of situations where resonance is useful, and THREE example of situations where
resonance is a problem. In each case, state what is the oscillating system and what forces it to
resonate.
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