Simple Harmonic Motion
Learning outcomes
Teaching challenges
Students must first be familiar with radians, trigonometry and the small angle approximation .
A double maths challenge:
A brief history & contexts
The study of SHM started with Galileo’s pendulum experiments (1638, Two new sciences).
Huygens, Newton & others developed further analyses.
Getting a feel for SHM 1
Careful observation of a big pendulum.
There is an equilibrium (rest) position. Displacement, s, is the distance from equilibrium.
Discuss, in pairs
Getting a feel for SHM 2
A circus of experiments (in small groups)
Answer the same set of questions about changing velocities, accelerations and forces.
Describing oscillations
The system goes through a recurring cycle.
Amplitude, A, is the maximum displacement.
If the cycle repeat time is independent of amplitude (the oscillations is isochronous), then you can define a periodic time, T, and frequency, f.
A restoring force tries to return system to equilibrium.
The system has inertia and overshoots equilibrium position.
Damping
As a system loses energy, the amplitude falls.
Trolley between springs
k is the constant relating restoring force to displacement, m is mass
The - sign indicates a and s have opposite directions.
The acceleration is
How does displacement vary?
[empirically]
PP experiment:
Broomstick pendulum, sinusoidal motion
Obtain a graph of displacement against time
How does velocity vary?�How does acceleration vary?�
[empirical result by datalogging]
PP experiment:
Investigating a mass-on-spring oscillator
The SHM auxiliary circle
An imaginary circular motion gives a mathematical insight into SHM. Its angular velocity is ω.
Physlets animation ‘The connection between SHM and circular motion’
The time period of the motion,
The frequency of the motion,
Displacement of the SHM,
SHM equations
Displacement
At t = 0, the object is released from its max displacement.
The velocity, v, (rate of change of displacement) is then given by:
The acceleration, a, (rate of change of velocity) is:
Note that
Other symbols, relationships
Displacement,
At t =0, the object passes through its equilibrium position.
Phase relationship, φ
Try the Geogebra simulation.
Maximum values of quantities
Simple harmonic motion:
maximum displacement = A
maximum velocity = Aω = 2πfA
maximum acceleration = Aω2 = (2πf)2A
Circular motion:
displacement = r
velocity = rω
acceleration = rω2
Problems session 1
Advancing Physics (AP) ‘Quick check’, Q 1 – 4 only.
Two particular systems
Mass-on-spring,
Simple pendulum,
(for small angle oscillations)
Finding a spring constant
Method A
Gradually load the spring with weights, and find the extension (x = l – l0) at each load. Do not go beyond the elastic limit! � , so plot F against x and find k from the gradient.
Method B
Time 10 oscillations for a range of masses. Work out the mean period of oscillation at each mass.�
, so plot T2 against m and find k from the gradient.
Did you get the same result both ways?
Which method do you prefer?
Forced vibrations & resonance
Millenium Bridge video clip
Amplitude of response depends on 3 factors:
Experiments:
simulation The pendulum driven by a periodic force
The kinetic energy of a vibrating object = ½ mv2. The maximum kinetic energy = ½ mvmax2 = ½ mω2A2 (since vmax = ωA).
This makes clear that the energy of an oscillator is proportional to the square of its amplitude, A.
Problems session 2
In order of difficulty:
AP Quick Check Q 5,6
Practice in Physics questions on SHM
AP Oscillator energy and resonance
Energy and pendulums TAP 305-5
Endpoints
Video clip: The Tacoma Narrows bridge disaster
Puget Sound, Washington state, USA, November 1940