1. 18 Simple Harmonic Motion
AQA Physics A Level 2nd Edition pp

2. Oscillations Examples All are movements about an equilibrium point
An object on a spring moving up and down
A pendulum swinging
A ball bearing rolling from side to side
A small boat rocking
A child on a swing
An oscillator in a microcomputer
A car going over a bump in the road
Rocking chair
All are movements about an equilibrium point

3. Oscillations
Regular movement about a point of minimum energy – the equilibrium point
Eventually energy is lost, motion reduces and the object comes to rest at the equilibrium point.
Amplitude – maximum displacement
Period – time taken to repeat motion, T [s]
Frequency = f =1/period=1/T [1/s = Hz] cycles per second
Free oscillation – when no frictional forces are present
Angular frequency – ω, is defined as [rad s-1]

4. Phase difference
Since the oscillation follows a wave pattern we can use the same Maths for both.
Angular Frequency, ω is in rad s-1
Use ωt as an angle – the phase angle ϴ
If we have a time difference ΔT the phase difference,
1 cycle difference -2π
½ cycle – π
¼ cycle - π/2
¾ cycle - 3π/2

5. Principles of SHM
The oscillating object speeds up as it moves towards equilibrium (gains KE, loses PE)
Slows down as it moves away from equilibrium
If there is no (or negligible) friction the amplitude is constant.
The displacement follows a sinusoidal pattern (cos or sin)

6. Variation of properties with time

7. Note Bene (NB)!
The velocity is the gradient of the displacement curve at the time t v(t) = ds(t)/dt
Velocity is greatest when the displacement is zero – when it pass equilibrium
The velocity is zero when the displacement is at its maximum ie (id est) when it is at amplitude
The acceleration is the gradient of the velocity curve at time t a(t) = dv(t)/dt
The acceleration is maximum when the displacement is maximum and the velocity is zero – and vice versa
The acceleration is in the opposite direction to the displacement

8. SHM definition SHM is defined as an oscillation in which:
the acceleration is proportional to the displacement
the acceleration is in the opposite direction to the displacement

9. Differentiating sine & cosine

10. Equations for waves
Displacement
Velocity
Acceleration

11. Circle and sine and cosine
The vertical and horizontal displacements of a point on a circular wheel are a sine and cosine
Pythagorus
Equation of a circle

12. Mass spring theory

13. Oscillations of mass spring systems
Often used to model solids
Modelling in buildings

14. Simple Pendulum

15. Simple harmonic motion
Acceleration is directly proportional to the displacement but in the opposite direction.
Resistive forces are always against motion.
Therefore for SHM there must be no resistive forces
Only achieved if swing is small for pendulums or displacement small for mass spring systems

16. Energy and SHM
An oscillating mass spring system has both potential , EP, and kinetic energy, EK.
If no work is done on the system we know the total energy, ET is constant.
Therefore ET = EP, + EK
But EP = ½ kx2 and EK = 0 at EP=ET= ½ kA2 where A is the maximum displacement or amplitude
Now we know that EK = ET – EP = ½k(A2-x2)
Now EK = ½ mv2 = ½k(A2-x2) but k/m=ω2
Therefore

17. Energy Displacement Graphs
Potential energy is a parabolic function ½ kx2
Similarly the kinetic energy function

18. Damped oscillations
If there are dissipative forces acting then energy is lost from the system. (N.B. If the energy loss is small then it still approximates to SHM)
Light damping
Heavy Damping
Critical damping

19. Car suspension Good example of critical damping.
No shock absorbers – continue to bounce up and down – light damping
Shock absorbers – critical damping – keeps force on the road.

20. Forced oscillations Free oscillations – no energy added, no loss
Forced oscillation – apply a periodic force
Natural Frequency 𝑓0 = frequency when free or when not driven
Driving Frequency , 𝑓
Energy is transfered to the oscillation preferentially at 𝑓 = 𝑓0
This condition is called resonance

21. Resonance As the applied frequency reaches the natural frequency
Amplitude increases
The phase difference between driving force and oscillation increases from zero to π/2

22. Barton’s Pendulum Watch the driving frequency and the phase.
Think about when the force is applied and the energy is transfered.

23. Examples of Resonance Fridge rattling Car panels/fitting rattling
Radio tuning
Water hammer
Guitar strings
Breaking a glass with sound (or possibly your compression drivers!)

24. Millennium Bridge
Originally natural frequency was near the walking frequency – resonance!
Fixed by adding more damping to change resonant frequency.