1. Advanced Higher Physics Unit 1
Simple Harmonic Motion
(SHM)

2. Oscillation
An oscillation is a regular to and fro movement around a zero point.
For example, a simple pendulum or a mass on a spring (horizontally or
Vertically).
zero point
zero point

3. Examples of SHM Example Ep stored as: Ek coming from: Pendulum
For each example of SHM, state how Ep is stored and where Ek is
coming from:
Example
Ep stored as:
Ek coming from:
Pendulum
Mass on a spring
Trolley moving between springs
Cork floating
Potential energy of mass
Moving mass
Elastic energy of spring
Moving mass
Elastic energy of springs
Moving trolley
Potential energy of cork
Moving cork

4. Period and frequency An oscillation has a:
period (T) which is the time taken for one complete cycle
Frequency (f) which is the number of cycles made per second.
Not in data booklet.
In data booklet
(higher part)
In data booklet.

5. Link between SHM and circular motion
turntable
shadow
SHM
Circular motion

6. Restoring force
For all oscillations there is an unbalanced force acting on the object,
and this force is opposite to the motion – so it is called a restoring
force.
If this restoring force is proportional to the displacement from the
zero position then the oscillation is called Simple Harmonic Motion. F
tension
Fun=0N
Zero position
weight F

7. Simple Harmonic motion
Where y is the displacement from the zero
position
Since F=ma and m is constant.
However as
and k is defined as w² for SHM
In data booklet

8. The constant is related to the period of the motion by
Therefore
In data booklet

9. Variation of period of oscillation with mass for a spring

10. Solution to the SHM equation
The expression for y which fits
requires
advanced calculus so we will limit ourselves to proving given solutions.
In data booklet
if y=0 at t=0
if y=A at t=0
In data booklet A
Zero
position
With A maximum amplitude of
Oscillation (from zero position)

11. Proof 1 You need to be able to derive this!
See data booklet for derivatives.

12. Proof 2 You need to be able to derive this!
See data booklet for derivatives.

13. Simple pendulum From s=rθ (circular motion) y=Lθ with θ in radians.
Hence: L θ y
The restoring force is mgsinθ.
Since F=ma, ma=-mgsinθ (as F and θ in
opposite directions)
a=-gsinθ
a=-gθ (if θ small) θ T
mgcosθ
(as )
mgsinθ mg

14. As and for SHM
hence
However
hence
You don’t need to be able to derive this!

15. Determination of “g”

16. Velocity at a given time
Consider the solution
Then
Rearranging these equations gives
and

17. Then:
See additional relationships in data booklet
You need to be able to derive this!
In data booklet.

18. Graphing SHM A
Zero position y A t -A

19. Graphing SHM A
Zero position y A t

20. Graphing SHM v A
Zero position y t A v t

21. Graphing v A
Zero position y t A
Maximum velocity is at y=0. v t

22. Graphing a A
Zero position y t A a t

23. Graphing a A Zero position y A t
Maximum acceleration is at y=A but is opposite in direction. a t

24. Energy and SHM
For mass shown: A
Zero
position
In data booklet

25. Assuming no energy loss, the total energy in the system is equal to the maximum kinetic energy.
This is when v is at a maximum; when y=0.
So the total energy is
Therefore, at displacement y the potential energy is:
In data booklet

26. Damping In reality, SHM oscillations come to rest because energy is
transferred from the system and so the amplitude of the oscillation
drops.
This is called DAMPING. y t

27. Critical damping
If the energy transfer is such that there is no displacement past
the zero point (assuming it starts at A) then the system is said to be
CRITICALLY DAMPED. y t
However, it is possible to overdamp the system (with very large
resistance) which means that the time it takes to stop moving is
actually larger than if there was no artificial damping and the
system is allowed to oscillate to rest.