1. Simple Harmonic Motion 3

2. Simple Harmonic Motion 2
Learning Objectives
Book Reference : Pages 36-37
To arrive at the relationship between displacement, velocity and acceleration for a system in SHM
To be able calculate the magnitude & direction of the acceleration at any time for an object in SHM

3. Circular Motion & SHM y
Consider an object in circular motion... At any given time the x and y coordinates are given by:
x = r cos 
y = r sin 
r cos 
r sin  r  x
Graphically the x coordinate changes as shown x +r
 / rad -r

4. Displacement 1
We have seen that the displacement of an object in SHM is described by a sinusoidal function. However, there are differences between where we start the motion at time t=0 x
Maximum displacement A at time t=0 (cosine) +A -A x
Zero displacement at time t=0 (sin) +A -A

5. Displacement 2
Providing that we operate in radians, the displacement x of an object in SHM is given by:
x = A cos 2ft (for displacement A at time t=0)
x = A sin 2ft (for zero displacement at time t=0)
Where A is the maximum displacement
Calculator must be in radians for this to work

6. Velocity The velocity of an object in SHM is given by:
v =  2f (A2 – x2)
Where v is the velocity, f is the frequency, A is the maximum displacement and x is the displacement
The velocity is considered positive if moving away from the equilibrium point and negative if moving towards it
Note this will be a maximum when x=0
vmax = 2fA

7. Maximum Acceleration
In a similar way the maximum value for acceleration given by the acceleration equation yesterday will be....
Acceleration = - (2f)2 x displacement
Accelerationmax = - (2f)2 A
Where f is the frequency, and A is the maximum displacement .

8. Problems 1
A spring oscillates in SHM with a period of 3s and an amplitude of 58mm. Calculate:
The frequency
The maximum acceleration
The displacement of an object oscillating in SHM changes with time and is described by
X (mm) = 12 cos 10t where t is the time in seconds after the object’s displacement was at its maximum value. Determine:
The amplitude
The time period
The displacement at t = 0.1s

9. Problems 2
An object on a spring oscillating in SHM has a time period of 0.48s and a maximum acceleration of 9.8m/s2. Calculate:
Its frequency
Its Amplitude
An object oscillates in SHM with an amplitude of 12mm and a period of 0.27s. Calculate
The frequency
Its displacement and direction of motion 0.1s, and 0.2s after the displacement was +12mm