Sample Problem: Simple Harmonic Motion
A particle of mass is performing simple harmonic motion with a maximum velocity of 4 m/s and maximum acceleration of 16 m/s^{2}. Find the amplitude and period of the motion.
Solution
The physical model in Fig. 1 shows the particle at the midpoint O, and at each end of its path, A and A?. Fig. 2 shows a table of the characteristics of the motion of the mass when at points A, O, A?
Characteristic 
At A

At O

At A?

Velocity 
0

Maximum

0

Direction of motion 
Towards O

Either direction

Towards O

Acceleration 
Maximum

0

Maximum

Direction in which acceleration is acting 
Towards O

Does not apply

Towards O

Mathematical equations are constructed involving relationships between mass, time, distance, velocity and acceleration:
From the standard simple harmonic motion equations:
(i) where w is a constant and x is the displacement from O
(ii) where a = amplitude = OA = OA?lt;/P>
(iii) T =
Given:
Maximum velocity = 4 m/s. This occurs when distance of the particle from the equilibrium point is 0 (i.e. when and acceleration =)
Equation (ii) becomes
The maximum acceleration is 16 m/s^{2}. This occurs when , i.e. when
x is a maximum Þ x = amplitude = a