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7. Simple Harmonic Motion
Problem
Solution
A particle moving with simple harmonic motion moves along a path with an acceleration which is:
- proportional to the distance of the particle from a fixed point in the path
- always directed towards the fixed point
Simple harmonic motion occurs in many mechanics problems eg those involving springs, elastic strings, pendulums, vibrating planes, etc.
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/s2. Find the amplitude and period of the motion.
Solution
The physical model in Fig. 1 shows the particle at the mid-point 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’:
Fig. 2
|
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’
(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/s2. This occurs when  , i.e. when 
x is a maximum Þ
x = amplitude = a
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