Simple Harmonic Motion

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.


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?


 At A

 At O

At A?





Direction of motion

Towards O

Either direction

Towards O





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 =


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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