Simple Harmonic Motion

This lesson covers: 

  1. The definition of simple harmonic motion (SHM) in relation to acceleration and displacement
  2. How to represent displacement, velocity, and acceleration in SHM through graphs
  3. The independence of frequency and period from amplitude in SHM
  4. Essential equations pertinent to SHM

Definition of simple harmonic motion

The acceleration of the object is proportional to the displacement and acts in the opposite direction to the displacement.

Graph showing the relationship between acceleration and displacement in simple harmonic motion with a negative gradient.

The negative gradient shows that the acceleration is always directed towards the equilibrium point position.

The acceleration in Simple harmonic motion is related to the displacement by the equation:


a = −ω2x


Where:

  • ω = angular frequency (rad s-1)
  • x = displacement from the midpoint (m)

Displacement during SHM

The variation of displacement with time can be plotted to show a sinusoidally repeating graph:

Graph showing the sinusoidal variation of displacement with time during simple harmonic motion.

The displacement can be expressed in terms of amplitude, angular velocity and time:


x = A cos(ωt)


Where:

x = displacement (m)

A = amplitude (m)

ω = angular frequency (rad s-1)

t = time (s)

Velocity during SHM

Graph showing the relationship between velocity and displacement in simple harmonic motion.

Velocity is 90° out of phase with displacement during simple harmonic motion.

  • velocity is maximum when displacement is zero
  • velocity is zero when the displacement is maximum


The velocity is related to the angular speed, displacement and amplitude by:


v = ±ω√A2−x2​


Where:

v = velocity (m s-1)

ω = angular frequency (rad s-1)

A = amplitude (m)

x = displacement (m)


The maximum velocity occurs when the displacement is zero:


vmax​ = ω A

Acceleration

Acceleration is 180° out of phase with displacement during simple harmonic motion.

Graph showing acceleration versus time in simple harmonic motion with maximum acceleration at amplitude.

The maximum acceleration occurs when displacement is equal to the amplitude:


amax​ = ω2 A


Where:

a = acceleration (m s-2)

ω = angular frequency (rad s-1)

A = amplitude (m)

Worked example - Determining maximum velocity in simple harmonic motion

Calculate the maximum velocity of an object undergoing SHM with an amplitude of 0.5 m.

The angular frequency is 10 rad s-1.


Step 1: Formula

vmax​ = ω A


Step 2: Substitution and correct evaluation

vmax​ = ω A = 10 × 0.5 = 5 m s−1