Calculating Acceleration & Displacement in SHM (CIE A Level Physics)

Calculating Acceleration

• The acceleration of an object oscillating in simple harmonic motion is:

$a = -{\omega }^{2}x$

• Where:
  • a = acceleration (m s^-2)
  • ⍵ = angular frequency (rad s^-1)
  • x = displacement (m)

• This is used to find the acceleration of an object in SHM with a particular angular frequency ⍵ at a specific displacement x
• The equation demonstrates:
  • The acceleration reaches its maximum value when the displacement is at a maximum ie. x = x₀ (amplitude)
  • The minus sign shows that when the object is displacement to the right, the direction of the acceleration is to the left

The Graph of a = −⍵²x

The acceleration of an object in SHM is directly proportional to the negative displacement

• The graph of acceleration against displacement is a straight line through the origin and a negative gradient (similar to y = − x)
• Key features of the graph:
  • The gradient is equal to − ⍵²
  • The maximum and minimum displacement x values are the amplitudes −x₀ and +x₀

Calculating Displacement

• A solution to the SHM acceleration equation is the displacement equation:

x = x₀sin(⍵t)

• Where:
  • x = displacement (m)
  • x₀ = amplitude (m)
  • t = time (s)

• This equation can be used to find the position of an object in SHM with a particular angular frequency and amplitude at a moment in time
  • Note: This version of the equation is only relevant when an object begins oscillating from the equilibrium position (x = 0 at t = 0)

The Graph of x = x₀sin(⍵t)

This graph shows that at t = 0 then x = 0 and the oscillation of the object is the same as that of a sine curve