Calculating Speed in SHM (CIE A Level Physics)

• The speed of an object in simple harmonic motion varies as it oscillates back and forth
  • Its speed is the magnitude of its velocity

SHM solution equation for speed

• The speed of an oscillator released from the equilibrium position in SHM is defined by:

v = v₀ cos(⍵t)

• Where:
  • v = speed (m s^-1)
  • v₀ = maximum speed (m s^-1)
  • ⍵ = angular frequency (rad s^-1)
  • t = time (s)

Interpreting the graph v = v₀ cos(⍵t)

The variation of the speed of a mass on a spring in SHM over one complete cycle

• This is a cosine function because the object starts oscillating from the equilibrium position (x = 0 when t = 0)
  • This is shown on the SHM graphs revision note
• Although the symbol v is commonly used to represent velocity, not speed, exam questions focus more on the magnitude of the velocity than its direction in SHM
• The maximum speed of an oscillator is the amplitude, v₀ of the velocity-time graph

SHM equation for speed

• The speed v changes with the oscillator’s displacement x according to the equation:

v = ±ω

• Where:
  • v = speed (m s^-1)
  • x₀ = amplitude (m)
  • ± = ‘plus or minus’, this value can be positive or negative
  • ⍵ = angular frequency (rad s^-1)
  • x = displacement (m)

• This equation shows that when an oscillator has a greater amplitude x₀, it has to travel a greater distance in the same amount of time and hence has greater speed v
  • Both speed equations will be given on your data sheet in the exam

Maximum speed

• An oscillator reaches its greatest speed at the equilibrium position i.e. when its displacement is 0 (x = 0)
• At maximum speed the SHM speed equation becomes:

v₀ = ⍵x₀