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

• The greatest speed of an oscillator is at the equilibrium position ie. when its displacement is 0 (x = 0)

Equation for Speed

• The speed of an oscillator 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)

• This is a cosine function if the object starts oscillating from the equilibrium position (x = 0 when t = 0)
• 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

Equation for Speed with Displacement

• How the speed v changes with the oscillator’s displacement x is defined by:

v = ±ω

• Where:
  • x = displacement (m)
  • x₀ = amplitude (m)
  • ± = ‘plus or minus’. The value can be negative or positive

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

Equation when at Maximum Speed

• When the speed is at its maximum (at x = 0), the equation becomes:

v₀ = ⍵x₀

Interpreting the Graph v = v₀ cos(⍵t)

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