Equations for Simple Harmonic Motion (Edexcel A Level Physics): Revision Note

Equations for Simple Harmonic Motion

Acceleration and SHM

• Acceleration a and displacement x can be represented by the defining equation of SHM:

a ∝ −x

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

a = −⍵²x

• Where:

  • a = acceleration (m s^-2)

  • ⍵ = angular frequency (rad s^-1)

  • x = displacement (m)

• This is used to find the acceleration of an object 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 i.e.. x = A (amplitude)

  • The minus sign shows that when the object is displaced to the right, the direction of the acceleration is to the left and vice versa (a and x are always in opposite directions to each other)

Displacement and SHM

• The graph of acceleration against displacement is a straight line through the origin sloping downwards (similar to y = −x)

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

• The key features of the graph are:

  • The gradient is equal to −⍵²

  • The maximum and minimum displacement x values are the amplitudes −A and +A

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

x = A cos (⍵t)

• Where:

  • A = amplitude (m)

  • t = time (s)

   

• This occurs when:

  • An object is oscillating from its amplitude position (x = A or x = −A at t = 0)

  • The displacement will be at its maximum when cos(⍵t) equals 1 or −1, when x = A

• 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

• If an object is oscillating from its equilibrium position (x = 0 at t = 0) then the displacement equation will be:

x = A sin (⍵t)

• The displacement will be at its maximum when sin(⍵t) equals 1 or −1, when x = A

• This is because the sine graph starts at 0, whereas the cosine graph starts at a maximum

These two graphs represent the same SHM. The difference is the starting position

Speed and SHM

• 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 x = 0

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

$v=\pm \omega \sqrt{\left(A^2-x^2\right)}$

• Where:

  • v = speed (m s^-1)

  • A = amplitude (m)

  • ± = ‘plus or minus’. The value can be negative or positive

  • ⍵ = angular frequency (rad s^-1)

  • x = displacement (m)

• This equation shows that when an oscillator has a greater amplitude A, it has to travel a greater distance in the same time and hence has greater speed v

• 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