Equations for Simple Harmonic Motion (SHM) (HL) (HL IB Physics)

Summary of SHM Equations

• For a body that begins oscillating from its equilibrium position (i.e.  when ), its displacement, velocity and acceleration can be described by the equations:

• For a body that begins oscillating from its amplitude position (i.e.  when ), its displacement, velocity and acceleration can be described by the equations:

• The variation of an oscillator's velocity with its displacement x is defined by:

• This equation shows that the larger the amplitude of an oscillation, the greater the distance it must travel in a given time period
  • Hence, the faster it travels, the closer it is to the equilibrium position

• In the above equations, the variables are as follows:
  •  = displacement of the oscillator (m)
  • = maximum displacement, or amplitude (m)
  • = velocity of the oscillator (m s⁻¹)
  • = acceleration of the oscillator (m s⁻²)
  • = angular frequency (rad s⁻¹)
  • = time (s)

Summary table of equations and graphs for displacement, velocity and acceleration

The Origin of the Displacement Equations

• The SHM displacement equation for an object oscillating from its equilibrium position (x = 0 at t = 0) is:

x = x₀ sin (⍵t + Φ)

• Where:
  • Φ = phase difference (radians) = 0

• Because:
  • The graph of x = sin (t) starts from amplitude x₀ = 0 when the pendulum is in the equilibrium position at t = 0
  • The displacement is at its maximum when sin(⍵t) equals 1 or −1, when x = x₀
• Use the IB revision notes on the graphs of trigonometric functions to aid your understanding of trigonometric graphs

The graph of y = cos (x) has maximum displacement when x = 0

• The SHM displacement equation for an object oscillating from its amplitude position (x = x₀ at t = 0) is:

x = x₀ cos (⍵t + Φ)

• The displacement will be at its maximum when cos(⍵t) equals 1 or −1, when x = x₀
• This is because the cosine graph starts at a maximum, whereas the sine graph starts at 0

These two graphs represent the same SHM. The difference is the starting position. In these graphs, the phase difference is Φ = 0.

The Origin of the Velocity Equations in Trigonometric Form

• The trigonometric equation for the velocity of an object starting from its equilibrium position (x = 0 at t = 0) is:

v = ωx₀ cos (⍵t + Φ)

• Where:
  • Φ = phase difference (radians) = 0
• This comes from the fact that velocity is the rate of change of displacement
  • It is the differential of the relevant displacement equation from above: x = x₀ sin (⍵t + Φ)
• The trigonometric equation for the velocity of an object starting from its amplitude position (x = x₀ at t = 0) is:

v = −ω x₀ sin (⍵t + Φ) 

• This is the differential of the relevant displacement equation from above: x = x₀ cos (⍵t + Φ) 

The Origin of the Displacement-Velocity Relation

• The velocity of an object in simple harmonic motion varies as it oscillates back and forth and is given by the equation:

• ± = ‘plus or minus’.
  • The value can be negative or positive
• This comes from the fact that acceleration is the rate of change of velocity
  • When the defining equation of simple harmonic motion is integrated using a differential equation the above equation for velocity is obtained
• 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

Equations for Calculating Energy Changes in SHM

• The revision note on Calculating Energy Changes in SHM explains the origin of these two equations for calculating energy changes in simple harmonic motion
• Potential energy:

• Total energy at the amplitude of oscillation: 

• Where:
  • m = mass (kg)
  • ω = angular frequency (rad s⁻¹)
  • x₀ = amplitude (m)