SHM Graphs (CIE A Level Physics)

• The displacement, velocity and acceleration of an object in simple harmonic motion can be represented graphically against time
• The graphs of all undamped SHM oscillations are represented as periodic functions
  • This means they can all be described by sine and cosine curves

• You need to know what each graph looks like and how it relates to the other graphs
• Remember that:
  • Velocity is the rate of change of displacement 
  • Acceleration is the rate of change of velocity 

• When oscillations start from the equilibrium position then:
  • The displacement-time graph is a sine curve
  • The velocity-time graph is the gradient of the displacement-time graph, so a cosine graph and 90^o out of phase with the displacement-time graph
  • The acceleration-time graph is the gradient of the velocity-time graph, so a negative sine graph and 90^o out of phase with the velocity-time graph
• More information on this can be found in the IGCSE Maths revision notes on differentiation and CIE A level Maths pure 3 revision notes on differentiating other functions

Graphs of displacement, velocity and acceleration against time

The displacement, velocity and acceleration graphs in SHM are all 90° out of phase with each other

Relationship between graphs

• Key features of the displacement-time graphs:
  • The amplitude of oscillations A is the maximum value of x
  • The time period of oscillations T is the time taken for one full wavelength cycle

• Key features of the velocity-time graphs:
  • The velocity of an oscillator at any time can be determined from the gradient of the displacement-time graph:

  

• • An oscillator moves the fastest at its equilibrium position
  • Therefore, the velocity is at its maximum when the displacement is zero (the amplitude of the graph is v_max)

• Key features of the acceleration-time graph:
  • The acceleration graph is a reflection of the displacement graph in the x-axis
  • This means when a mass has positive displacement (to the right) the acceleration is in the opposite direction (to the left) and vice versa (from a = −ω²x)
  • The acceleration of an oscillator at any time can be determined from the gradient of the velocity-time graph:

  

• • The maximum value of the acceleration is when the oscillator is at its maximum displacement