6.2.2 SHM Graphs

• The displacement, velocity and acceleration of an object in simple harmonic motion can be represented by graphs against time
• All undamped SHM graphs are represented by 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 A-level Maths revision notes on trigonometric differentiation

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

• Key features of the displacement-time graph:
  • 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
  • The graph might not always start at x = 0
  • If the oscillations start at the positive or negative amplitude, then initial displacement x = A

• Key features of the velocity-time graph:
  • 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 on 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

Step 1: The velocity is at its maximum when the displacement x = 0

Step 2: Reading value of time when x = 0

From the graph, this is equal to 0.2 s