17.1.5 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

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 x₀ can be found from the maximum value of x
  • The time period of oscillations T can be found from reading the time taken for one full cycle
  • The graph might not always start at 0
  • If the oscillations starts at the positive or negative amplitude, the displacement will be at its maximum

• Key features of the velocity-time graph:
  • It is 90^o out of phase with the displacement-time graph
  • Velocity is equal to the rate of change of displacement
  • So, 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

• 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
  • It is 90^o out of phase with the velocity-time graph
  • Acceleration is equal to the rate of change of velocity
  • So, 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