SHM Graphs (AQA A Level Physics)

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 

Graphs that Start at the Equilibrium Position

• 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

Graphs that Start at the Amplitude Position

• When oscillations start from the amplitude position then:

  • The displacement-time graph is a cosine curve

  • The velocity-time graph is the gradient of the displacement-time graph, so a negative sine 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 cosine graph and 90^o out of phase with the velocity-time graph

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