SHM Graphs (AQA A Level Physics): Revision Note

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

• Each graph needs to be recognised, along with how it relates to the other graphs

• Remember that:

  • velocity is the rate of change of displacement $v=\frac{s}{t}$

  • acceleration is the rate of change of velocity $a=\frac{∆v}{t}$

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$°$ 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$°$ 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

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$°$ 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$°$ 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:

    $v=\frac{∆x}{∆t}$

  • 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 $a=-{\omega }^{2}x$)

  • The acceleration of an oscillator at any time can be determined from the gradient of the velocity-time graph:

    $a=\frac{∆v}{∆t}$

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