19.1.5 SHM Graphs

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
• 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 90o 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 90o 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

Worked example: Using SHM graph data

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

Exam Tip

These graphs might not look identical to what is in your textbook, depending on where the object starts oscillating from at t = 0 (on either side of the equilibrium, or at the equilibrium). However, if there is no damping, they will all always be a general sine or cosine curves.

Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
Close