# 19.1.3 Calculating Acceleration & Displacement in SHM

### Acceleration of an Oscillator

• Simple harmonic motion (SHM) is a specific type of oscillation
• SHM is defined as:

A type of oscillation in which the acceleration on a body is proportional to its displacement, but acts in the opposite direction

• This means for an object to oscillate specifically in SHM, it must satisfy the following conditions:
• Periodic oscillations
• Acceleration proportional to its displacement
• Acceleration in the opposite direction to its displacement

• Acceleration a and displacement x can be represented by the defining equation of SHM:

a ∝ −x

• An object in SHM will also have a restoring force to return it to its equilibrium position
• This restoring force will be directly proportional, but in the opposite direction, to the displacement of the object from the equilibrium position
• Note: the restoring force and acceleration act in the same direction

### Calculating Acceleration & Displacement of an Oscillator

• The acceleration of an object oscillating in simple harmonic motion is:

a = −⍵2x

• Where:
• a = acceleration (m s-2)
• ⍵ = angular frequency (rad s-1)
• x = displacement (m)
• This is used to find the acceleration of an object in SHM with a particular angular frequency ⍵ at a specific displacement x
• The equation demonstrates:
• The acceleration reaches its maximum value when the displacement is at a maximum ie. x = x0 (amplitude)
• The minus sign shows that when the object is displacement to the right, the direction of the acceleration is to the left
• The graph of acceleration against displacement is a straight line through the origin sloping downwards (similar to y = − x)
• Key features of the graph:
• The gradient is equal to − ⍵2
• The maximum and minimum displacement x values are the amplitudes −x0 and +x0
• A solution to the SHM acceleration equation is the displacement equation:

x = x0sin(⍵t)

• Where:
• x = displacement (m)
• x0 = amplitude (m)
• t = time (s)
• This equation can be used to find the position of an object in SHM with a particular angular frequency and amplitude at a moment in time
• Note: This version of the equation is only relevant when an object begins oscillating from the equilibrium position (x = 0 at t = 0)
• The displacement will be at its maximum when sin(⍵t) equals 1 or − 1, when x = x0
• If an object is oscillating from its amplitude position (x = x0 or x = − x0 at t = 0) then the displacement equation will be:

x = x0cos(⍵t)

• This is because the cosine graph starts at a maximum, whilst the sine graph starts at 0

#### Worked example: Calculating SHM displacement

Step 1:

Write down the SHM displacement equation

Since the mass is released at t = 0 at its maximum displacement, the displacement equation will be with the cosine function:

x = x0 cos(⍵t)

Step 2:

Calculate angular frequency

Remember to use the value of the time period given, not the time where you are calculating the displacement from

Step 3:

Substitute values into the displacement equation

x = 4.3cos (7.85 × 0.3) = –3.0369… = –3.0 cm (2 s.f)

Make sure the calculator is in radians mode

The negative value means the mass is 3.0 cm on the opposite side of the equilibrium position to where it started (3.0 cm above it)

#### Exam Tip

Since displacement is a vector quantity, remember to keep the minus sign in your solutions if they are negative, you could lose a mark if not!

Also remember that your calculator must be in radians mode when using the cos and sine functions. This is because the angular frequency ⍵ is calculated in rad s-1, not degrees.

### 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.
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