A LevelPhysicsCIERevision Notes17. Oscillations17.1 Simple Harmonic OscillationsCalculating Acceleration & Displacement in SHM

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Calculating Acceleration & Displacement in SHM (CIE A Level Physics)

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Calculating acceleration & displacement of an oscillator

Calculating acceleration

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

a space equals space minus omega squared x

  • Where:
    • a = acceleration (m s-2)
    • ⍵ = angular frequency (rad s-1)
    • x = displacement (m)

  • The equation demonstrates:
    • Acceleration reaches its maximum value when the displacement is at a maximum i.e. x = x0 at its amplitude
    • The minus sign shows that when the object is displaced to the right, the direction of the acceleration is to the left and vice versa (a and x are always in opposite directions to each other)

The graph representing a = −⍵2x

  • The graph of acceleration against displacement is a straight line through the origin and a negative gradient (similar to the line y = − x)

The graph of a = −⍵2x

Graph of acceleration and displacement, downloadable AS & A Level Physics revision notes

The acceleration of an object in SHM is directly proportional to the negative displacement

  • Key features of the acceleration-displacement graph:
    • The gradient is equal to − ⍵2
    • The maximum and minimum displacement x values are the amplitudes −x0 and +x0

Calculating displacement

  • A solution to the SHM acceleration equation is the displacement equation
    • For an object that begins oscillating from the equilibrium position (x = 0 at t = 0):

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 specific moment in time
  • The displacement will be at its maximum when sin(⍵t) equals 1 or −1, when x = x0

The graph of x = x0sin(⍵t)

17-1-displacement-shm-graphs

This graph shows that at t = 0 then x = 0 and the oscillation of the object is the same as that of a sine curve

Worked example

A mass of 55 g is suspended from a fixed point by means of a spring. The stationary mass is released from the central equilibrium position at t = 0 and reaches a maximum displacement of 4.3 cm.

The mass is observed to perform simple harmonic motion with a period of 0.8 s.

Calculate the displacement x in cm of the mass at time t = 0.3 s.

Answer:

Step 1: Write down the SHM displacement equation

    • Since the mass is released at t = 0 from its equilibrium position then the displacement is calculated using:

x = x0 sin(⍵t)

Step 2: Calculate angular frequency

omega space equals space fraction numerator 2 straight pi over denominator T end fraction space equals space fraction numerator 2 straight pi over denominator 0.8 end fraction space equals space 7.85 space rad space straight s to the power of negative 1 end exponent

    • Remember to use the value of the time period given and not any other values of time

Step 3: Substitute values into the displacement equation

x = 4.3 sin (7.85 × 0.3) = 3.044… = 3.0 cm (2 s.f)

    • Make sure the calculator is in radians mode
    • The positive value means the mass is 3.0 cm is on the same side of the equilibrium position to where it started (3.0 cm above it)

Examiner 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 cosine and sine functions. This is because the angular frequency ⍵ is calculated in rad s-1, not degrees.

You often have to convert between time period T, frequency f and angular frequency ⍵ for many exam questions – so make sure you revise the equations relating to these.

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    Ann H

    Author: Ann H

    Expertise: Physics

    Ann obtained her Maths and Physics degree from the University of Bath before completing her PGCE in Science and Maths teaching. She spent ten years teaching Maths and Physics to wonderful students from all around the world whilst living in China, Ethiopia and Nepal. Now based in beautiful Devon she is thrilled to be creating awesome Physics resources to make Physics more accessible and understandable for all students no matter their schooling or background.