# 19.1.4 Calculating Speed in SHM

### Calculating Speed of an Oscillator

• The speed of an object in simple harmonic motion varies as it oscillates back and forth
• Its speed is the magnitude of its velocity
• The greatest speed of an oscillator is at the equilibrium position ie. when its displacement is 0 (x = 0)
• The speed of an oscillator in SHM is defined by:

v = v0 cos(⍵t)

• Where:
• v = speed (m s-1)
• v0 = maximum speed (m s-1)
• ⍵ = angular frequency (rad s-1)
• t = time (s)
• This is a cosine function if the object starts oscillating from the equilibrium position (x = 0 when t = 0)
• Although the symbol v is commonly used to represent velocity, not speed, exam questions focus more on the magnitude of the velocity than its direction in SHM
• How the speed v changes with the oscillator’s displacement x is defined by:

• Where:
• x = displacement (m)
• x0 = amplitude (m)
• ± = ‘plus or minus’. The value can be negative or positive
• This equation show that when an oscillator has a greater amplitude x0, it has to travel a greater distance in the same time and hence has greater speed v
• Both equations for speed will be given on your formulae sheet in the exam
• When the speed is at its maximums (at x = 0), the equation becomes:

v0 = ⍵x0

#### Worked example: Calculating the speed of a pendulum

Step 1:

Write out the known quantities

Amplitude of oscillations, x0 = 15 cm = 0.15 m

Displacement at which the speed is to be found, x = 12 cm = 0.12 m

Frequency, f = 6.7 Hz

Step 2:

Oscillator speed with displacement equation

Since the speed is being calculated, the ± sign can be removed as direction does not matter in this case

Step 3:

Write an expression for the angular frequency

Equation relating angular frequency and normal frequency:

⍵ = 2πf = 2π× 6.7 = 42.097…

Step 4:

Substitute in values and calculate

v = 3.789 = 3.8 m s-1 (2 s.f)

#### Exam Tip

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.

### 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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