# 5.5.6 Velocity

## Velocity of an Oscillator

• The velocity of an object in simple harmonic motion varies as it oscillates back and forth
• Since velocity is a vector, the velocity of the oscillator is its speed in a certain direction

• The maximum velocity of an oscillator is at the equilibrium position i.e. when its displacement is 0 (x = 0)
• The velocity of an oscillator in SHM is defined by:

v = v0 cos(⍵t)

• Where:
• v = velocity (m s-1)
• v0 = maximum velocity (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)
• How the velocity 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 shows that when an oscillator has a greater amplitude x0, it has to travel a greater distance in the same time and hence has greater velocity, v
• When the velocity is at its maximum (at x = 0), the equation becomes:

v0 = ⍵x0 The variation of the speed of a mass on a spring in SHM over one complete cycle

#### Worked example

A simple pendulum oscillates with simple harmonic motion with an amplitude of 15 cm. The frequency of the oscillations is 6.7 Hz.

Calculate the speed of the pendulum at a position of 12 cm from the equilibrium position.

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:  ### Get unlimited access

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