# 6.2.3 Calculating Maximum Speed & Acceleration

### Maximum Speed

• The speed v of an oscillator will vary in SHM. It is:
• Maximum at the equilibrium position (x = 0)
• Zero at the amplitude (x = A)

• The maximum speed, vmax, is given by the equation:

vmax = ωA

• Where:
• vmax  = maximum velocity (m s-1)
• ω = angular frequency (rad s-1)
• A = amplitude (m) The variation of the speed of a mass on a spring in SHM over one complete cycle

#### Worked Example

Calculate the frequency of an oscillator with a maximum velocity of 12 m s-1 and amplitude of 1.4 m.

Step 1: State the known values

• Maximum velocity, vmax = 12 m s-1
• Amplitude, A = 1.4 m

Step 2: Write down the equation

vmax = ωA

Step 3: Rewrite angular velocity in terms of frequency f

ω = 2πf

vmax = 2πfA

Step 4: Rearrange for frequency, f Step 5: Substitute in the values ### Maximum Acceleration

• The acceleration a of an oscillator will also vary in SHM. It is:
• Maximum at the amplitude (x = A)
• Zero at the equilibrium position (x = 0)

• This is because the acceleration is directly proportional to the displacement of an oscillator
• The maximum acceleration is given by the equation:

amax = ω2A

• Where:
• amax = maximum acceleration (m s2)
• ω = angular frequency (rad s-1)
• A = amplitude (m)

• Although at the amplitude, the speed is zero, the oscillator has changed direction
• This means that it has a non–zero velocity, and since acceleration is the rate of change of velocity, the oscillator has an acceleration at the amplitude too

#### Worked Example

Calculate the maximum acceleration of an oscillator with a time period of 0.4 s and amplitude of 2.8 m.

Step 1: State the known values

• Time period, T = 0.4 s
• Amplitude, A = 2.8 m

Step 2: Write down the equation

amax = ω2A

Step 3: Rewrite maximum acceleration with time period T Step 4: Substitute in the values #### Exam Tip

Make sure not to get mixed up with lower case a (acceleration) and upper case A (amplitude)

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