Q6.5 Angular acceleration (S)

A flywheel completes 40 revolutions as it slows from an angular speed of 1.5 rad s-1 to a complete stop. Assuming uniform acceleration
  1. What is the angular acceleration?
  2. What is the time required for it to come to rest?
  3. How much time is required to complete the first 20 of the 40 revolutions?
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Hint

Use the constant acceleration equations (Key Point 5.11).
 

Solution

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Solution

The angular acceleration is constant, therefore we can use the constant acceleration equations (Key Point 5.11).
  1. The angular acceleration can be calculated using the third constant acceleration equation, with ω=0, ω0=1.5 rad s-1 and the displacement Δθ=40 revolutions

    \[ \omega^2 = \omega_0^2 + 2 \alpha \Delta \theta. \]

    This gives α=-4.47×10-3 rad s-2 (3.s.f)

  2. Using the first constant acceleration equation

    ω=ω0+αt,

    and solving for t we get t=336 s (3 s.f).

  3. To complete the first 20 of the 40 revolutions we need to use the second constant acceleration equation. The total displacement is Δθ=20×2π.

    \[ \Delta \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

    Solving this quadratic for t (and choosing the smallest root, the one less than t=336 s) we obtain t=98.1 s (3.s.f).