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A wheel has an initial angular velocity of 39.1 rad/s. It starts to slow down uniformly and comes to rest, making 76.8 revolutions during the process.

a) What is its angular acceleration (in rad/s\(^2\))?

Answer :

Answer:

Approximately [tex]-1.58\; \rm rad \cdot s^{-2}[/tex].

Explanation:

This question suggests that the rotation of this object slows down "uniformly". Therefore, the angular acceleration of this object should be constant and smaller than zero.

This question does not provide any information about the time required for the rotation of this object to come to a stop. In linear motions with a constant acceleration, there's an SUVAT equation that does not involve time:

[tex]v^2 - u^2 = 2\, a\, x[/tex],

where

  • [tex]v[/tex] is the final velocity of the moving object,
  • [tex]u[/tex] is the initial velocity of the moving object,
  • [tex]a[/tex] is the (linear) acceleration of the moving object, and
  • [tex]x[/tex] is the (linear) displacement of the object while its velocity changed from [tex]u[/tex] to [tex]v[/tex].

The angular analogue of that equation will be:

[tex](\omega(\text{final}))^2 - (\omega(\text{initial}))^2 = 2\, \alpha\, \theta[/tex], where

  • [tex]\omega(\text{final})[/tex] and [tex]\omega(\text{initial})[/tex] are the initial and final angular velocity of the rotating object,
  • [tex]\alpha[/tex] is the angular acceleration of the moving object, and
  • [tex]\theta[/tex] is the angular displacement of the object while its angular velocity changed from [tex]\omega(\text{initial})[/tex] to [tex]\omega(\text{final})[/tex].

For this object:

  • [tex]\omega(\text{final}) = 0\; \rm rad\cdot s^{-1}[/tex], whereas
  • [tex]\omega(\text{initial}) = 39.1\; \rm rad\cdot s^{-1}[/tex].

The question is asking for an angular acceleration with the unit [tex]\rm rad \cdot s^{-1}[/tex]. However, the angular displacement from the question is described with the number of revolutions. Convert that to radians:

[tex]\begin{aligned}\theta &= 76.8\; \rm \text{revolution} \\ &= 76.8\;\text{revolution} \times 2\pi\; \rm rad \cdot \text{revolution}^{-1} \\ &= 153.6\pi\; \rm rad\end{aligned}[/tex].

Rearrange the equation [tex](\omega(\text{final}))^2 - (\omega(\text{initial}))^2 = 2\, \alpha\, \theta[/tex] and solve for [tex]\alpha[/tex]:

[tex]\begin{aligned}\alpha &= \frac{(\omega(\text{final}))^2 - (\omega(\text{initial}))^2}{2\, \theta} \\ &= \frac{-\left(39.1\; \rm rad \cdot s^{-1}\right)^2}{2\times 153.6\pi\; \rm rad} \approx -1.58\; \rm rad \cdot s^{-1}\end{aligned}[/tex].

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