A figure skater is spinning with an angular velocity of + 12.8 rad/s. She then comes to a stop over a brief period of time. During this time, her angular displacement is 4.38 rad. Determine (a) her average angular acceleration and (b) the time during which she comes to rest.
(a) The average angular acceleration can be determined using the formula:
$\text{average angular acceleration} = \dfrac{\text{change in angular velocity}}{\text{change in time}}$
Since the skater has come to a stop, her change in angular velocity is equal to her initial angular velocity of +12.8 rad/s. Thus, the average angular acceleration is:
$\text{average angular acceleration} = \dfrac{0 - 12.8}{t} = -12.8/t$
(b) The angular displacement can be related to the angular velocity and the time using the formula:
$\text{angular displacement} = \text{angular velocity} \times \text{time}$
In this case, the angular displacement is given as 4.38 rad and the angular velocity is given as 12.8 rad/s. Solving for time, we get:
$\text{time} = \dfrac{\text{angular displacement}}{\text{angular velocity}} = \dfrac{4.38}{12.8} \approx 0.342\ \text{s}$
Therefore, the time during which the skater comes to rest is approximately 0.342 seconds.