A figure skater is spinning with an angular velocity of +10 rad/s. She then comes to a stop over a brief period of time. During this time, her angular displacement is +5.8 rad. Determine (a) her average angular acceleration and (b) the time during which she comes to rest.
To find the average angular acceleration, we can use the formula:
Average angular acceleration (α) = Change in angular velocity (ω) / Time taken (t)
In this case, the change in angular velocity is from +10 rad/s to 0 rad/s. So, the change in angular velocity is:
Change in angular velocity (ω) = 0 rad/s - (+10 rad/s) = -10 rad/s
Now, we need to determine the time taken to go from an angular velocity of +10 rad/s to 0 rad/s. Since we know the change in angular displacement and the initial angular velocity, we can use the formula:
Angular displacement (θ) = Initial angular velocity (ω₀) * Time taken (t) + (1/2) * Average angular acceleration (α) * Time taken (t)²
Plugging in the values, we get:
5.8 rad = +10 rad/s * t + (1/2) * α * t²
Simplifying the equation, we get:
5.8 rad = 10t + (1/2) αt²
Rearranging the equation, we get:
(1/2) αt² + 10t - 5.8 = 0
Now, we can solve this quadratic equation for t using any suitable method such as factoring, completing the square, or using the quadratic formula. Once we find the value(s) of t, we can substitute it back into the equation for average angular acceleration (α) to find the answer.
Note: Since the time taken for the figure skater to come to rest is not specified, there may be multiple solutions to the quadratic equation. Make sure to check which value(s) of t make sense in the context of the problem.