Question

A flywheel slows from 583 rev/min to 377 rev/min while rotating through 48 revolutions. (a) What is the angular acceleration (in rad/s2) of the flywheel? (Assume the flywheel is rotating in the positive direction. Indicate the direction with the sign of your answer.)

Answer 1

To find the angular acceleration of a slowing flywheel, convert the angular velocities to radians per second, use the appropriate rotational kinematics equations, and solve for angular acceleration. The sign will indicate the direction of the acceleration.

Step-by-step explanation:

1. First, we need to find the change in angular velocity. The initial angular velocity
`(\(\omega_i\)) is given as \(583 \, \text{rev/min}\`), and the final angular velocity (\
`(\omega_f\)) is \(377 \, \text{rev/min}\)`. Convert these values to radians per second by multiplying by \(\frac{2\pi}{60}\):

`\[\omega_i = 583 * (2\pi)/(60) \, \text{rad/s}\]`

`\[\omega_f = 377 * (2\pi)/(60) \, \text{rad/s}\]`

2. Now, compute the change in angular velocity (\(\Delta\omega\)):

`\[\Delta\omega = \omega_f - \omega_i\]`

3. Next, find the change in time (\(\Delta t\)). The number of revolutions (\(\theta\)) is given as \(48\), and the relation between angular displacement, angular velocity, and time is \(\theta = \omega_i \cdot \Delta t + \frac{1}{2} \cdot \alpha \cdot (\Delta t)^2\), where \(\alpha\) is angular acceleration. Since we know \(\omega_i\), \(\theta\), and \(\omega_f\), we can solve for \(\alpha\).

4. Rearrange the kinematic equation to solve for \(\Delta t\):

`\[\Delta t = (\omega_f - \omega_i)/(\alpha)\]`

5. Substitute the values and solve for \(\Delta t\).

6. Now, we can use the value of \(\Delta t\) to find the angular acceleration (\(\alpha\)):

`\[\alpha = (\Delta\omega)/(\Delta t)\]`

7. Substitute the known values and calculate \(\alpha\).

8. The negative sign in the angular acceleration indicates deceleration in the positive direction.