A flywheel slows from 643 rev/min to 387 rev/min while rotating through 53 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.)

  rad/s2

(b)

How much time elapses (in s) during the 53 revolutions?

To solve this problem, we will use the following kinematic equations for rotational motion:

1. **Angular acceleration** (\(\alpha\)):
   \[
   \omega_f^2 = \omega_i^2 + 2\alpha\theta
   \]
   where:
   - \(\omega_f\) is the final angular velocity,
   - \(\omega_i\) is the initial angular velocity,
   - \(\alpha\) is the angular acceleration,
   - \(\theta\) is the angular displacement in radians.

2. **Time elapsed** (\(t\)):
   \[
   \omega_f = \omega_i + \alpha t
   \]
   where:
   - \(t\) is the time elapsed.

### Step 1: Convert revolutions per minute (rev/min) to radians per second (rad/s)
- \( \omega_i = 643 \, \text{rev/min} \)
- \( \omega_f = 387 \, \text{rev/min} \)
- \( \theta = 53 \, \text{revolutions} \)

Since \(1 \, \text{revolution} = 2\pi \, \text{radians}\) and \(1 \, \text{minute} = 60 \, \text{seconds}\):

\[
\omega_{\text{rad/s}} = \omega_{\text{rev/min}} \times \frac{2\pi}{60}
\]

Now, let's compute the angular acceleration \(\alpha\) and the time elapsed \(t\).(a) The angular acceleration of the flywheel is approximately \(-4.34 \, \text{rad/s}^2\). The negative sign indicates that the flywheel is decelerating.

(b) The time elapsed during the 53 revolutions is approximately \(6.17 \, \text{seconds}\).

Would you like further details, or do you have any other questions?

Here are some related questions to consider:
1. How would the angular acceleration change if the flywheel were initially rotating faster?
2. What is the significance of a negative angular acceleration?
3. How would the time change if the flywheel decelerated at a slower rate?
4. How many revolutions would occur if the flywheel continued to decelerate at the same rate?
5. What would happen if there were a sudden increase in friction during the rotation?

**Tip:** When dealing with rotational motion, always pay attention to the direction of acceleration and velocities, as they can significantly affect the results.

Learn how to calculate angular acceleration and time elapsed in a rotational motion problem where a flywheel decelerates from 643 rev/min to 387 rev/min over 53 revolutions. Includes step-by-step solutions using kinematic equations.

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