An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. Find the angular acceleration in rev/s² and the number of revolutions made by the motor in the 4.00-s interval.

Verified step by step guidance

1. Step 1: Convert the initial and final angular velocities from revolutions per minute (rev/min) to revolutions per second (rev/s). Use the conversion factor: 1 minute = 60 seconds. For example, \( \omega_{i} = \frac{500}{60} \) rev/s and \( \omega_{f} = \frac{200}{60} \) rev/s.
2. Step 2: Calculate the angular acceleration \( \alpha \) using the formula for uniform angular acceleration: \( \alpha = \frac{\omega_{f} - \omega_{i}}{\Delta t} \), where \( \Delta t \) is the time interval (4.00 s). Substitute the values of \( \omega_{i} \), \( \omega_{f} \), and \( \Delta t \) to find \( \alpha \) in rev/s².
3. Step 3: Determine the number of revolutions made by the fan during the 4.00-s interval. Use the kinematic equation for rotational motion: \( \theta = \omega_{i} \Delta t + \frac{1}{2} \alpha \Delta t^2 \), where \( \theta \) is the angular displacement in revolutions. Substitute the values of \( \omega_{i} \), \( \alpha \), and \( \Delta t \) to calculate \( \theta \).
4. Step 4: To find the additional time required for the fan to come to rest, use the formula \( \Delta t = \frac{\omega_{f} - \omega_{i}}{\alpha} \), where \( \omega_{f} = 0 \) (since the fan comes to rest) and \( \omega_{i} \) is the final angular velocity from part (a). Substitute the values of \( \omega_{i} \) and \( \alpha \) to calculate the additional time.
5. Step 5: Combine the results from part (a) and part (b) to understand the total behavior of the fan's motion. Ensure all units are consistent and verify the calculations for accuracy.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in revolutions per minute (rev/min) or radians per second. In this problem, the fan's angular velocity decreases from 500 rev/min to 200 rev/min, indicating a change in its rotational speed over time.

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time, usually expressed in rev/s² or rad/s². It can be calculated by taking the difference in angular velocity and dividing it by the time interval during which the change occurs. In this case, it helps determine how quickly the fan slows down.

Kinematics of Rotational Motion

The kinematics of rotational motion involves equations that describe the motion of rotating objects, similar to linear motion but in terms of angular quantities. Key equations relate angular displacement, angular velocity, and angular acceleration, allowing us to calculate the total revolutions made by the fan during its deceleration and the time required to come to a complete stop.