An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s^2 and the number of revolutions made by the motor in the 4.00-s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

Verified step by step guidance

1. To find the angular acceleration, we first need to 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. Thus, \( \omega_i = \frac{500}{60} \text{ rev/s} \) and \( \omega_f = \frac{200}{60} \text{ rev/s} \).
2. The formula for angular acceleration \( \alpha \) when the angular velocity changes uniformly is \( \alpha = \frac{\omega_f - \omega_i}{\Delta t} \), where \( \Delta t \) is the time interval. Substitute the values to find \( \alpha \) in rev/s².
3. To find the number of revolutions made by the motor, use the formula for angular displacement \( \theta \) in terms of initial angular velocity, time, and angular acceleration: \( \theta = \omega_i \cdot \Delta t + \frac{1}{2} \alpha \cdot (\Delta t)^2 \). Calculate \( \theta \) in revolutions.
4. For part (b), to find the additional time required for the fan to come to rest, use the formula \( \omega_f = \omega_i + \alpha \cdot t \), where \( \omega_f = 0 \) (since the fan comes to rest). Solve for \( t \) using the angular acceleration found in part (a).
5. Add the time calculated in part (b) to the initial 4.00 seconds to find the total time from when the fan was turned off until it comes to rest.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves relative to another point, typically the center of a circle. It is expressed in units such as revolutions per minute (rev/min) or radians per second (rad/s). In this problem, the fan's angular velocity changes from 500 rev/min to 200 rev/min, indicating a deceleration.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is expressed in units like revolutions per second squared (rev/s^2) or radians per second squared (rad/s^2). In this scenario, the fan's angular velocity decreases uniformly, meaning the angular acceleration is constant and can be calculated using the change in angular velocity over the given time interval.

Kinematic Equations for Rotational Motion

Kinematic equations for rotational motion describe the relationships between angular displacement, angular velocity, angular acceleration, and time. These equations are analogous to linear motion equations and are essential for solving problems involving rotational dynamics. In this problem, they help determine the number of revolutions during deceleration and the time required for the fan to come to rest.