Problem 16 - College Physics

Question

An electric fan is turned off, and its angular velocity decreases uniformly from 500.0 rev/min to 200.0 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s' 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)?

Step-by-Step Solution

Answer

(a) Angular acceleration: -1.25 rev/s², Revolutions: 23.32. (b) Additional time to stop: 2.67 seconds.

Step 1: Convert Angular Velocities to rev/s

The initial angular velocity is given as 500.0 rev/min and the final angular velocity as 200.0 rev/min. We need to convert these values to revolutions per second (rev/s) using the conversion factor \(1 \text{ min} = 60 \text{ s}\). \[\omega_i = \frac{500.0 \text{ rev/min}}{60} = 8.33 \text{ rev/s}\]\[\omega_f = \frac{200.0 \text{ rev/min}}{60} = 3.33 \text{ rev/s}\]

Step 2: Calculate Angular Acceleration

Use the formula for angular acceleration \( \alpha \) which is defined as the change in angular velocity \( \Delta \omega \) divided by the time \( t \) taken for that change.\[\alpha = \frac{\omega_f - \omega_i}{t} = \frac{3.33 \text{ rev/s} - 8.33 \text{ rev/s}}{4.00 \text{ s}} = -1.25 \text{ rev/s}^2\]

Step 3: Calculate Number of Revolutions During Deceleration

Use the equation for angular displacement \( \theta \) during uniform angular acceleration: \[ \theta = \omega_i \cdot t + \frac{1}{2} \alpha \cdot t^2 \]Substitute the values calculated:\[ \theta = 8.33 \cdot 4.00 + \frac{1}{2} \cdot (-1.25) \cdot 4.00^2 \]\[ \theta = 33.32 - 10.00 = 23.32 \text{ revolutions} \]

Step 4: Time to Come to Rest

We need to find the additional time \( t' \) it takes for the fan to stop completely (\( \omega_f = 0 \text{ rev/s} \)) using the angular acceleration calculated.\[t' = \frac{\omega_f - \omega_i}{\alpha} = \frac{0 - 3.33}{-1.25} = 2.67 \text{ seconds}\]

Key Concepts

Understanding Angular Velocity

Angular velocity is a key concept when dealing with rotational motion. It refers to how quickly an object rotates or spins around a point or axis.
For example, in the problem of an electric fan, its angular velocity changes over time as it slows down.
If angular velocity is high, the fan blades spin rapidly. If it's low, they turn slowly. To standardize this measurement, we often convert angular velocity from revolutions per minute (rev/min) to revolutions per second (rev/s). This is done with the conversion factor:

• 1 minute = 60 seconds.

Therefore, knowing how to convert correctly is crucial for solving problems involving angular motion. In cases of deceleration, like with the fan, the final angular velocity is less than the initial, which indicates that the system is slowing down.

Exploring Angular Acceleration

Angular acceleration describes how quickly the angular velocity changes with time. It is represented by the symbol \( \alpha \) and measured in revolutions per second squared (rev/s\(^2\)).
In our example, the fan's angular velocity decreases uniformly, meaning it decelerates at a constant rate.
The formula used is:

• \( \alpha = \frac{\omega_f - \omega_i}{t} \)

This formula tells us the rate of change of angular velocity over a period. In essence, it shows the uniform drop in speed, computed as \(-1.25 \) rev/s\(^2\) for this fan, indicating that every second, the velocity drops by 1.25 rev/s.
This notion of uniform or constant angular acceleration is central when predicting future states of the motion, such as when the fan will come to a stop if it continues at this rate.

Considering Angular Displacement

Angular displacement measures the total angle turned by an object, often measured in revolutions or radians.
For the fan section of the problem, the fan's total angular displacement during its deceleration is computed as approximately 23.32 revolutions.
The formula used is:

• \( \theta = \omega_i \cdot t + \frac{1}{2} \alpha \cdot t^2 \)

This key equation includes both the initial velocity contribution and the part due to continuous acceleration (or deceleration) over time.
Understanding angular displacement helps visualize how far or the extent of rotation a spinning object undergoes before coming to rest or reaching a certain speed.

Grasping Uniform Motion

Uniform motion signifies that the motion of an object is constant, either not changing in speed (uniform linear motion) or not changing in rotational speed (uniform angular motion).
With the electric fan, the uniform aspect refers to the uniform angular acceleration, rather than uniform velocity since the fan is decelerating and not maintaining a constant speed.
It means the rate at which the speed changes remains constant, as verified in the exercise. While the fan blades initially have a high angular velocity, they reduce uniformly over time with a consistent angular acceleration until they stop.
Such uniformity simplifies the calculation and prediction of future motion, making it essential in physics when evaluating how long systems will take to change or stop.