Problem 10 - University Physics with Modern Physics

Question

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' and the number of revolutions made by the motor in the \(4.00-\) - 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

Angular acceleration is -1.25 rev/s², 13.33 revolutions made, 2.67 seconds to stop.

Step 1: Understand the Problem

We need to find the angular acceleration and the number of revolutions during a time interval where the angular velocity decreases uniformly. Then, using the same acceleration, we find the time required for the fan to come to a stop.

Step 2: Calculate Angular Acceleration

Angular acceleration \( \alpha \) is given by the change in angular velocity \( \Delta \omega \) over time \( \Delta t \). Convert velocities from revolutions per minute to revolutions per second: \( \omega_i = \frac{500}{60} \text{ rev/s} \), \( \omega_f = \frac{200}{60} \text{ rev/s} \). Apply the formula \( \alpha = \frac{\omega_f - \omega_i}{\Delta t} \). Substitute and calculate:\( \alpha = \frac{\frac{200}{60} - \frac{500}{60}}{4} = \frac{-300}{240} = -1.25 \text{ rev/s}^2 \).

Step 3: Find the Number of Revolutions

Use the equation for revolutions in terms of initial velocity, time, and acceleration: \( \Delta \theta = \omega_i \cdot t + \frac{1}{2} \alpha t^2 \).Substitute the known values: \( \Delta \theta = \frac{500}{60} \cdot 4 + \frac{1}{2} \cdot (-1.25) \cdot 4^2 \).Calculate \( \Delta \theta = \frac{2000}{60} + \frac{1}{2} \cdot (-1.25) \cdot 16 = \frac{2000}{60} - 10 = 23.3333 - 10 = 13.3333 \) revolutions.

Step 4: Calculate Time to Rest

To find the time \( t \) required for the fan to stop from 200 rev/min, use the equation \( \omega_f = \omega_i + \alpha t_{stop} \) with \( \omega_f = 0 \). Convert \( \omega_i = \frac{200}{60} \text{ rev/s} \), and use \( \alpha = -1.25 \text{ rev/s}^2 \):\( 0 = \frac{200}{60} + (-1.25) \cdot t_{stop} \).Solve for \( t_{stop} \): \( t_{stop} = \frac{\frac{200}{60}}{1.25} = \frac{200}{75} = \frac{8}{3} \approx 2.67 \) seconds.

Key Concepts

Uniform Angular Motion

When dealing with uniform angular motion, it's crucial to understand that it involves an object rotating at a constant rate. This motion implies that both the angular velocity and angular acceleration remain unchanged over time.
In the exercise, as an electric fan slows down, it exhibits uniform angular motion by decreasing its speed at a steady rate. This is because the fan's angular velocity decreases linearly, which signals a constant angular acceleration or deceleration. When the fan's speed drops from 500 revolutions per minute (rev/min) to 200 rev/min, this change occurs uniformly over 4 seconds.
Think of uniform angular motion as similar to a car gradually slowing down at a fixed pace when approaching a red light. In this scenario:

• The angular velocity refers to how fast the fan spins.
• The angular acceleration tells us how quickly the fan slows down.

Understanding this concept allows us to compute how many spins the fan completes during this slowing period.

Angular Velocity Conversion

Angular velocity conversion is a necessary step for solving problems involving rotational motion. Angular velocity is usually expressed in revolutions per minute (rev/min) or revolutions per second (rev/s), depending on the context of the problem.
In many exercises, you need to convert between these units to perform calculations correctly. For this exercise, the fan's angular velocity begins at 500 rev/min, which must be converted to rev/s to find the angular acceleration. The conversion process involves dividing the angular velocity by 60, since there are 60 seconds in a minute.
Here's a quick conversion breakdown:

• Start with the initial angular velocity of 500 rev/min.
• Convert it to rev/s by dividing by 60: \( \omega_i = \frac{500}{60} \text{ rev/s} \).
• Repeat this for the final angular velocity: \( \omega_f = \frac{200}{60} \text{ rev/s} \).

This step ensures that all units are consistent, simplifying calculations and allowing for the determination of angular acceleration using increments in seconds.

Revolution Calculation

Calculating the number of revolutions completed by a rotating object is essential for understanding the total distance it covers during a specific period. When dealing with problems like calculating how far a fan blade spins as it slows down, the formula involving initial angular velocity, time, and angular acceleration comes into play.
In our case, the equation \(\Delta \theta = \omega_i \cdot t + \frac{1}{2} \alpha t^2\) aids in determining the total revolutions. This formula takes into account both the initial velocity and the decrease in speed due to constant angular acceleration.
Steps to calculate revolutions:

• Identify the initial angular velocity \( \omega_i\), which is \(\frac{500}{60} \text{ rev/s} \).
• Determine the time interval, 4 seconds.
• Use the calculated angular acceleration \( \alpha = -1.25 \text{ rev/s}^2 \).

By substituting these values into the formula, the total number of revolutions the fan blade completes is determined, which in this case is approximately 13.33 revolutions. This shows how even as the fan slows uniformly, it still makes several complete spins.