Question

The flywheel of an engine is rotating at \(25.2 \mathrm{rad} / \mathrm{s}\). When the engine is turned off, the flywheel decelerates at a constant rate and comes to rest after \(19.7 \mathrm{~s}\). Calculate \((a)\) the angular acceleration (in \(\mathrm{rad} / \mathrm{s}^{2}\) ) of the flywheel, (b) the angle (in rad) through which the flywheel rotates in coming to rest, and \((c)\) the number of revolutions made by the flywheel in coming to rest.

Short answer

The angular acceleration of the flywheel is \(-1.28\) rad/s\(^2\), the angle through which the flywheel rotates in coming to rest is \(497.04\) rad, and the number of revolutions made by the flywheel in coming to rest is \(79.1\) revolutions.

Step-by-step solution

Calculate the Angular Acceleration

The angular acceleration can be found by using the formula for acceleration which is \(\alpha = (ω_f - ω_i)/ t\). Here, \(ω_f\) (final angular velocity) is \(0\) rad/s, \(ω_i\) (initial angular velocity) is \(25.2\) rad/s and \(t\) (time) is \(19.7\) s. Hence, \(\alpha = (0 - 25.2) / 19.7 = -1.28 \) rad/s\(^2\). The negative sign indicates that it is a deceleration.

Calculate the Angle

The angle or displacement can be found by using the kinematic equation \(\theta = ω_i t + 0.5*α*t^2\). Here, \(\theta\) is the angle, \(ω_i\) is \(25.2\) rad/s, \(α\) is \(-1.28\) rad/s\(^2\) and \(t\) is \(19.7\) s. Hence, \(\theta = 25.2 * 19.7 + 0.5*(-1.28)*(19.7)^2 = 497.04\) rad.

Calculate the Number of Revolutions

The number of revolutions can be calculated by understanding that \(1\) revolution equals to \(2π\) radians. To convert, divide the total angle of \(497.04\) rad by \(2π\). Hence, the number of revolutions \(N\) is \(497.04 / 2π = 79.1\) revolutions.

Key concepts

Angular Acceleration

Angular acceleration is a key concept when dealing with rotating objects. It measures how quickly an object's angular velocity changes over time. In simple terms, angular acceleration (\( \alpha \)) tells us how fast something is speeding up or slowing down its spin.
To find angular acceleration, we use the formula:

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

where:

• \(\omega_f\): final angular velocity
• \(\omega_i\): initial angular velocity
• \(t\): time

In this exercise, the flywheel was initially spinning at an angular velocity of 25.2 rad/s. Over 19.7 seconds, it came to a halt, making the final angular velocity 0 rad/s. Therefore, the angular acceleration was calculated as:

• \(\alpha = \frac{0 - 25.2}{19.7} = -1.28 \text{ rad/s}^2 \)

The negative sign indicates that this is a deceleration, which simply means that the object is slowing down.

Angular Displacement

Angular displacement (\( \theta \) ) is a measure of the angle through which an object moves on a circular path. It is very much like linear distance, but for rotational motion.
To find angular displacement, one can use the kinematic equation:

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

where:

• \(\theta\): angular displacement
• \(\omega_i\): initial angular velocity
• \(\alpha\): angular acceleration
• \(t\): time

In this context, the flywheel started with an initial velocity of 25.2 rad/s and was slowed down by an acceleration of -1.28 rad/s² for 19.7 seconds. Plugging these values in, we get:

• \( \theta = 25.2 \times 19.7 + \frac{1}{2} \times (-1.28) \times (19.7)^2 = 497.04 \text{ radians} \)

This means the flywheel rotated 497.04 radians before it stopped.

Kinematic Equations

The kinematic equations are used to relate the five most common motion variables: displacement, initial velocity, final velocity, acceleration, and time. These equations are essential for solving problems related to linear motion as well as angular motion.
For angular motion, these equations are slightly modified, using angular quantities like angular velocity (\( \omega \)), angular displacement (\( \theta \)), and angular acceleration (\( \alpha \)). Here's the equation used in this exercise:

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

This formula helps to determine how far a rotating object travels in a given time period when experiencing constant angular acceleration. Knowing these equations, you can easily determine crucial aspects of an object's rotational dynamics.
The flexibility of kinematic equations makes them powerful tools for solving problems in physics, especially for students facing complex exercises.