Problem 16 - College Physics

Question

A flywheel in a motor is spinning at 500.0 rpm when a power failure suddenly occurs. The flywheel has mass \(40.0 \mathrm{~kg}\) and diameter \(75.0 \mathrm{~cm} .\) The power is off for \(30.0 \mathrm{~s},\) and during this time the flywheel slows down uniformly due to friction in its axle bearings. During the time the power is off, the flywheel makes 200.0 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on? (b) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on, and how many revolutions would the wheel have made during this time?

Step-by-Step Solution

Answer

(a) The flywheel spins at \( \frac{20\pi}{3} \) rad/s when power returns. (b) It would take 45 s to stop, covering 187.5 revolutions.

Step 1: Convert Initial Angular Velocity

The initial angular velocity \( \omega_0 \) of the flywheel needs to be converted from revolutions per minute (rpm) to radians per second (rad/s). Start by converting 500 rpm to revolutions per second: \( \frac{500}{60} \text{ rev/s} = \frac{25}{3} \text{ rev/s} \). Since 1 revolution is \( 2\pi \) radians, \( \omega_0 = \frac{25}{3} \times 2\pi = \frac{50\pi}{3} \text{ rad/s} \).

Step 2: Calculate Total Displacement

The flywheel makes 200 complete revolutions while slowing down. Convert this to radians because each revolution is \( 2\pi \) radians. Thus, the total angular displacement \( \theta = 200 imes 2\pi = 400\pi \text{ rad} \).

Step 3: Use Kinematic Equation to Find Final Angular Velocity

Since the flywheel comes to rest temporarily due to the power failure and then continues at a reduced speed, use the equation \( \theta = \frac{1}{2} (\omega_0 + \omega) t \) where \( t = 30 \). Substitute the known values: \( 400\pi = \frac{1}{2} \left( \frac{50\pi}{3} + \omega \right) 30 \). Solve for \( \omega \), resulting in \( \omega = \frac{20\pi}{3}\text{ rad/s} \).

Step 4: Find Angular Deceleration

Use the formula \( \omega = \omega_0 + \alpha t \) to find the angular deceleration \( \alpha \). Substitute \( \omega = \frac{20\pi}{3} \) rad/s, \( \omega_0 = \frac{50\pi}{3} \) rad/s, and \( t = 30 \) s. Solve to find \( \alpha = -\frac{10\pi}{9} \text{ rad/s}^2 \).

Step 5: Calculate Time to Stop

Find how long it would take for the flywheel to stop completely using the formula \( \omega = \omega_0 + \alpha t \) set \( \omega = 0 \). Solve for \( t \): \( 0 = \frac{50\pi}{3} + ( -\frac{10\pi}{9} ) t \). Thus, \( t = \frac{50\pi}{3} \times -\frac{9}{10\pi} = 45 \text{ s} \).

Step 6: Calculate Total Revolutions to Stop

Use the formula \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \) with total \( t = 45 \) s. Calculate \( \theta = \frac{50\pi}{3} \times 45 + \frac{1}{2} (-\frac{10\pi}{9}) \times 45^2 \). Simplifying, \( \theta = 375\pi \text{ radians} \) which is \( \frac{375\pi}{2\pi} = 187.5 \text{ revolutions} \).

Key Concepts

Angular Velocity

When talking about rotational motion, angular velocity is a key component. Angular velocity describes how fast an object rotates or spins around a particular axis. Similar to how linear velocity is measured in meters per second (m/s), angular velocity is expressed in radians per second (rad/s). This is because we often measure the angle of rotation in radians. For example, in the given problem, the flywheel initially spins at 500 revolutions per minute (rpm). To work with angular velocity in the relevant equations, we first convert this to radians per second using the conversion factor that one revolution equals \(2\pi\) radians.
Thus, the initial angular velocity \(\omega_0\) is converted to \(\frac{50\pi}{3}\) rad/s. Understanding angular velocity helps us determine how quickly something is spinning, which is crucial in various applications like turbines, engines, and even hard drives.

Angular Deceleration

Angular deceleration occurs when a rotating object's speed decreases over time due to opposing forces such as friction. It's like the brakes in your car, but for rotating objects. In our exercise, the flywheel loses speed because of friction in its axle bearings during a power outage. Deceleration is often depicted by a negative acceleration value, illustrating that it's slowing down.
To find the angular deceleration \(\alpha\), the formula \(\omega = \omega_0 + \alpha t\) is used. Knowing the initial and final angular velocities and the time involved, you can solve for \(\alpha\). For the flywheel, the deceleration was \(-\frac{10\pi}{9} \text{ rad/s}^2\), indicating the speed is decreasing. Grasping angular deceleration helps us manage equipment durability and efficiency, ensuring machines are slowing down safely.

Kinematic Equations

Kinematic equations are pivotal in solving problems of rotational motion, especially when it comes to constant acceleration or deceleration. They help relate a body's angular displacement, initial and final angular velocities, and angular acceleration. In this exercise, these equations are crucial to finding how the flywheel behaves during a power outage.
The equation \(\theta = \frac{1}{2} (\omega_0 + \omega) t\) is used to determine the angular displacement during the deceleration period. It helped us calculate that the flywheel completes 200 rotations or \(400\pi\) radians as it slows down initially. Similarly, the time to stop completely if the power hadn't returned was found using another kinematic equation. These equations are indispensable tools in physics for predicting the motions of rotating objects.

Flywheel Dynamics

Flywheels are fascinating components found in numerous machines. They store rotational energy efficiently and release it later when needed, helping smooth out power delivery in mechanical systems. Flywheel dynamics involve analyzing how these spinning masses react to engagements like friction, as seen in our exercise.
In this scenario, the dynamics of the flywheel are put to the test during a power failure, wherein it continues to rotate due to its inertia. It initially spins at 500 rpm before slowing due to friction in its bearings. Understanding flywheel dynamics aids in designing systems requiring energy stability, like engines, where the flywheel ensures smooth operations by absorbing and releasing energy as needed.