Problem 60 - College Physics

Question

A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. Whenever the bus was stopped at a station, the wheel was brought up to speed with the use of an electric motor that could then be attached to the electric power lines. The flywheel was a solid cylinder with a mass of 1000 \(\mathrm{kg}\) and a diameter of \(1.80 \mathrm{m} ;\) its top angular speed was 3000 \(\mathrm{rev} / \mathrm{min.}\) At this angular speed, what was the kinetic energy of the flywheel?

Step-by-Step Solution

Answer

The kinetic energy of the flywheel is approximately 19,987,980 joules.

Step 1: Convert Angular Speed from RPM to Radians per Second

First, you need to convert the angular velocity from revolutions per minute (rpm) to radians per second. Use the conversion factor, knowing that one revolution is \(2\pi\) radians, and there are 60 seconds per minute.\[\omega = 3000 \times \frac{2\pi}{60} = 100\pi \, \text{rad/s}\]

Step 2: Calculate the Moment of Inertia for the Flywheel

The moment of inertia \(I\) for a solid cylinder is given by the formula:\[I = \frac{1}{2} m r^2\]Where:- \(m\) is the mass of the flywheel: 1000 kg,- \(r\) is the radius of the flywheel: \( \frac{1.80}{2} = 0.90 \, \text{m} \).Substitute the values:\[I = \frac{1}{2} \times 1000 \times (0.90)^2 = 405 \, \text{kg}\cdot\text{m}^2\]

Step 3: Calculate the Kinetic Energy of the Flywheel

The kinetic energy \(K\) of a rotating object is given by:\[K = \frac{1}{2} I \omega^2\]Substitute \(I = 405\, \text{kg}\cdot\text{m}^2\) and \(\omega = 100\pi \, \text{rad/s}\):\[K = \frac{1}{2} \times 405 \times (100\pi)^2\]Simplify and calculate:\[K = 0.5 \times 405 \times 10000 \times \pi^2\]\[K = 2,025,000\pi^2 \, \text{Joules}\]Approximating \(\pi^2 \approx 9.8696\), we find:\[K \approx 19,987,980 \, \text{J}\]

Step 4: Summarize the Result

The final kinetic energy of the flywheel is approximately 19,987,980 joules. This energy represents the amount of rotational kinetic energy stored in the flywheel at its peak angular speed.

Key Concepts

Angular Speed

Angular speed is how fast something spins around an axis. It's similar to how velocity is the speed of how fast something moves forward. When describing angular speed, we often use revolutions per minute (RPM) or radians per second (rad/s).
For calculations, rad/s is preferred because it connects so beautifully with other formulas in physics. In rotational motion, one full revolution equals 2π radians.
So, to switch from RPM to rad/s, you multiply the RPM by 2π and then divide by 60, since there are 60 seconds in a minute. In the given problem, the angular speed of the flywheel was converted into rad/s, which allowed us to find how quickly the flywheel is spinning in a way that's useful for further calculations, like finding kinetic energy.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. For linear motion, you think about mass, but for rotational motion, you think about the moment of inertia. The farther the mass is from the axis of rotation, the larger the inertia.
For a solid cylinder like the flywheel in the exercise, the formula to calculate moment of inertia is \[I = \frac{1}{2} m r^2\] where \(m\) is the mass and \(r\) is the radius. This formula highlights that not only the mass but also the distribution of this mass (how far it is from the axis) matters a lot in rotational dynamics.
The moment of inertia was calculated as 405 kg·m² for the given flywheel, indicating how its mass and shape influence its rotational properties.

Rotational Motion

Rotational motion involves objects spinning around a central point or axis. Unlike linear motion, rotational motion takes into account angular quantities like angular speed, angular acceleration, and angular displacement. Just like in linear motion, where Newton's laws can be applied, rotational motion has analogous principles.
The key aspects of rotational motion include the relationship between torque, which is a force that causes rotation, and the moment of inertia which we discussed earlier. In the context of the exercise, the flywheel's rotation allows it to store energy in the form of rotational kinetic energy, which is later used to drive the bus.
This stored energy is what makes flywheels valuable in applications where consistent power over a stretched time is appreciated.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed. In the context of the flywheel, the energy from the electric motor converts into kinetic energy as the flywheel spins faster.
The equation for rotational kinetic energy is \[K = \frac{1}{2} I \omega^2\] In this formula, the kinetic energy \(K\) increases with both the moment of inertia \(I\) and the square of the angular speed \(\omega\). The stored energy in the flywheel can then be used to power the bus when the electric current isn't available, making use of energy already stored without wasting it. This aspect of energy conservation is a fundamental concept in physics and engineering, optimizing resources by transforming energy from one form to another efficiently.