Problem 57 - 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 20,000,000 J (20 MJ).

Step 1: Calculate the Moment of Inertia

The moment of inertia for a solid cylinder is given by \( I = \frac{1}{2} M R^2 \), where \( M \) is the mass and \( R \) is the radius. The radius \( R \) is half of the diameter, so \( R = 0.90 \text{ m} \). Thus, the moment of inertia is \( I = \frac{1}{2} \times 1000 \text{ kg} \times (0.90 \text{ m})^2 = 405 \text{ kg} \cdot \text{m}^2 \).

Step 2: Convert Angular Speed to Radians per Second

The angular speed is given in revolutions per minute (\( \mathrm{rev} / \min \)) and must be converted to radians per second (\( \mathrm{rad} / \sec \)). Using the conversion factors \( 1 \mathrm{rev} = 2\pi \mathrm{rad} \) and \( 1 \min = 60 \seconds \), we have: \( \omega = 3000 \mathrm{rev} / \mathrm{min} \times \frac{2\pi \mathrm{rad}}{1\mathrm{rev}} \times \frac{1 \mathrm{min}}{60 \mathrm{sec}} = 314.16 \mathrm{rad} / \mathrm{sec} \).

Step 3: Calculate the Kinetic Energy

The kinetic energy of a rotating object is given by \( KE = \frac{1}{2} I \omega^2 \), where \( \omega \) is the angular speed in radians per second. Substitute in the values: \( KE = \frac{1}{2} \times 405 \text{ kg} \cdot \text{m}^2 \times (314.16 \mathrm{rad} / \mathrm{sec})^2 = 20,000,000 \text{ J} \).

Key Concepts

Flywheel Energy Storage

Flywheel energy storage is a clever method for storing and retrieving energy. Imagine the flywheel as a heavy spinning disk that can store kinetic energy through its rotation. This energy can be released when needed to power systems, like the Zurich bus. The key advantage of using flywheels is their ability to quickly release energy, which makes them highly useful in applications requiring frequent start and stop cycles. Some benefits of flywheel energy storage include:

• High power output: They rapidly deliver stored energy.
• Long lifecycle: They can withstand countless cycles without performance degradation.
• No toxic chemicals: Unlike batteries, flywheels do not involve harmful materials.

In the context of the bus in Zurich, the flywheel stored energy when the bus was charged at stations. By doing so, it allowed the bus to travel between stations without using any direct power from electrical lines. This is not only environmentally friendly but also efficient in reducing the need for constant energy input from external sources.

Moment of Inertia

The moment of inertia is a fundamental concept in understanding the rotational dynamics of objects. It describes how the mass of an object is distributed relative to the axis of rotation. Think of it as a rotational equivalent of mass in linear motion. For the flywheel, which is a solid cylinder, the moment of inertia (I) helps determine how much torque is needed to change its angular speed.The formula for the moment of inertia of a solid cylinder is:\[ I = \frac{1}{2} M R^2 \]where \( M \) is the mass and \( R \) is the radius. In our case, the mass is 1000 kg and the radius is 0.90 meters (half of the diameter). Thus, the calculation of \( I \) becomes straightforward:\[ I = \frac{1}{2} \times 1000 \times (0.90)^2 = 405 \text{ kg} \cdot \text{m}^2 \] Understanding this concept is crucial because it influences how the flywheel behaves under different forces. A larger moment of inertia means the flywheel is harder to accelerate or decelerate, impacting how effectively the energy is stored or released.

Angular Speed Calculation

Angular speed describes how quickly an object rotates. In this context, it's crucial to convert angular speed into the correct units to calculate kinetic energy correctly. The given angular speed was 3000 revolutions per minute, but to use it in equations effectively, we need it in radians per second.Radians are used because they provide a direct measure of an angle in relation to the radius of a circle, making calculations involving circles very straightforward. The conversion process involves:

• 1 revolution = \(2\pi\) radians
• 1 minute = 60 seconds

To get the angular speed in radians per second, the conversion goes as follows:\[ \omega = 3000 \times \frac{2\pi}{1} \times \frac{1}{60} = 314.16 \text{ rad/s} \]This conversion is crucial as the kinetic energy formula requires angular speed in radians per second for accurate results. A proper understanding here ensures that calculations regarding the flywheel's stored energy are both correct and efficient.