Problem 88 - University Physics with Modern Physics

Question

A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. The wheel was brought up to speed periodically, when the bus stopped at a station, by an electric motor, which could then be attached to the electric power lines. The flywheel was a solid cylinder with mass 1000 \(\mathrm{kg}\) and diameter \(1.80 \mathrm{m} ;\) its top angular speed was 3000 \(\mathrm{rev} / \mathrm{min}\) . (a) At this angular speed, what is the kinetic energy of the flywheel? (b) If the average power required to operate the bus is \(1.86 \times 10^{4} \mathrm{W},\) how long could it operate between stops?

Step-by-Step Solution

Answer

(a) The flywheel's kinetic energy is approximately 20,000,000 J. (b) The bus could operate for about 1075 seconds between stops.

Step 1: Calculate Moment of Inertia

The flywheel is a solid cylinder, so we use the formula for the moment of inertia of a solid cylinder, which is \( I = \frac{1}{2} m r^2 \). First, we find the radius \( r \) from the diameter: \( r = \frac{1.80 \mathrm{m}}{2} = 0.90 \mathrm{m} \). We know the mass \( m = 1000 \mathrm{kg} \). Now we compute: \[ I = \frac{1}{2} \times 1000 \times (0.90)^2 = 405 \, \mathrm{kg \cdot m^2} \]

Step 2: Convert Angular Speed to Radians per Second

The maximum angular speed is given in revolutions per minute (\( \mathrm{rev/min} \)), so we convert to radians per second (\( \mathrm{rad/s} \)) using the conversion factor \( 2\pi \) radians per revolution and 60 seconds per minute:\[ \omega = 3000 \times \frac{2\pi}{60} = 3000 \times \frac{\pi}{30} = 100 \pi \approx 314.16 \mathrm{rad/s} \]

Step 3: Calculate Kinetic Energy

The kinetic energy of a rotating object is given by \( KE = \frac{1}{2} I \omega^2 \). Substitute the moment of inertia \( I = 405 \mathrm{kg \cdot m^2} \) and angular velocity \( \omega \approx 314.16 \mathrm{rad/s} \) into the formula:\[ KE = \frac{1}{2} \times 405 \times (314.16)^2 \approx 20,000,000 \mathrm{J} \]

Step 4: Determine Operating Time Between Stops

To find how long the bus can operate between stops, use the formula \( t = \frac{E}{P} \), where \( E \) is the kinetic energy and \( P \) is the power consumption. The kinetic energy \( E \approx 20,000,000 \) J from Step 3 and the power requirement \( P = 1.86 \times 10^4 \) W. Therefore:\[ t = \frac{20,000,000}{1.86 \times 10^4} \approx 1075 \mathrm{s} \]

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In the case of rotational motion, such as that of the flywheel used in the bus, kinetic energy depends on how fast it spins around its axis. This type of energy can be calculated with the formula: \( KE = \frac{1}{2} I \omega^2 \).
Here, \( I \) is the moment of inertia, and \( \omega \) is the angular speed. In our example of the flywheel bus, the kinetic energy is calculated using the flywheel's moment of inertia (405 kg·m²) and its angular speed (approximately 314.16 rad/s). Substituting these values into the formula gives us the kinetic energy of approximately 20,000,000 joules.
- **Key takeaway**: The kinetic energy tells us how much energy is stored in the flywheel at its top speed. The greater the kinetic energy, the more potential motion the flywheel contains.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. For a solid cylinder, like the flywheel in the bus, the moment of inertia can be calculated with the formula: \( I = \frac{1}{2} m r^2 \). It depends on the mass and the radius of the object.
In the given exercise, we found the moment of inertia of the bus's flywheel. Here, the flywheel has a mass of 1000 kg and a radius of 0.90 m, leading to a moment of inertia of 405 kg·m².
- **Key insight**: A larger moment of inertia means it's harder to change the rotational speed of the flywheel. It's a key factor in the amount of kinetic energy the flywheel can store.

Angular Speed

Angular speed describes how fast an object rotates or revolves. It's often measured in revolutions per minute (rev/min) but can be converted to radians per second (rad/s) for calculations in physics. The relation is given by: \( \omega = \text{revolution rate} \times \frac{2\pi}{60} \).
In this scenario, the flywheel's top speed is 3000 rev/min, which converts to about 314.16 rad/s.
- **Key consideration**: Higher angular speed results in more kinetic energy in the system. Converting angular speed into radians per second is crucial for using standard physics equations.

Power Consumption

Power consumption refers to the rate at which energy is used. In this exercise, the bus's average power requirement is 18,600 watts, or 18.6 kW. To find out how long the bus can keep running before needing to recharge the flywheel, we use the formula \( t = \frac{E}{P} \), where \( E \) is the energy supplied by the flywheel, and \( P \) is the power consumption.
With a kinetic energy of 20,000,000 joules and a power consumption of 18,600 watts, the bus can operate for about 1075 seconds between stops. - **Key concept**: Power consumption allows you to determine how long the stored energy will last, which is crucial for assessing the bus's operational efficacy.