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Chapter 10: Problem 77

You're skeptical about a new hybrid car that stores energy in a flywheel. The manufacturer claims the flywheel stores \(12 \mathrm{MJ}\) of energy and can supply \(40 \mathrm{kW}\) of power for 5 minutes. You dig deeper and find that the flywheel is a 39 -cm-diameter ring with mass \(48 \mathrm{kg}\) that rotates at 30,000 rpm. Are the specs correct?

Short Answer

Expert verified
After calculating, compare the calculated energy and power values with the claimed ones. If they match, the manufacturer's specs are correct. Otherwise, they are not.

Step by step solution

01

Calculate the Energy Stored in the Flywheel

The energy stored in a flywheel can be calculated using the formula for the kinetic energy of a rotating object, which is \( KE = \frac{1}{2}I蠅^2 \), where \( I \) is the moment of inertia and \( 蠅 \) is the angular velocity. First, we need to calculate \( I \) and \( 蠅 \). For a ring-shaped object, \( I = MR^2 \), where \( M \) is the mass and \( R \) is the radius (half the diameter). Given \( M = 48 \, \mathrm{kg} \) and \( D = 39 \, \mathrm{cm} \), we have \( R = \frac{39 \, \mathrm{cm}}{2} = 0.195 \, \mathrm{m} \), so \( I = (48 \, \mathrm{kg})(0.195 \, \mathrm{m})^2 \). 蠅 can be found by converting the given rpm to radians per second, using the conversion factor \( \frac{2蟺 \, \mathrm{rad}}{1 \, \mathrm{rpm}} \). Given \( \mathrm{rpm} = 30000 \), we have \( 蠅 = 30000\frac{2蟺 \, \mathrm{rad}}{60 \, \mathrm{s}} \). Substituting \( I \) and \( 蠅 \) into the formula for kinetic energy gives the total energy of the flywheel in Joules.
02

Compare the Calculated Energy with the Claimed Energy

The claimed energy stored by the flywheel is \( 12 \, \mathrm{MJ} = 12 脳 10^6 \, \mathrm{J} \). After calculating the actual energy stored in the flywheel, compare this value with the claimed value to verify the manufacturer's claim.
03

Calculate the Flywheel's Power Output over 5 Minutes

Power is defined as the energy supplied per unit time, i.e., \( P = \frac{E}{t} \), where \( E \) is the energy and \( t \) is the time. Using the calculated total energy and the given time of 5 minutes (which is 300 seconds), we can calculate the power output of the flywheel.
04

Compare the Calculated Power with the Claimed Power

The claimed power that the flywheel can supply is\( 40 \, \mathrm{kW} = 40 脳 10^3 \, \mathrm{W} \). Compare the calculated power with the claimed value to verify the manufacturer's claim.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy of Rotating Objects
Kinetic energy is the energy possessed by an object due to its motion. For rotating objects like flywheels, kinetic energy is not just based on the speed the object travels through space, but also on how fast it is spinning. The mathematical expression for the kinetic energy (KE) of a rotating object is given by the formula:

\[ KE = \frac{1}{2}I\omega^2 \]

Here, \( I \) stands for the moment of inertia, which represents how mass is distributed in the object relative to the axis of rotation, and \( \omega \) is the angular velocity, a measure of the object's angular speed. Understanding this concept is crucial since the actual energy stored in a flywheel, such as the one mentioned in the exercise, derives entirely from its rotational motion.
Moment of Inertia
The moment of inertia (\( I \)) is a fundamental concept when analyzing the dynamics of rotating systems. It is the rotational equivalent of mass for linear motion, playing a key role in kinetic energy storage in flywheels. The moment of inertia depends on both the mass of the object and how this mass is distributed with respect to the axis of rotation. For a ring-shaped object like the flywheel in the exercise, the moment of inertia is calculated using the formula:

\[ I = MR^2 \]

where \( M \) is the mass of the object and \( R \) is the radius. The radius is particularly important - if more mass is located further from the axis, the moment of inertia increases, enhancing the flywheel's energy storage capacity.
Angular Velocity
Angular velocity (\( \omega \)) is a measure of how quickly an object rotates around a particular axis. It's not to be confused with linear velocity, which measures how fast an object is moving along a path. Angular velocity is particularly important for flywheels since their energy storage capability is proportional to the square of angular velocity. Angular velocity is typically measured in radians per second (rad/s). To find it from revolutions per minute (rpm), as in the exercise, you can use the conversion factor \( \frac{2\pi \, \text{rad}}{60 \, \text{s}} \) because there are \( 2\pi \) radians in one revolution and 60 seconds in one minute. The equation is:

\[ \omega = rpms \times \frac{2\pi}{60} \]
Power Output Calculation
To understand the performance of a flywheel energy storage system, one needs to calculate the power output. Power is the rate at which energy is transferred or transformed. In terms of a flywheel, it is the rate of energy release. To determine this, we use the formula:

\[ P = \frac{E}{t} \]

where \( P \) is the power in watts, \( E \) is the energy in joules, and \( t \) is the time in seconds. The exercise's car manufacturer claims a power output for a given duration, which can be verified by checking whether the energy stored in the flywheel can sustain the claimed power level for the specified time frame. It's important to convert time to seconds and power to watts to ensure the correct units are being used when performing these calculations.

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