Chapter 9: Problem 60
You are designing a rotating metal flywheel that will be used to store energy. The flywheel is to be a uniform disk with radius 25.0 cm. Starting from rest at \(t =\) 0, the flywheel rotates with constant angular acceleration 3.00 rad/s\(^2\) about an axis perpendicular to the flywheel at its center. If the flywheel has a density (mass per unit volume) of 8600 kg/m\(^3\), what thickness must it have to store 800 J of kinetic energy at \(t =\) 8.00 s?
Short Answer
Step by step solution
Understanding the Physical Problem
Kinetic Energy of the Flywheel
Moment of Inertia for a Uniform Disk
Calculating Angular Velocity at t = 8s
Solving for Mass Using Kinetic Energy Formula
Solve for Volume and Ultimately Thickness
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rotational Kinetics
A key concept in rotational kinetics is **torque**, which is the rotational equivalent of linear force. Torque causes an angular acceleration, which changes the angular velocity of the rotating body. The relationship between torque (\( \tau \)) and angular acceleration (\( \alpha \)) is given by the equation:\[ \tau = I \alpha \]where \( I \) is the moment of inertia.
**Rotational kinetic energy** is another critical aspect of rotational kinetics. It's the energy an object possesses due to its rotation and is given by:\[ K = \frac{1}{2} I \omega^2 \]where \( \omega \) is the angular velocity. This equation tells us that a higher angular velocity or a larger moment of inertia will lead to more kinetic energy stored in a rotating object. Understanding rotational kinetics is essential for designing machines and engines, such as flywheels, which require precise control over rotational motion.
Moment of Inertia
For different shapes, the moment of inertia is calculated differently. Specifically for a uniform disk, like the flywheel in the exercise, the formula to find the moment of inertia is:\[ I = \frac{1}{2} m R^2 \]where \( m \) is the mass of the disk and \( R \) is the radius. This means that increasing either the mass or the radius of a disk will increase its moment of inertia, making it harder to start or stop its rotation.
The moment of inertia plays a crucial role in determining the rotational kinetic energy. A higher moment of inertia implies that more energy is needed to achieve a particular angular velocity, which is a key consideration in both designing and analyzing rotating systems like flywheels.
Angular Velocity
Angular velocity can be calculated using the equation:\[ \omega = \alpha t \]where \( \alpha \) is the angular acceleration and \( t \) is time. In the given problem, the flywheel starts from rest and, given an angular acceleration of 3.00 rad/sĀ², it reaches an angular velocity of 24.00 rad/s after 8 seconds.
This understanding of angular velocity is fundamental in rotational dynamics because it directly influences the rotational kinetic energy. When the angular velocity increases, the kinetic energy of the rotating body also increases, demonstrating the importance of precise control over rotational speeds in mechanical designs.
Energy Storage in Flywheels
The energy stored in a flywheel is determined by its rotational kinetic energy, which is given by the formula:\[ K = \frac{1}{2} I \omega^2 \]where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. A flywheel's ability to store energy efficiently makes it an important component in various applications, such as in energy recovery systems and mechanical batteries.
To maximize energy storage in a flywheel, you can either increase its moment of inertia or its angular velocity. Engineers often modify the flywheel's geometry, such as its thickness or radius, to optimize energy storage without making the device too heavy or occupying too much space. As in the exercise, calculating the required dimensions and materials is key to designing an effective flywheel.