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
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 (
**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:
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:
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:
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:
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.