Chapter 9: Problem 36
A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at \(t =\) 0, the wheel turns through 8.20 revolutions in 12.0 s. At \(t =\) 12.0 s the kinetic energy of the wheel is 36.0 J. For an axis through its center, what is the moment of inertia of the wheel?
Short Answer
Step by step solution
Convert Revolutions to Radians
Use Angular Displacement Formula
Calculate Final Angular Velocity
Use Kinetic Energy Formula to Find Moment of Inertia
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Displacement
In our example, the wheel turns through 8.20 revolutions, which needs to be translated into radians for most calculations. Since one full revolution equals \(2\pi\) radians, the angular displacement \(\theta\) becomes \(16.40\pi\) radians.
- Revolutions to radians conversion is essential in these problems.
- Angular displacement helps understand the extent of rotation.
Angular Velocity
- \(\omega = \omega_0 + \alpha t\)
- For objects starting from rest: \(\omega = \alpha t\)
Angular Acceleration
- This represents constant acceleration in rotational motion.
- Critical for calculating the subsequent angular velocity.
Kinetic Energy
- Shows how the wheel's ability to resist rotational changes relates to its energy.
- Relates energy and rotational dynamics.