Different Shapes Rolling Down an Incline
Unit: Torque and Rotational Dynamics
Later Topics
Multi-Step Problem Preview
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Part 1
A hoop, solid cylinder, solid sphere, and thin spherical shell, each with the same mass and radius , are released from rest at the top of a rough incline of angle and length .
Rank these objects from highest to lowest moment of inertia about their central axis.
Correct!
The moment of inertia depends on how far each mass element is from the rotation axis. Mass located farther from the axis contributes more to the moment of inertia.
Hoop: All mass is concentrated at distance from the axis. This maximizes the moment of inertia.
Thin spherical shell: Mass is distributed over the surface of a sphere. Some mass lies on the rotation axis (at the poles), while mass at the equator is at distance . On average, mass is closer to the axis than in a hoop.
Solid cylinder: Mass is distributed uniformly from the axis out to radius . The average distance of mass elements from the axis is less than , placing it below the thin spherical shell.
Solid sphere: Mass is distributed throughout the interior, including regions very close to the axis. This places the average mass element closer to the rotation axis than in a solid cylinder.
Therefore, ranking from highest to lowest moment of inertia:
Hoop, thin spherical shell, solid cylinder, solid sphere
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