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Ch 09: Rotation of Rigid Bodies
All textbooksYoung & Freedman Calc 14th EditionCh 09: Rotation of Rigid BodiesProblem 9
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Chapter 9, Problem 9

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass M and radius R about an axis perpendicular to the hoop's plane at an edge.

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Identify the given parameters: mass of the hoop (M) and radius (R).
Recognize that the moment of inertia for a hoop about an axis through its center perpendicular to its plane is given by the formula $I_{center} = MR^2$.
Apply the parallel axis theorem to find the moment of inertia about an axis at the edge of the hoop. The parallel axis theorem states that $I = I_{center} + Md^2$, where $d$ is the distance from the new axis to the original axis through the center.
Since the axis is now at the edge of the hoop, the distance $d$ from the center of the hoop to the edge is equal to the radius of the hoop, $R$.
Substitute $d = R$ into the parallel axis theorem formula to get $I = MR^2 + MR^2 = 2MR^2$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It depends on the mass distribution relative to that axis. For a hoop, the moment of inertia can be calculated using the formula I = Σ(m * r²), where m is the mass of the particles and r is their distance from the axis of rotation.
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Parallel Axis Theorem

The parallel axis theorem allows us to find the moment of inertia of an object about any axis parallel to an axis through its center of mass. It states that I = I_cm + Md², where I_cm is the moment of inertia about the center of mass, M is the mass of the object, and d is the distance between the two axes. This theorem is essential for calculating the moment of inertia of the hoop about an edge.
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Thin-Walled Hoop

A thin-walled hoop is a hollow circular object with negligible thickness compared to its radius. Its mass is uniformly distributed along its circumference. The moment of inertia for a thin-walled hoop about an axis through its center is given by I = MR². However, when calculating for an edge, the parallel axis theorem must be applied to adjust this value.
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