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.

검증된 단계별 안내

Step 1: Recall the formula for the moment of inertia of a hoop about an axis perpendicular to its plane and passing through its center. For a hoop of mass \( M \) and radius \( R \), the moment of inertia is \( I = M R^2 \). However, in this problem, the axis is at the edge of the hoop, not the center.

Step 2: Use the parallel axis theorem to account for the shift in the axis. The parallel axis theorem states \( I = I_{\text{center}} + M d^2 \), where \( I_{\text{center}} \) is the moment of inertia about the center, \( M \) is the mass, and \( d \) is the distance between the center and the new axis.

Step 3: Determine the distance \( d \) between the center of the hoop and the edge. Since the hoop has a radius \( R \), the distance \( d \) is equal to \( R \).

Step 4: Substitute \( I_{\text{center}} = M R^2 \) and \( d = R \) into the parallel axis theorem formula. This gives \( I = M R^2 + M R^2 \).

Step 5: Combine the terms to express the total moment of inertia. The result is \( I = 2 M R^2 \). This is the moment of inertia of the hoop about the specified axis.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its 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 each particle and r is the distance from the axis of rotation.

Parallel Axis Theorem

The parallel axis theorem allows us to calculate 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 total mass, and d is the distance between the two axes. This theorem is essential for finding the moment of inertia of the hoop about an edge.

Geometry of the Hoop

A hoop is defined as a thin-walled, hollow ring with all its mass concentrated at a constant distance (the radius R) from the axis of rotation. Understanding the geometry of the hoop is crucial for applying the moment of inertia formulas correctly, as it directly influences the calculations and the distribution of mass relative to the chosen axis.