A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door’s moment of inertia for rotation about a vertical axis inside the door, 15 cm from one edge?

검증된 단계별 안내

Step 1: Understand the problem. The moment of inertia depends on the axis of rotation and the distribution of mass relative to that axis. Here, the door rotates about a vertical axis located 15 cm from one edge. The door can be treated as a rectangular solid with uniform mass distribution.

Step 2: Use the parallel axis theorem to calculate the moment of inertia. The theorem states: \( I = I_{cm} + Md^2 \), where \( I_{cm} \) is the moment of inertia about the center of mass, \( M \) is the mass of the door, and \( d \) is the distance from the center of mass to the new axis of rotation.

Step 3: Calculate \( I_{cm} \) for a rectangular solid rotating about an axis parallel to one of its edges. The formula is \( I_{cm} = \frac{1}{3} M h^2 \), where \( h \) is the width of the door (91 cm or 0.91 m). Substitute \( M = 25 \, \text{kg} \) and \( h = 0.91 \, \text{m} \) into the formula.

Step 4: Determine the distance \( d \) from the center of mass to the new axis of rotation. The center of mass is located at the midpoint of the door's width, which is \( \frac{91}{2} = 45.5 \, \text{cm} \). The new axis is 15 cm from one edge, so \( d = |45.5 - 15| = 30.5 \, \text{cm} \) or \( 0.305 \, \text{m} \).

Step 5: Combine the results using the parallel axis theorem. Substitute \( I_{cm} \) and \( d \) into \( I = I_{cm} + Md^2 \). Perform the necessary algebraic operations to express the moment of inertia in terms of the given values.

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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 an axis. It depends on the mass distribution relative to the axis of rotation. For rigid bodies, it is calculated by integrating the mass elements multiplied by the square of their distance from the axis. The greater the distance of mass from the axis, the higher the moment of inertia.

Parallel Axis Theorem

The parallel axis theorem allows us to calculate the moment of inertia of a body about any axis parallel to an axis through its center of mass. It states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass axis plus the product of the mass and the square of the distance between the two axes. This theorem is essential for determining the moment of inertia when the axis of rotation is not through the center of mass.

Geometry of the Door

Understanding the geometry of the door is crucial for calculating its moment of inertia. The door can be approximated as a rectangular solid, and its dimensions (height and width) will influence the distribution of mass. The specific axis of rotation, in this case, 15 cm from one edge, requires careful consideration of how the mass is distributed relative to that axis, impacting the final calculation of the moment of inertia.