A 12-cm-diameter DVD has a mass of 21 g. What is the DVD’s moment of inertia for rotation about a perpendicular axis through its center?

검증된 단계별 안내

Step 1: Identify the shape of the DVD. The DVD can be approximated as a uniform solid disk. The formula for the moment of inertia of a solid disk rotating about an axis perpendicular to its center is \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass and \( r \) is the radius.

Step 2: Convert the given diameter of the DVD into its radius. The diameter is 12 cm, so the radius \( r \) is half of the diameter: \( r = \frac{12}{2} = 6 \, \text{cm} \). Convert this to meters: \( r = 0.06 \, \text{m} \).

Step 3: Convert the mass of the DVD from grams to kilograms. The mass is given as 21 g, and since \( 1 \, \text{kg} = 1000 \, \text{g} \), the mass in kilograms is \( m = \frac{21}{1000} = 0.021 \, \text{kg} \).

Step 4: Substitute the values of \( m \) and \( r \) into the formula for the moment of inertia: \( I = \frac{1}{2} m r^2 \). This becomes \( I = \frac{1}{2} \times 0.021 \times (0.06)^2 \).

Step 5: Perform the calculation to find the moment of inertia. Ensure that the units are consistent (mass in kilograms and radius in meters) and simplify the expression to obtain the final value of \( I \).

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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. It depends on the mass distribution relative to the axis of rotation. For a solid disk, the moment of inertia can be calculated using the formula I = (1/2) m r², where m is the mass and r is the radius.

Rotational Axis

The rotational axis is the line about which an object rotates. In this case, the axis is perpendicular to the plane of the DVD and passes through its center. The choice of axis significantly affects the moment of inertia, as it determines how the mass is distributed relative to that axis.

Mass and Radius

The mass and radius of an object are critical parameters in calculating its moment of inertia. The mass (21 g) indicates how much matter is present, while the radius (6 cm, derived from the 12 cm diameter) affects how far the mass is distributed from the axis of rotation. Both factors are essential for determining the object's resistance to rotational acceleration.