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 (a) through its center?

Verified step by step guidance

1. Identify the shape of the object and the axis about which it is rotating. In this case, the DVD is a circular disk, and the rotation is about an axis perpendicular to the disk through its center.
2. Use the formula for the moment of inertia (I) of a circular disk rotating about an axis perpendicular to the disk through its center: I = \(\frac{1}{2}MR^2\), where M is the mass of the disk and R is the radius.
3. Convert the diameter of the DVD to radius by dividing it by 2. Since the diameter is 12 cm, the radius R = 6 cm.
4. Convert the radius from centimeters to meters for consistency in SI units. Since 1 cm = 0.01 m, R = 0.06 m.
5. Substitute the values of M and R into the formula. Remember to convert the mass from grams to kilograms (since 1 g = 0.001 kg), so M = 0.021 kg.

Key Concepts

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.

Axis of Rotation

The axis of rotation is an imaginary line around 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

Mass and radius are critical parameters in calculating the moment of inertia. The mass of the DVD is given as 21 g, and the radius can be derived from its diameter of 12 cm, which is 6 cm. These values are essential for applying the moment of inertia formula accurately.