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 (b) through the edge of the disk?

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 disk, and it is rotating about an axis perpendicular to its surface through its edge.
2. Use the formula for the moment of inertia of a disk rotating about an axis through its center, which is $I_{center} = \frac{1}{2} M R^2$, where $M$ is the mass of the disk and $R$ is its radius.
3. Convert the diameter of the DVD to radius by dividing it by 2. Since the diameter is 12 cm, the radius $R$ is 6 cm.
4. Apply the parallel axis theorem to find the moment of inertia about the new axis (through the edge). The theorem states $I = I_{center} + M d^2$, where $d$ is the distance from the center of mass to the new axis. Here, $d$ equals the radius $R$ of the disk.
5. Substitute the values into the modified moment of inertia formula and solve for $I$. Remember to convert the mass from grams to kilograms for consistency in units (1 g = 0.001 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 disk, the moment of inertia can be calculated using the formula I = (1/2) m r^2 for rotation about its center, but it changes when the axis is shifted, requiring the parallel axis theorem.

Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia about any axis parallel to an axis through the center of mass can be found by adding the product of the mass and the square of the distance between the two axes to the moment of inertia about the center of mass. This is crucial for calculating the moment of inertia when the axis of rotation is not through the center.

Rotational Dynamics

Rotational dynamics is the study of the motion of rotating bodies and the forces that cause this motion. It encompasses concepts such as torque, angular momentum, and the relationship between linear and angular quantities. Understanding these principles is essential for analyzing how objects like the DVD behave when subjected to rotational forces.