Question

The minute and hour hands of a clock have a common axis of rotation and equal mass. The minute hand is long, thin, and uniform; the hour hand is short, thick, and uniform. Is the moment of inertia of the minute hand greater than, less than, or equal to the moment of inertia of the hour hand?

a. greater than
b. less than
c. equal to

Answer 1

The moment of inertia of the minute hand is greater than that of the hour hand. This is because the longer structure of the minute hand means more of its mass is distributed farther from the axis, leading to a larger moment of inertia compared to the shorter, thicker hour hand.

Step-by-step explanation:

The moment of inertia depends on how the mass of an object is distributed in relation to its axis of rotation. Given that the minute hand is longer, the majority of its mass is farther from the axis of rotation, resulting in a greater moment of inertia compared to the shorter hour hand, where the mass is closer to the axis of rotation. Considering both hands have an equal mass, the longer minute hand will have a greater moment of inertia because for a given shape, the moment of inertia increases as the square of the distance from the axis.

To answer the given question, the moment of inertia of the minute hand is greater than that of the hour hand. This is because the minute hand's length causes more mass to be farther from the axis of rotation as outlined in the rotational inertia concepts, analogous to a long rod spun around an axis through one end which has the moment of inertia ML² /3, greater than if the mass were concentrated at the center of mass (ML² /4).