Question

part a

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?

less than
greater than
equal to

part b

choose the best explanation from among the following:

the hands have equal mass, and hence equal moments of inertia.
the more compact hour hand concentrates its mass and has the greater moment of inertia.
having mass farther from the axis of rotation results in a greater moment of inertia.

Explanation:

Analyze the physical system

Using the Moment of Inertia and Mass Distribution in Rotation knowledge points

$$ I = \int r^2 dm $$

The minute hand is long and thin, while the hour hand is short and thick. Both hands have equal mass \(M\). Because the minute hand is longer, its mass is distributed farther from the common axis of rotation on average compared to the shorter hour hand.

Compare the moments of inertia

Using the Mass Distribution in Rotation knowledge point

$$ R_{\text{minute}} > R_{\text{hour}} \implies I_{\text{minute}} > I_{\text{hour}} $$

Since moment of inertia increases as mass is distributed farther from the rotational axis, the moment of inertia of the minute hand is greater than that of the hour hand.

Select the best explanation

Using the Moment of Inertia knowledge point

$$ I \propto \text{distance}^2 $$

Distributing the same mass farther from the axis of rotation increases the rotational inertia. Therefore, having mass farther from the axis of rotation results in a greater moment of inertia.

Answer:

Question 1

• less than
• greater than (Correct answer)
• equal to

Question 2

• The hands have equal mass, and hence equal moments of inertia.
• The more compact hour hand concentrates its mass and has the greater moment of inertia.
• Having mass farther from the axis of rotation results in a greater moment of inertia. (Correct answer)