Questions: Moment of Inertia

Question 1

A solid disk and a hollow ring have identical mass M and radius R. Which has the greater moment of inertia about its central axis?

• A. The solid disk (I = ½MR²)
• B. The hollow ring (I = MR²)
• C. They are equal
• D. It depends on the angular velocity

Answer: (B)

The hollow ring concentrates all its mass at radius R, giving I = MR². The solid disk spreads mass from 0 to R, so the average r² is less than R², yielding I = ½MR². Because I = ∫r² dm, mass farther from the axis contributes disproportionately more (it scales as r²), so the ring wins.

Question 2

The moment of inertia of an object is a fixed property of that object, independent of the chosen rotation axis.

• T. True
• F. False

Answer: False

I depends on both the mass distribution and the choice of rotation axis. The same rod has I = (1/12)ML² when spun about its center but I = (1/3)ML² when spun about one end — a factor of four difference. Moment of inertia is always defined relative to a specific axis.

Question 3

You know a solid disk's moment of inertia about its central axis is I_cm = ½MR². Using the parallel axis theorem, what is I about an axis parallel to the center but passing through the rim?

Model answer: (3/2)MR²

The parallel axis theorem states I = I_cm + Md², where d is the perpendicular distance from the center-of-mass axis to the new axis. Here d = R, so I = ½MR² + MR² = (3/2)MR². A common error is applying the theorem with d measured from an arbitrary axis rather than specifically from the center-of-mass axis.