A solid disc and a solid sphere have the same mass and same radius. Which one has the higher moment of inertia about its centre of mass?
- A.The disc
- B.The sphere
- C.Both have the same moment of inertia
- D.The information provided is not sufficient to answer the question
Correct answer
Explanation
The correct answer is A, the disc. About an axis through the centre, a solid disc has I = ½MR² while a solid sphere has I = ⅖MR², and ½ is larger than ⅖.
Moment of inertia measures how hard it is to change a body's rotation, and it depends on how far the mass lies from the axis. In a flat disc all the mass is spread out in one plane at distances up to R from the axis. In a sphere much of the mass sits near the poles, close to the axis, so with the same mass and radius it has less moment of inertia: 0.4MR² against 0.5MR². This is why, rolling down the same slope, a solid sphere reaches the bottom before a solid disc.
B is wrong because ⅖MR², that is 0.4MR², is smaller than 0.5MR².
C is wrong because the two shapes distribute their mass differently.
D is wrong because equal mass and radius are enough to compare them.
Exam tip: ring MR², disc ½MR², hollow sphere ⅔MR², solid sphere ⅖MR²; the smaller I, the faster it rolls.
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