A thin disc and a thin ring, both have mass M and radius R. Both rotate about axes through their center of mass and are perpendicular to their surfaces at the same angular velocity. Which of the following is true?
- (a)The ring has higher kinetic energy
- (b)The disc has higher kinetic energy
- (c)The ring and the disc have the same kinetic energy
- (d)Kinetic energies of both the bodies are zero since they are not in linear motion
Answer
Why
Correct — A, the ring has higher kinetic energy. For rotation about the central axis perpendicular to the plane, the moment of inertia of a ring is I = MR² while that of a disc is I = ½MR². Rotational kinetic energy is ½Iω², so at the same M, R and angular velocity ω the ring, with the larger moment of inertia, stores twice the rotational kinetic energy of the disc.
Why the others are wrong
- (b)The disc has higher kinetic energy — The disc has the smaller moment of inertia (½MR² < MR²), so at equal ω it stores less kinetic energy, not more.
- (c)The ring and the disc have the same kinetic energy — Equal kinetic energy would need equal moments of inertia, but the ring carries all its mass at radius R while the disc spreads mass from centre to rim, so their I values differ (MR² vs ½MR²).
- (d)Kinetic energies of both the bodies are zero since they are not in linear motion — A spinning body has rotational kinetic energy ½Iω² even without any translation; the energy is zero only if ω = 0.
Concept
Moment of inertia measures a body's resistance to angular acceleration and depends on how the mass is distributed about the axis (I = Σmr²): mass farther from the axis counts more. A ring keeps all its mass at radius R, while a disc has mass spread from the centre outward, so the ring's moment of inertia is the larger of the two.
Reduce the comparison to moments of inertia, since ½Iω² has the same ω for both. Ring = MR², disc = ½MR², so the ring's kinetic energy is exactly twice the disc's. The distractor about 'no linear motion' is meant to make you forget that rotation alone carries energy.
Key facts
- Moment of inertia of a ring about its central perpendicular axis is MR².
- Moment of inertia of a disc about the same axis is ½MR².
- Rotational kinetic energy is ½Iω².
- At equal M, R and ω the ring stores twice the disc's rotational kinetic energy.
Same mass, radius and ω, but the ring's larger moment of inertia gives it the higher kinetic energy.
Study next
Common traps
- Assuming the more solid-looking disc must store more energy — kinetic energy tracks moment of inertia, not bulk.
- Thinking a spinning body has zero energy because it is not moving in a straight line.
Two shapes are compared at equal M, R and ω — recall each shape's moment-of-inertia factor and plug into ½Iω².
Related PYQs
No directly related past PYQ was found.
Practice
- practice — not a real PYQ
About an axis through its centre, the moment of inertia of a solid sphere of mass M and radius R is
- (a)MR²
- (b)½MR²
- (c)⅔MR²
- (d)⅕MR²
Answer(d) ⅕MR² — the standard moment of inertia of a solid sphere about a central axis. - practice — not a real PYQ
A ring and a disc of the same mass and radius roll down an incline; which reaches the bottom first?
- (a)The ring
- (b)The disc
- (c)Both together
- (d)Cannot be decided
Answer(b) The disc — its smaller moment of inertia gives it greater linear acceleration.