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
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.
- (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.
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.
- 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.
- 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.
No directly related past PYQ was found.
- 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.