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 - A, the disc. For the same mass M and radius R, a solid disc rotating about the axis through its centre and perpendicular to its plane has moment of inertia I = (1/2)MR^2, whereas a solid sphere about a diameter through its centre has I = (2/5)MR^2. Since 1/2 (0.5) is larger than 2/5 (0.4), the disc has the higher moment of inertia. Physically, the disc keeps more of its mass out near the rim, while the sphere packs mass closer to the centre.
- (b)The sphere — The solid sphere's I = (2/5)MR^2 = 0.4MR^2 is smaller than the disc's 0.5MR^2, because its mass sits nearer the rotation axis. It therefore has the lower, not the higher, moment of inertia.
- (c)Both have the same moment of inertia — The shape coefficients differ (1/2 for the disc versus 2/5 for the sphere), so equal mass and equal radius do not make the moments of inertia equal - the mass distribution decides it.
- (d)The information provided is not sufficient to answer the question — Mass M and radius R are all a standard-body moment of inertia needs; the formulae (1/2)MR^2 and (2/5)MR^2 give a definite comparison, so the data are sufficient.
Moment of inertia measures a body's resistance to angular acceleration and depends on how its mass is distributed about the axis (I = sum of m*r^2), so mass farther from the axis counts more. For identical M and R, a flat disc spreads mass to its rim while a solid sphere concentrates mass towards the centre, giving different standard coefficients.
The trap is assuming 'same mass and radius means same moment of inertia'. Rotational inertia is fixed by the mass distribution, not by M and R alone. Comparing the standard coefficients - 1/2 for a disc or cylinder, 2/5 for a solid sphere - settles it at once.
- Solid disc or cylinder about its central axis: I equals one-half M R squared.
- Solid sphere about a diameter: I equals two-fifths M R squared, which is 0.4 M R squared.
- Thin ring or hoop about its central axis: I equals M R squared, the maximum for a given M and R.
- For equal M and R the ordering of I is ring (1) greater than disc (0.5) greater than solid sphere (0.4).
The disc (0.5 M R^2) beats the solid sphere (0.4 M R^2), so the disc has the higher moment of inertia.
- Assuming equal mass and radius force equal moment of inertia - the mass distribution decides it.
- Confusing the disc's central-axis value (1/2 M R^2) with its diameter value (1/4 M R^2).
Asked by ranking the moment of inertia of standard bodies of equal mass and radius, or by quoting a single formula.
No directly related past PYQ was found.
- practice — not a real PYQ
For the same mass and radius, which body has the greatest moment of inertia about its central axis?
- (a)Solid sphere
- (b)Solid disc
- (c)Thin ring
- (d)Thin rod
Answer(c) Thin ring - I = M R^2, the largest for a given mass and radius.
- practice — not a real PYQ
The moment of inertia of a solid sphere of mass M and radius R about a diameter is
- (a)M R^2
- (b)(1/2) M R^2
- (c)(2/5) M R^2
- (d)(2/3) M R^2
Answer(c) (2/5) M R^2 - the standard result for a solid sphere.