A uniform circular disc of radius 'R' and mass 'M' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius $\mathrm{R} / 2$ is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

(D) $\frac{13}{32} \mathrm{MR}^{2}$

(A) $\frac{7}{32} \mathrm{MR}^{2}$

(B) $\frac{9}{32} \mathrm{MR}^{2}$

(C) $\frac{17}{32} \mathrm{MR}^{2}$

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Rotational Mechanics

A uniform circular disc of radius 'R' and mass 'M' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius

R /2

is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

(A) $\frac{7}{32} MR^{2}$
(B) $\frac{9}{32} MR^{2}$
(C) $\frac{17}{32} MR^{2}$
(D) $\frac{13}{32} MR^{2}$ ✓ Correct

Answer: (D)

Solution

The full disc has mass

M

, radius

R

, and moment of inertia about the central axis

I_{f u l l} = \frac{1}{2} M R^{2}

.

The removed piece is a disc of radius

R /2

whose centre lies at distance

R /2

from O, tangent to the rim of the original disc as shown. Since mass is proportional to area for a uniform disc, its mass is

m = M (\frac{R /2}{R})^{2} = \frac{M}{4} .

Its moment of inertia about its own centre is

\frac{1}{2} m (\frac{R}{2})^{2} = \frac{1}{2} \cdot \frac{M}{4} \cdot \frac{R ^{2}}{4} = \frac{M R ^{2}}{32}

.

Shifting to the axis through O with the parallel axis theorem (offset

d = R /2

I_{r e m o v e d, O} = \frac{M R ^{2}}{32} + \frac{M}{4} (\frac{R}{2})^{2} = \frac{M R ^{2}}{32} + \frac{M R ^{2}}{16} = \frac{3 M R ^{2}}{32} .

The moment of inertia of the remaining part is

I = \frac{M R ^{2}}{2} - \frac{3 M R ^{2}}{32} = \frac{16 M R ^{2} - 3 M R ^{2}}{32} = \frac{13}{32} M R^{2} .

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Single correct option

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JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Rotational Mechanics

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