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NEET PHYSICSRotational motionMedium

Question

From a disc of radius RR and mass MM, a circular hole of diameter RR, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?

A

13MR232\frac{13MR^2}{32}

B

11MR232\frac{11MR^2}{32}

C

9MR232\frac{9MR^2}{32}

D

15MR232\frac{15MR^2}{32}

Step-by-Step Solution

Mass per unit area of the original disc is σ=MπR2\sigma = \frac{M}{\pi R^2}. The diameter of the cut-out hole is RR, so its radius is R2\frac{R}{2}. Mass of the cut-out portion, m′=σ×Area=MπR2×π(R2)2=M4m' = \sigma \times \text{Area} = \frac{M}{\pi R^2} \times \pi \left(\frac{R}{2}\right)^2 = \frac{M}{4}. The hole's rim passes through the centre of the original disc, which means the distance between the centre of the original disc and the centre of the cut-out portion is d=R2d = \frac{R}{2}. Moment of inertia of the original disc about an axis passing through its centre and perpendicular to its plane is I0=12MR2I_0 = \frac{1}{2}MR^2. Moment of inertia of the cut-out portion about an axis passing through its own centre and perpendicular to its plane is Icm′=12m′(R2)2=12(M4)(R24)=MR232I'_{cm} = \frac{1}{2}m'\left(\frac{R}{2}\right)^2 = \frac{1}{2}\left(\frac{M}{4}\right)\left(\frac{R^2}{4}\right) = \frac{MR^2}{32}. Using the parallel axis theorem, the moment of inertia of the cut-out portion about the centre of the original disc is I0′=Icm′+m′d2=MR232+(M4)(R2)2=MR232+MR216=3MR232I'_0 = I'_{cm} + m'd^2 = \frac{MR^2}{32} + \left(\frac{M}{4}\right)\left(\frac{R}{2}\right)^2 = \frac{MR^2}{32} + \frac{MR^2}{16} = \frac{3MR^2}{32}. Moment of inertia of the remaining part is Irem=I0−I0′=12MR2−3MR232=16MR2−3MR232=13MR232I_{rem} = I_0 - I'_0 = \frac{1}{2}MR^2 - \frac{3MR^2}{32} = \frac{16MR^2 - 3MR^2}{32} = \frac{13MR^2}{32}.

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from Rotational motion. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSRotational motionradiuscirculardiameterpassesthrough

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