question_answer

The moment of inertia of uniform semi-circular disc of mass M and radius r about a line perpendicular to the plane of the disc through the centre is                                                                                     [AIEEE 2005]

A)   \[\frac{1}{4}M{{r}^{2}}\]             

B)        \[\frac{2}{5}M{{r}^{2}}\]

C)   \[M{{r}^{2}}\]               

D)        \[\frac{1}{2}M{{r}^{2}}\]

Correct Answer: D

Solution :

[d] The mass of complete (circular) disc is

The moment of inertia of disc about the given axis is

\[l=\frac{2M{{r}^{2}}}{2}=M{{r}^{2}}\]

The disc may be assumed as combination of two semi-circular parts.

Thus, the moment of inertia of a semi-circular part of disc,

\[{{l}_{1}}-l-{{l}_{1}}\]

\[\therefore \,\,\,\,\,\,\,\,\,\,{{l}_{1}}=\frac{l}{2}=\frac{M{{r}^{2}}}{2}\]