Question

A square of a side $a$ is cut from a square of side $2a$ as shown in the figure. Mass of this square with a hole is $M$ . Then its moment of inertia about an axis passing through its centre of mass and perpendicular to its plane will be

• A. $\frac{M a^{2}}{6}$
• B. $\frac{2 M a^{2}}{6}$
• C. $\frac{4 M a^{2}}{6}$
• D. $\frac{5 M a^{2}}{6}$

Answer 1

Answer: (D)

Given plate has mass M and area $3a^{2}$
A square plate of side $2a$ (made of the same material) has mass $\frac{4 M}{3}$ and
Therefore moment of inertia about its axis is
$I_{1}=\frac{1}{6}\left(\frac{4 M}{3}\right)\left(2 a\right)^{2}$
A square plate of side $a$ (made of the same material) has mass $\frac{M}{3}$ and
Therefore moment of inertia about its axis is $I_{2}=\frac{1}{6}\left(\frac{M}{3}\right)a^{2}$
Moment of inertia of the given plate is
$I=I_{1}-I_{2}=\frac{5}{6}Ma^{2}$