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

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 $\frac{4M}{3}$ and
Therefore moment of inertia about its axis is
$I_1=\frac{1}{6}\left(\frac{4M}{3}\right){\left(2a\right)}^2$
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
$I=I_1-I_2=\frac{5}{6}M{a}^2$