Question

A thin uniform square lamina of side $a$ is placed in the $XY$ plane with its sides parallel to $X$ and $Y$ axes and with its centre coinciding with the origin. Its moment of inertia about an axis passing through a point on the $Y$axis at a distance $y=2a$ and parallel to $X$ axis is equal to its moment of inertia about an axis passing through a point on the $X$ axis at a distance $x=d$ and perpendicular to $XY$ plane. The value of $d$ is:

• A. $\frac{7}{3}a$
• B. $\sqrt{\frac{47}{12}}a$
• C. $\frac{9}{5}a$
• D. $\sqrt{\frac{51}{12}}a$

Answer 1

Answer: (B)

$\sqrt{\frac{47}{12}}a$

The Let mass of plate be $M$.

So MI about the axis in plane of square, parallel to x-axis and passing through its center is $M{a}^2/12$

Using parallel axis theorem, MI about axis parallel to x-axis and at distance of $2a$ from center is $M{a}^2/12+M(2a{)}^2=49M{a}^2/12-(1)$

In second case, MI about the axis perpendicular to square, passing through its center is $M{a}^2/6$

Using parallel axis theorem, MI about axis perpendicular to square, and at distance of $d$ from its center on x-axis is $M{a}^2/6+M(d{)}^2-(2)$

As $\left(1\right)=\left(2\right)$

$\Rightarrow M{a}^2/6+M{d}^2=49M{a}^2/12$

$\Rightarrow M{d}^2=47M{a}^2/12\Rightarrow d=\sqrt{47/12}a$