Question

Two rods equal mass $m$ and length $\ell$ lie along the $x$ axis and $y$ axis with their centres origin. What is the moment of inertia of both about the line $x=y$ :

• A. $\frac{m{\ell }^2}{3}$
• B. $\frac{m{\ell }^2}{4}$
• C. $\frac{m{\ell }^2}{12}$
• D. $\frac{m{\ell }^2}{6}$

Answer 1

The correct option is C $\frac{m{\ell }^2}{12}$

Moment of inertial of a rod of about an axis passing through centre and perpendicular to length :

$I=\frac{M{l}^2}{12}$

moment of inertia of the rod about an axis inclined an angle $\theta$ to the original axis:

$I^{\prime }=I{\sin }^2\theta$

The line $x=y$ makes an angle of ${45}^{\circ }$ about $x$ and $y$ axes.

Hence $I^{\prime }=I{\sin }^2{45}^{\circ }=\frac{I}{2}$

moment of inertia of both rods about the line passing through $x=y$

$I_{two/,rods}=2I^{\prime }$

$=2\times \frac{I}{2}=I$

$=\frac{M{l}^2}{12}$