Question

Two rods $OA$ and $OB$ of equal length and mass are lying on the $xy$ plane as shown in figure. Let $I_x,I_y$ and $I_z$ be the moment of inertia of the system about $x,y$ and $z$ axis respectively. Then,

• A. $I_x=I_y>I_z$
• B. $I_x=I_y<I_z$
• C. $I_x>I_y>I_z$
• D. $I_z>I_y>I_x$

Answer 1

The correct option is B $I_x=I_y<I_z$

Let $OA=OB=l$ (in metres)
$m_A=m_B=m\text{kg}$
Moment of inertia of the system about $x$ axis
$I_x=2\left(\frac{m{l}^2}{3}{\sin }^2{45}^{\circ }\right)$
$=\frac{m{l}^2}{3}$
(effective length of both rods is $l\sin {45}^{\circ }$)

Moment of inertia of the system about $y$ axis
$I_y=2\left(\frac{m{l}^2}{3}{\cos }^2{45}^{\circ }\right)$
$=\frac{m{l}^2}{3}$

Moment of inertia of the system about $z-$ axis
$I_z=2\left(\frac{m{l}^2}{3}\right)=\frac{2m{l}^2}{3}$
$\Rightarrow I_x=I_y<I_z$