Question

Two rods of equal lengths(l) and equal mass M are kept along x and y-axis respectively such that their centre of mass lie at the origin. The moment of inertia about a line y = x is

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

Answer 1

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

Moment of inertia of a rod of about an axis passing through centre and perpendicular to length:
$I=\frac{1}{12}M{L}^2$
Moment of inertia of a rod of about an axis inclined an angle theta to the original axis:
$I^{\prime }=Isi{n}^2\theta$
The line x = y makes an angle of 45 degrees x and y-axis.
Hence,
$I^{\prime }=Isi{n}^{2}45$
$=\frac{1}{2}$
Moment of inertia of both rods about the line passing through x = y:
$I_{tworods}=2I^{\prime }=2\left(\frac{1}{2}\right)=I=\frac{1}{12}M{L}^2$