Question

A thin rod of linear pass density $\lambda$ at right angle at its mid point $\left(C\right)$ and fixed to points $A$ and $B$ such that it can rotate about an axis passing through $AB$ .The moment of inertia about an axis passing through $AB$ is :

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

Answer 1

The correct option is A $\frac{\lambda l^3}{6\sqrt{2}}$

We have length of each rod $=\frac{1}{\sqrt{2}}$

Let a small part $dx$ be at a distance $x$ from the of first rod it's small mass $dm=\lambda dx$ (Linear density of rod $=\lambda$)

distance of dm from axis $=x\cos {45}^o=\frac{x}{\sqrt{2}}$

So moment of interia of $dm$ about the axis is :

$dI=dm{\left(\frac{x}{\sqrt{2}}\right)}^2=\frac{\lambda x^{2}dx}{2}$

So $I={\int }_0^{\frac{L}{\sqrt{2}}}\frac{\lambda x^{2}dx}{2}=\frac{\lambda }{2}{\frac{x^3}{3}}_0^{\frac{L}{\sqrt{2}}}=\frac{\lambda L^3}{3\times 4\sqrt{2}}$

Now the second rod will also have same moment of inertia about the axis

Hence net moment of inertia $=2\frac{\lambda L^3}{3\times 4\sqrt{2}}=\frac{\lambda L^3}{6\sqrt{2}}$