Question

A rod of mass $m$ and length $2l$ hangs by two identical light threads tied to its ends. An insect of mass $\frac{3m}{8}$ lands on the rod gently at a distance $\frac{l}{3}$ from the centre of the rod as shown and sticks to it. As a result one of the thread breaks. The acceleration of the insect (in ${\text{m/s}}^2$) just after the thread breaks given that the insect remains at rest with respect to the rod is

Assume $g=10{\text{m/s}}^2$

Answer 1

Net torque from one end of rod where thread is uncut can be written as change in angular momentum.
$\Rightarrow mgl+\frac{3mg}{8}(l+\frac{l}{3})=[m\left(\frac{4l^2}{3}\right)+\frac{3}{8}m{\left(\frac{4l}{3}\right)}^2]\alpha$
$\Rightarrow \alpha =\frac{3g}{4l}$
So acceleration of insect will be
$a=\frac{3g}{4l}\times \frac{4l}{3}=g$
$\therefore \left(d\right)$ is the correct option.