Question

A homogeneous rod AB of length L and mass M is pivoted at the centre O in such a way that it can rotate freely in the vertical plane. The rod is initially in the horizontal position. An insect S of the same mass M falls vertically with speed V on the point C mid-way between O and B. The initial angular velocity $\omega$ in terms of V and L is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Consider the rod and insect as a single system.
We can conserve the angular momentum of the system w.r.t centre 'O' of the rod.

$L_i$ = Initial angular momentum = $MV\left(\frac{L}{4}\right)$
$L_f$ = Final angular momentum = $\left(\frac{M{L}^2}{12}\right)\omega +M(\frac{L}{4}{)}^2\omega$
$\Rightarrow MV\left(\frac{L}{4}\right)=\omega (\frac{M{L}^2}{12}+\frac{M{L}^2}{16})$
$\Rightarrow \omega =\frac{12V}{7L}$