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A thin uniform rod, pivoted at $O$, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At time, $t =0$, a small insect starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reaches the end of the rod at $t = T$ and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $(|\vec{\tau}|)$ on the system about $O$, as a function of time is best represented by which plot?
Solution
$ L =\left[ m ( vt )^2\right] \omega $
$L $$ = mv ^2 \omega t ^2 $
$\text { So } \quad \tau $$=\frac{ dL }{ dt }=2 m v^2 \omega t $
$ \tau \propto t $
$\Rightarrow \quad \begin{array}{l}\tau \propto t \\ \text { straight line passing through }(0,0)\end{array}$
