Question

A thin uniform rod,pivoted at O, is rotating in the horizontal plane with constant angular speed ω, 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 other end. It reaches the end of the rod at t = T and stops.The angular speed of the system remains ω throughout. The magnitude of the torque $\left(\left|\overrightarrow{\tau }\right|\right)$ about O,as a function of time is best represented by which plot?

• A.
• B.
• C.
• D.

Answer 1

Answer: (B)

angular momentum
$$\begin{gathered} \mathrm{J}=\mathrm{IW}\mathrm{z}=\frac{\mathrm{dJ}}{\mathrm{dt}}=\mathrm{w}\frac{\mathrm{dI}}{\mathrm{Dt}} \\ [\mathrm{I}={\mathrm{I}}_{\mathrm{rod}}+{\mathrm{I}}_{\mathrm{insect}} \\ =\mathrm{C}+{\mathrm{mr}}^2\Rightarrow \mathrm{c}+\mathrm{m}(\mathrm{vt}{)}^2] \\ \tau =\omega \frac{\mathrm{dI}}{\mathrm{dt}}=\omega \frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{C}+{\mathrm{mv}}^2{t}^2\right) \\ =\mathrm{m}\omega {\mathrm{v}}^{2}2t \end{gathered}$$
So, the graph between $|\tau |$ and $t$ is straight line.