Question

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed $\omega$, as shown int he 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 $\left(\overrightarrow{\tau }\right)$ on the system about O, as a function of time is best represented by which plot?

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let M and l be the mass and length of the rod respectively and m be the mass of the insect. Let insect be at a distance x from O at any instant of time t.
$\therefore x=vt$ (i)
Angular momentum of the system about O,

$L=[\frac{M{l}^2}{3}+m{x}^2]\omega$
$=[\frac{M{l}^2}{3}+m(vt{)}^2]\omega$ (using (i))
$=[\frac{M{l}^2}{3}+m{v}^2{t}^2]\omega$
As $|\overrightarrow{\tau }|=\frac{dL}{dt}=\frac{d}{dt}[\frac{M{l}^2}{3}+m{v}^2{t}^2]\omega$
As $\omega$ and v remain constant
$\therefore |\overrightarrow{\tau }|=2m\omega v^{2}t$
$|\overrightarrow{\tau }|\alpha t$
Hence, the graph and t is a straight line passing through the (0, 0). Option (b) represents correct plot.