Question

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 $|\tau |$ on the system about $O$, as a function of time is best represented by which plot?

• A.
• B.
• C.
• D.

Answer 1

Answer: (B)

$|L|$ or $L=I\omega$ (about axis of rod)
$I=I_{rod}+m{x}^2=I_{rod}+m{v}^2{t}^2$
Here $m$ = mass of insect
$\therefore L=(I_{rod}+m{v}^2{t}^2)\omega$
Now $|\tau |=\frac{dL}{dt}=(2m{v}^{2}t\omega )$ or $|\tau |\propto t$
i.e. the graph is straight line passing through origin.
After time $T,L$ = constant
$\therefore |\tau |$ or $\frac{dL}{dt}=0$