1. Tardigrade
  2. Question
  3. Physics
  4. 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 the 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 |τ| on the system about O, as a function of time is best represented by which plot?

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Q. 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?

IIT JEEIIT JEE 2012System of Particles and Rotational Motion

Solution:

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