Torque on Rotating Rod with Crawling Insect

To determine the variation of the magnitude of the torque ($|\vec{\tau}|$) acting on the system about $O$ as a function of time $t$, we analyze the angular momentum of the system.

1. Angular Momentum of the System

The total angular momentum $\vec{L}$ of the system about the pivot $O$ consists of the angular momentum of the rod and the angular momentum of the crawling insect: $\vec{L} = \vec{L}_{\text{rod}} + \vec{L}_{\text{insect}}$

• For the rod: Since the rod is uniform and rotates with a constant angular velocity $\vec{\omega} = \omega \hat{k}$, its angular momentum about $O$ is constant: $\vec{L}_{\text{rod}} = I_{\text{rod}} \omega \hat{k}$

• For the insect during $0 \le t < T$: The insect moves with a constant relative speed $v$ along the rod. Its position vector relative to $O$ at time $t$ is: $\vec{r}(t) = (vt) \hat{r}$

  The velocity of the insect in polar coordinates is: $\vec{v}_{\text{insect}} = \frac{d\vec{r}}{dt} = v \hat{r} + (vt \omega) \hat{\theta}$

  The angular momentum of the insect about $O$ is: $\vec{L}_{\text{insect}} = \vec{r} \times m \vec{v}_{\text{insect}} = (vt \hat{r}) \times m \left( v \hat{r} + vt \omega \hat{\theta} \right)$ $\vec{L}_{\text{insect}} = m v^2 \omega t^2 \hat{k}$

Thus, for $0 \le t < T$, the total angular momentum of the system is: $\vec{L}(t) = \left( I_{\text{rod}} \omega + m v^2 \omega t^2 \right) \hat{k}$

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2. Calculation of Torque

The net torque $\vec{\tau}$ required to maintain the rotation at a constant angular speed $\omega$ is given by the rate of change of total angular momentum: $\vec{\tau} = \frac{d\vec{L}}{dt}$

• For $0 \le t < T$: $\vec{\tau}(t) = \frac{d}{dt} \left[ \left( I_{\text{rod}} \omega + m v^2 \omega t^2 \right) \hat{k} \right] = 2 m v^2 \omega t \hat{k}$

  The magnitude of the torque is: $|\vec{\tau}(t)| = (2 m v^2 \omega) t$

  This shows that $|\vec{\tau}|$ varies linearly with time $t$ during the interval $0 \le t < T$, starting from $0$ at $t = 0$ and reaching a maximum value of $2 m v^2 \omega T$ at $t \to T^-$.

• For $t > T$: At $t = T$, the insect reaches the end of the rod and stops crawling relative to the rod. Thereafter, its distance from $O$ remains constant ($r = vT$). The total moment of inertia of the system becomes constant: $I_{\text{total}} = I_{\text{rod}} + m(vT)^2$

  Since $I_{\text{total}}$ and $\omega$ are both constant, the total angular momentum $\vec{L}$ becomes constant, which implies: $\vec{\tau}(t) = \frac{d\vec{L}}{dt} = 0 \quad \text{for } t > T$

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Conclusion

• From $t = 0$ to $t = T$, $|\vec{\tau}|$ increases linearly with $t$.
• For $t > T$, $|\vec{\tau}|$ drops immediately to zero.

This behavior is correctly depicted by the graph in option (B).

Correct Option: B