A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed , as shown in the figure. [2012]

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 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 $(| \vec{τ} |)$ about O, as a function of time is best represented by which plot [2012]

Smart Achievers · Accepted Answer

(2)

Angular momentum, $| L |$ or $L = I ω$ (about axis of rod)

Moment of inertia of the rod-insect system

$I = I_{rod} + m x^{2} = I_{rod} + m v^{2} t^{2}$

Here, $m$ = mass of insect

$∴$ $L = (I_{rod} + m v^{2} t^{2}) ω$

$Now | τ | = \frac{d L}{d t}$

$= (2 m v^{2} t ω) o r | τ | \propto t$

i.e., the graph is a straight line passing through origin.

After time $T, L$ = constant

$∴$ $| τ | or \frac{d L}{d t} = 0$

i.e., when the insect stops moving, L does not change and therefore $T$ becomes constant.

Solution

1

2

3

4

Want Us to Email you About Special Offers & Updates?