JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Rotational Mechanics

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 ?

Answer: (B)

Solution

The torque about

O

is given by

τ = ω \frac{d I}{d t}

. As the insect moves from

O

to the end of the rod, the moment of inertia of the system increases. At time

t

, the insect is at distance

v t

from

O

, so

I = C + m (v t)^{2}

where

C

is the moment of inertia of the rod. Thus

τ = ω \frac{d}{d t} (C + m v^{2} t^{2}) = 2 mω v^{2} t

. This is a linear function of time, starting from zero at

t = 0

and increasing linearly until

t = T

, then dropping to zero when the insect stops. This corresponds to option (B).

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Single correct option