A thin uniform rod, pivoted at `O`, is rotating in the horizontal plane with constant angular speed `omega`, as shown in the figure. At tune `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?

Options

A.
B.
C.
D.

Answer

B

Explanation

Let M and l be the mass and length of the rod respectively and m be the mass of the insect. Let insect be at a distance x from O at any instant of time t.
x= vt
Angular momentum of the system abot O.
`L=[(MI^(2))/3 + mx^(2)] omega = [(MI^(2))/3 + m(vt)^(2)]omega`

`=[(MI^(2))/3 + mv^(2)t^(2)]omega , As |vectau| = (dL)/(dt) = d/(dt)[(MI^(2))/3 + mV^(2)t^(2)]omega`
As `omega` and v remain constant. `therefore |vectau|=2m omega v^(2)t`
`therefore |vectau| propto t`
Hence, the graph is a straight line passing through the (0,0). Option (b) represents correct plot.