Question

A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now the platform is given an angular velocity ${\omega }_0$ When the tortoise move along a chord of the platform with a constant velocity (with respect to the platform). The angular velocity of the platform $\omega (t)$ will vary with time t as

• A.
• B.
• C.
• D.

Answer 1

Answer: (C)

Since, there is no external torque, angular momentum will remain conserved. The moment o f inertia will first decrease till the tortoise moves from A to C and then increase as it moves from C and D. Therefore, (0 w ill initially increase and then decrease.
Let R be the radius o f platform, m the mass o f disc and M is the mass o f platform.
Moment o f inertia when the tortoise is at A
$I_1=m{R}^2+\frac{M{R}^2}{2}$
and moment o f inertia when the tortoise is at B
$I_2=m{r}^2+\frac{M{R}^2}{2}$
Here , $r^2=a^2+[\sqrt{R^2-a^2}-vt{]}^2$
From conservation o f angular momentum
${\omega }_0{I}_1=\omega (t)I_2$
Substituting the values we can see that variation o f $\omega (t)$ is non-linear.