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 moves along a chord of the platform with a constant velocity (w.r.t. the platform), which of the following graphs shows properly the variation of angular velocity of the platform with the time $t$ ?

• A.
• B.
• C.
• D.

Answer 1

Answer: (C)

As there is no external torque, angular momentum will remain constant. When the tortoise moves from $A$ to $C$ , figure, moment of inertia of the platform and tortoise decreases. Therefore, angular velocity of the system increases. When the tortoise moves from $C$ to $B$ , moment of inertia increases. Therefore, angular velocity decreases

If, $M$ =mass of platform
$R=$ radius of platform
$m=$ mass of tortoise moving along the chord $AB$
$a=$ perpendicular distance of $O$ from $AB$
Initial angular momentum, $I_1=m{R}^2+\frac{M{R}^2}{2}$
At any time $t$ , let the tortoise reach $D$ moving with velocity $v$
$\therefore AD=vt$
$AC=\sqrt{R^2-a^2}$
As $DC=AC-AD=(\sqrt{R^2-a^2}-vt)$
$\therefore OD=r=a^2+{\left[\sqrt{R^2-a^2}-vt\right]}^2$
Angular momentum at time $t$
$I_2=m{r}^2+\frac{M{R}^2}{2}$
As angular momentum is conserved
$\therefore I_1{\omega }_0=I_2\omega (t)$
This shows that variation of $\omega (t)$ with time is nonlinear. Choice