Q. 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
IIT JEEIIT JEE 2002System of Particles and Rotational Motion
Solution:
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 = mR^2 + \frac{MR^2}{2}$
and moment o f inertia when the tortoise is at B
$ I_2 = mr^2 + \frac{MR^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.
