A circular stage is free to rotate about vertical axis passing through centre. $A$ tortoise is  sitting at corner of stage. Stage is provided angular velocity $\omega_0$. If tortoise start  moving along one chord at constant speed with respect to stage then how the angular  velocity of stage $\omega(t)$ vary with time $t$ :-

Two particles $A$ and $B$ initially at rest move towards each other under a mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2v$, the speed of centre of mass of the system is

We have two spheres, one of which is hollow shell and the other solid. They have identical masses and moment of inertia about their respective diameters. The ratio of their radius is given by

What is the moment of inhertia of a solid sphere of radius $R$ and density $\rho $ about its diameter ?

A $T$ shaped object with dimensions shown in the figure, is lying a smooth floor. A force $'\vec F'$ is applied at the point $P$ parallel to $AB,$ such that the object has only the translational motion without rotation. Find the location of $P$ with respect to $C$

$ ABC$  is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. $I_{AB}, I_{BC}, I_{CA}$  are the moment of inertia of the plate about $AB, BC$  and $CA$  respectively. Which one of the following relations is correct