If a solid sphere is rolling, the ratio of its rotational energy to the total kinetic energy is given by

A disc of mass $M$ and radius $R$ rolls on a horizontal surface and then rolls up an inclined plane as shown in the figure. If the velocity of the disc is $v,$ the height to which the disc will rise will be

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

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 $t$ :-

In the $HCl$ molecule, the separation between the nuclei of the two atoms is about $1.27\,\mathop A\limits^o \left( {1\,\mathop A\limits^o  = {{10}^{ - 10}}\,m} \right)$. The approximate location of the centre of mass of the molecule from hydrogen atom, assuming the chlorine atom to be about $35.5$ times massive as hydrogen is ....... $\mathop A\limits^o $

A thin circular ring of mass $M$  and radius $R$  is rotating about its axis with a constant angular velocity $\omega .$ Two objects, each of mass $m,$  are attached gently to the opposite ends of a diameter of the ring. The ring rotates now with an angular velocity