If the equation for the displacement of a particle moving on a circular path is given by:

                               $\theta  = 2t^3 + 0.5$

Where $\theta $ is in radian and $t$ in second, then the angular velocity of the particle at $t = 2\,sec$ is $t=$ ....... $rad/sec$

Two racing cars of masses $m_1$ and $m_2$ are moving in circles of radii $r_1$ and $r_2$ respectively. Their speeds are such that each makes a complete circle in the same time $t$. The ratio of the angular speeds of the first to the second car is

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 sphere of diameter $r$ is cut from a sphere of radius $r$ such that the centre of mass of the remaining mass be at maximum distance from original centre; then the distance 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

A disc is performing pure rolling on a smooth stationary surface with constant angular velocity as shown in figure. At any instant, for the lower most point of the disc -