The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta (t) = 2t^3 -6t^2$. The torque on the wheel becomes zero at $t$ $=$ ........ $\sec.$

In a gravity free space, a man of mass $M$ standing at a height $h$ above the floor, throws a ball of mass $m$ straight down with a speed $u$ . When the ball reaches the floor, the distance of the man above the floor will be

A solid cylinder and a disc of same radii are allowed to roll down a rough inclined plane from the top of a plane. The ratio of their times taken to reach the bottom of the inclined plane is

In free space, a shell moving with velocity $60\,m/s$ along the positive $x$ -axis. When it  passes through origin, it explodes into two peices of mass ratio $1 : 2.$ After the  explosion, the velocity of the centre of mass is ........ $m/s$.

A solid cylinder rolls without slipping down an inclined plane of height $h$. The velocity of the cylinder when it reaches the bottom is

In the above problem the angular velocity of the system after the particle sticks to it will be ....... $rad/s$