The angular velocity of a body is $\mathop \omega \limits^ \to   = 2\hat i + 3\hat j + 4\hat k$ and a torque $\mathop \tau \limits^ \to   = \hat i + 2\hat j + 3\hat k$ acts on it. The rotational power will be .......... $W$

Write the formula of work done by torque in rotational rigid body about a the fixed axis.

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights $h_{sph}$ and $h_{cyl}$ on the incline. The radio $\frac{{{h_{sph}}}}{{{h_{cyl}}}}$ is given by

A stick of length $L$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in any ways. A ball of mass $m$ moving with speed $v$ collides elastically with the stick as shown in the figure. If after the collision the ball comes to rest, then what should be the mass of the ball ?

A tangential force $F$ is applied on a disc of radius $R$, due to which it deflects through an angle $\theta $ from its initial position. The work done by this force would be

A small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $\frac{3 \mathrm{v}^2}{4 \mathrm{~g}}$ with respect to the initial position. The object is