At time $t = 0$,a $2\, kg$ particle has position vector $\vec{r} = (4\hat{i} - 2\hat{j})\, m$ relative to the origin. Its velocity is given by $\vec{v} = 2t^2\hat{i}\, m/s$. The torque acting on the particle about the origin at $t = 2\, s$ is ........ $\hat{k}\, N\cdot m$.

$A$ particle of mass $m$ moves in a circular path of radius $r$,under the action of a force which delivers constant power $P$ and increases its speed. The angular acceleration of the particle at time $t$ is proportional to:

Which of the following relations is incorrect?

$A$ solid sphere of radius $4\text{ cm}$ and mass $5\text{ kg}$ is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of $1200\text{ rpm}$. It is brought to rest in $10\text{ s}$ by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are . . . . . . and . . . . . . respectively.

The variation of angular acceleration $\alpha$ of a point on a rigid body rotating about a fixed axis with time $t$ is shown in the figure. What is the sense of rotation of the rod? (Assume clockwise as negative)

$A$ slender uniform rod of mass $M$ and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The rod is released from a position where it makes an angle $\theta$ with the vertical. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical is