$A$ solid body rotates about a fixed axis such that its angular velocity depends on $\theta$ as $\omega = k \theta^{-1}$,where $k$ is a positive constant. At $t = 0$,$\theta = 0$. Then,the time dependence of $\theta$ is given as ..........

What is the value of linear velocity if $\overrightarrow{r} = 3\widehat{i} + 4\widehat{j} + 6\widehat{k}$ and $\overrightarrow{\omega} = -5\widehat{i} + 3\widehat{j} + 5\widehat{k}$?

$A$ rigid rod of length $\ell$ is sliding. At some instant,the position of the rod is as shown in the figure. End $A$ has a constant velocity $v_0$ along the $x$-axis. At $t = 0$,the rod is vertical with end $B$ at $y = \ell$. Which of the following statements is correct?

$A$ ladder of length $L$ is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment,the speed of the end in contact with the horizontal floor is $v$ and the ladder makes an angle $\alpha = 30^o$ with the horizontal. If $dv/dt = 0$,then the angular acceleration of the ladder when $\alpha = 45^o$ is:

Let $\vec{A}$ be a unit vector along the axis of rotation of a purely rotating body and $\vec{B}$ be a unit vector along the velocity of a particle $P$ of the body away from the axis. The value of $\vec{A} \cdot \vec{B}$ is

The left end of a massless stick with length $l$ is placed on the corner of a table,as shown in the figure. $A$ point mass $m$ is attached to the center of the stick,which is initially held horizontal. It is then released. Immediately afterward,what normal force does the table exert on the stick?