$A$ spherical bob of mass $250 \ g$ is attached to the end of a string having length $50 \ cm$. The bob is rotated on a horizontal circular path about a vertical axis. The maximum tension that the string can bear is $72 \ N$. The maximum possible value of angular velocity of the bob (in $rad/s$) is

$A$ particle moves in space according to the equation $\vec{r} = (\sin t \hat{i} + \cos t \hat{j} + t \hat{k}) \text{ m}$. Find the time $t$ when the position vector and acceleration vector are perpendicular to each other.

$A$ particle is moving in a circular path. The acceleration and momentum of the particle at a certain moment are $\vec{a} = (4\hat{i} + 3\hat{j}) \, m/s^2$ and $\vec{P} = (8\hat{i} - 6\hat{j}) \, kg \cdot m/s$. The motion of the particle is:

Three particles start simultaneously from a point on a horizontal smooth plane. The first particle moves with speed $v_1$ towards the east,the second particle moves towards the north with speed $v_2$,and the third one moves towards the north-east. The velocity of the third particle,such that the three particles always lie on a straight line,is

Two bodies are thrown up at angles of $45^\circ$ and $60^\circ$,respectively,with the horizontal. If both bodies attain the same vertical height,then the ratio of the velocities with which these are thrown is:

Two stones of masses $m$ and $2\,m$ are whirled in horizontal circles,the heavier one in a radius $\frac{r}{2}$ and the lighter one in radius $r$. The tangential speed of the lighter stone is $n$ times that of the heavier stone when they experience the same centripetal force. The value of $n$ is