Two bodies of masses $m$ and $3m$ are rotating in horizontal circles of radii $r$ and $\frac{r}{3}$ respectively. The tangential speed of the body of mass $m$ is $n$ times that of the heavier body. If the centripetal force is the same for both,the value of $n$ is:

$A$ projectile crosses two walls of equal height $H$ symmetrically as shown. If the horizontal distance between the two walls is $d = 120\, m$ and the times at which the projectile crosses the walls are $t_1 = 2\, s$ and $t_2 = 6\, s$,then the range of the projectile is ........ $m$.

State whether the following statements are true or false:
$(a)$ If the angular velocity is constant on a circular path,the linear velocity is also constant.
$(b)$ In projectile motion,the velocity vector of the projectile is always perpendicular to the acceleration vector.
$(c)$ For the maximum horizontal range $R$ of a projectile,the maximum height attained is $\frac{R}{4}$.
$(d)$ If $|\vec{A} \times \vec{B}| = AB$,then the angle between $\vec{A}$ and $\vec{B}$ is zero.

If a particle of mass $m$ covers half of the horizontal circle with constant speed $v$,then the change in its kinetic energy is

An object of mass $m$ is projected with a momentum $p$ at such an angle that its maximum height is $\frac{1}{4}$th of its horizontal range. Its minimum kinetic energy in its path will be

$A$ 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