$A$ body performing uniform circular motion of radius '$R$' has frequency '$n$'. Its centripetal acceleration per unit radius is proportional to $(n)^x$. The value of $x$ is

An insect trapped in a circular groove of radius $12 \; cm$ moves along the groove steadily and completes $7$ revolutions in $100 \; s$.
$(a)$ What is the angular speed,and the linear speed of the motion?
$(b)$ Is the acceleration vector a constant vector? What is its magnitude?

Two stones are projected with the same speed but making different angles with the horizontal. Their ranges are equal. If the angle of projection of one is $\pi / 3$ and its maximum height is $h_1$,then the maximum height of the other will be:

At $t = 0$,a projectile is fired from a point $O$ (taken as origin) on the ground with a speed of $50 \ m/s$ at an angle of $53^{\circ}$ with the horizontal. It passes through two points $A$ and $B$,each at a height of $75 \ m$ above the horizontal,as shown in the figure. The horizontal separation between the points $A$ and $B$ is . . . . . . $m$.

$A$ particle moves along a circle of radius $R$ with kinetic energy $K = as^2$,where $s$ is the distance traveled. What is the net force acting on the particle?

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: