$A$ particle is kept at rest at the top of a sphere of diameter $42\, m$. When disturbed slightly,it slides down. At what height $H$ from the bottom will the particle leave the sphere? $(...\, m)$

$A$ stone of mass $1 \,kg$ is tied with a string and it is whirled in a vertical circle of radius $1 \,m$. If tension at the highest point is $14 \,N$,then velocity at the lowest point will be ............ $m/s$.

$A$ stone of mass $m$,tied to a string,is being whirled in a vertical circle with a uniform speed. The tension in the string is

$A$ block of mass $m$ is given a velocity $20\ m/s$ on a horizontal rough surface of length $x\ m$. After the rough surface,there is a vertical circular path of radius $R = 2\ m$ which is frictionless. The block enters the circular path with the minimum velocity required to complete the path. Find the length of the rough surface $x$ if the coefficient of friction is $\mu = 0.5$.

If a particle completes a vertical circular motion,what is the ratio of its velocities at points $A$,$B$,and $C$?

$A$ steel wire can withstand a load up to $2940 \ N$. $A$ load of $150 \ kg$ is suspended from a rigid support. The maximum angle through which the wire can be displaced from the mean position,so that the wire does not break when the load passes through the position of equilibrium,is (in $^{\circ}$)