$A$ 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? $(...\, m)$

$A$ heavy mass is attached at one end of a thin wire and whirled in a vertical circle. The chances of breaking the wire are maximum when:

$A$ weightless thread can bear a tension of up to $3.7 \, kg \, wt$. $A$ stone of mass $500 \, g$ is tied to it and revolved in a circular path of radius $4 \, m$ in a vertical plane. If $g = 10 \, m/s^2$,then the maximum angular velocity of the stone will be ........ $rad/s$.

$A$ small bob tied at one end of a thin string of length $1\, m$ is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio $5: 1$. The velocity of the bob at the highest position is ............. $m/s$ (Take $g=10\, m/s^2$).

$A$ 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 $2 \,kg$ tied to a light inextensible string of length $\frac{5}{3} \,m$ is whirling in a circular path in a vertical plane. If the ratio of the maximum tension to the minimum tension in the string is $4$, then the speed of the stone at the highest point of the circle is $\left(g=10 \,ms^{-2}\right)$.