$A$ particle is tied to a $20 \ cm$ long string. It performs circular motion in a vertical plane. What is the angular speed of the particle when the tension in the string at the top is zero? (Take $g = 9.8 \ m/s^2$)

$A$ bob of mass $m$ is tied to a string of length $l$. What is the minimum kinetic energy that must be provided to the bob at the lowest point so that it completes a full vertical circle (in $;mgl$)?

$A$ mass $m$ is released from point $A$ as shown in the figure. The tension in the string at point $B$ will be:

$A$ bucket full of water is revolved in a vertical circle of radius $2\,m$. What should be the maximum time-period of revolution so that the water does not fall off the bucket? (Take $g = 10\,m/s^2$) ......... $\sec$.

$A$ particle with charge $Q$ coulomb,tied at the end of an inextensible string of length $R$ metre,revolves in a vertical plane. At the centre of the circular trajectory,there is a fixed charge of magnitude $Q$ coulomb. The mass of the moving charge $M$ is such that $Mg = \frac{Q^2}{4 \pi \epsilon_0 R^2}$. If at the highest position of the particle,the tension of the string just vanishes,the horizontal velocity at the lowest point has to be

$A$ bob of mass $m$ suspended by a light string of length $L$ is whirled in a vertical circle as shown in the figure. What will be the trajectory of the particle if the string is cut at $(a)$ point $B$,$(b)$ point $C$,and $(c)$ point $X$?