$A$ bucket filled with water is tied to a string of length $4 \ m$ and rotated in a vertical circle. To ensure that the water does not spill,the time period of rotation should be ........ $s$.

$A$ bob of mass $m$ is suspended by a light string of length $L$. It is imparted a minimum horizontal velocity at the lowest point $A$ such that it just completes a full vertical circle,reaching the topmost position $B$. The ratio of kinetic energies $\frac{(\text{K.E.})_A}{(\text{K.E.})_B}$ is:

Consider a particle moving with a minimum speed $v$ at the highest point of a vertical circle of radius $R$. If the radius of the circle is doubled, the corresponding minimum speed will be:

$A$ bucket filled with water is revolved in a vertical circle of radius $4 \ m$ and the water just does not fall down. The time period of revolution will be ........ $s$.

$A$ ball attached to one end of a string swings in a vertical plane such that its acceleration at point $A$ (extreme position) is equal to its acceleration at point $B$ (mean position). The angle $\theta$ is :-

$A$ stone of mass $1 \ kg$ tied to a light inextensible string of length $L = \frac{5}{3} \ m$ is rotating in a circular path of radius $L$ in a vertical plane. If the ratio of maximum tension in the string to the minimum tension in the string is $3$,the speed of the stone at the highest point of the circle is ($g =$ acceleration due to gravity).