$A$ bucket containing water is revolved in a vertical circle of radius $r$. To prevent the water from falling down, the period of revolution required is ($g =$ gravitational acceleration)

$A$ ball whose size is slightly smaller than the width of a tube of radius $R = 2.5 \, m$ is projected from the bottommost point of a smooth tube fixed in a vertical plane with a velocity of $v_0 = 10 \, m/s$. Let $N_1$ be the normal reaction exerted by the inner side of the tube and $N_2$ be the normal reaction exerted by the outer side of the tube on the ball. Which of the following statements is correct?

$A$ 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 $m$ 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 $X$?

As per the given figure,to complete the circular loop,what should be the radius if the initial height is $5 \, m$ (in $, m$)?

$A$ car starts from rest at height $H$ and moves along a track to complete a vertical loop of radius $r$. If the car just completes the loop,find the ratio $H/h$,where $h$ is the height of the loop above the starting point of the loop track.