$A$ pendulum consisting of a small sphere of mass $m$ suspended by an inextensible and massless string of length $l$ is made to swing in a vertical plane. If the breaking strength of the string is $2mg$,then the maximum angular amplitude of the displacement from the vertical can be ....... $^o$

For a particle moving in a vertical circle,the total energy at different positions along the path (the motion is under gravity) is:

$A$ 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?

$A$ small cube with mass $M$ starts at rest at point $1$ at a height $4R$,where $R$ is the radius of the circular part of the track. The cube slides down the frictionless track and around the loop. The force that the track exerts on the cube at point $2$ is nearly ......... times the cube's weight $mg$.

Length of a simple pendulum is $1 \,m$. When its bob is at its lowest point its velocity is $7 \,ms^{-1}$. If the bob leaves its circular path at a height $h$ above the centre of the circle, then the value of $h$ is $(g=10 \,ms^{-2})$. (in $\,m$)

$A$ small block slides down from the top of a hemisphere of radius $R=3\, m$ as shown in the figure. The height $h$ at which the block will lose contact with the surface of the sphere is $............. \; m$. (Assume there is no friction between the block and the hemisphere)