When a disc is rotating with angular velocity $\omega$,a particle situated at a distance of $4 \ cm$ just begins to slip. If the angular velocity is doubled,at what distance will the particle start to slip (in $cm$)?

$A$ block of $200\, g$ mass moves with a uniform speed in a horizontal circular groove,with vertical side walls of radius $20\, cm$. If the block takes $40\, s$ to complete one round,the normal force by the side walls of the groove is

$A$ string of length $L$ is fixed at one end and carries a mass $M$ at the other end. The string makes $\frac{\pi}{2}$ revolutions per second around the vertical axis through the fixed end as shown in the figure. The tension in the string is:

$A$ thin circular loop of radius $R$ rotates about its vertical diameter with an angular frequency $\omega$. Show that a small bead on the wire loop remains at its lowermost point for $\omega \leq \sqrt{g / R}$. What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for $\omega = \sqrt{2g / R}$? Neglect friction.

$A$ ball of mass $0.25 \, kg$ attached to the end of a string of length $1.96 \, m$ is moving in a horizontal circle. The string will break if the tension is more than $25 \, N$. The maximum speed with which the ball can be moved is .......... $m/s$.

$A$ man carrying a monkey on his shoulder is cycling smoothly on a circular track of radius $9 \,m$ and completes $120$ revolutions in $3$ minutes. The magnitude of the centripetal acceleration of the monkey is (in $m/s^2$):