$A$ particle moves in a circular orbit under the action of a central attractive force inversely proportional to the distance $r$. The speed of the particle is

$A$ particle of mass $m$ is suspended from a ceiling through a string of length $L$. The particle moves in a horizontal circle of radius $r$ such that $r = \frac{L}{\sqrt{2}}$. The speed of the particle will be:

$A$ 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 .......... $m/s$.

$A$ coin,placed on a rotating turntable,slips when it is placed at a distance of $9 \, cm$ from the center. If the angular velocity of the turntable is tripled,it will just slip if its distance from the center is ......... $cm$.

One end of a massless spring of spring constant $k$ and natural length $l_{0}$ is fixed,while the other end is connected to a small object of mass $m$ lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $\omega$ about an axis passing through the fixed end,then the elongation of the spring will be

$A$ ball of mass $m = 0.5 \ kg$ is attached to the end of a string having length $L = 0.5 \ m$. The ball is rotated on a horizontal circular path about a vertical axis. The maximum tension that the string can bear is $324 \ N$. The maximum possible value of angular velocity of the ball (in $rad/s$) is