$A$ weightless thread can support a tension of up to $30 \, N$. $A$ stone of mass $0.5 \, kg$ is tied to it and is revolved in a circular path of radius $2 \, m$ in a vertical plane. If $g = 10 \, m/s^2$,then the maximum angular velocity of the stone will be ........ $rad/s$.

$A$ simple pendulum consisting of a light inextensible string of length $\ell$ attached to a heavy small bob of mass $m$ is at rest. The bob is imparted a horizontal impulsive force which gives it a speed of $u = \sqrt{4 g \ell}$. The speed of the bob at its highest point is ($g$ is the acceleration due to gravity).

In a vertical circle of radius $r$,at what point in its path does a particle have tension equal to zero if it is just able to complete the vertical circle?

$A$ stone of mass $40$ g is tied to a light inextensible string of length $50$ cm and whirled in a vertical circle at the rate of $30$ revolutions per minute. The tension in the string when it is at the lowest point of the circle is (Take gravitational acceleration, $g = 10 \text{ m/s}^2$ and $\pi^2 = 10$) (in $N$)

An object is at the top of a smooth sphere which is kept fixed. As the object slides down after being given a negligible side push,what is the behavior of the magnitude of the acceleration of the object during its motion until it reaches the ground?

The string of a pendulum of length $L$ is displaced through $90^{\circ}$ from the vertical and released. The maximum tension the string must withstand as the pendulum passes through the mean position is ($m =$ mass of the pendulum,$g =$ acceleration due to gravity).