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).

$A$ small bob tied at one end of a thin string of length $1\, m$ is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio $5: 1$. The velocity of the bob at the highest position is ............. $m/s$ (Take $g=10\, m/s^2$).

$A$ stone of mass $1 \ kg$ is tied to one end of a light string and is whirled in a vertical circle as shown in the figure. The tension in the string when the string becomes horizontal is ......... $N$.

$A$ stone tied to a string of length $L$ is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed $u$. The magnitude of the change in its velocity, as it reaches a position where the string is horizontal, is $\sqrt{x(u^{2}-gL)}$. The value of $x$ is .............

$A$ stone of mass $m$ $kg$ is tied to a string of length $L$ $m$ and moved in a vertical circle of radius $49$ $cm$ in a vertical plane. If it completes $30$ revolutions per minute,the tension in the string when it is at the lowermost point is nearly. [Take $\pi^2=10$ and acceleration due to gravity,$g=10$ $m/s^2$]

$A$ weightless thread can bear a tension of up to $3.7 \, kg \, wt$. $A$ stone of mass $500 \, g$ is tied to it and revolved in a circular path of radius $4 \, m$ in a vertical plane. If $g = 10 \, m/s^2$,then the maximum angular velocity of the stone will be ........ $rad/s$.