$A$ string breaks if its tension exceeds $10 \, N$. $A$ stone of mass $250 \, g$ tied to this string of length $10 \, cm$ is rotated in a horizontal circle. The maximum angular velocity of rotation can be .......... $rad/s$.

$A$ train is moving towards north. At one place it turns towards north-east. Here,we observe that:

$A$ particle starts from rest and performs circular motion of constant radius with speed given by $v = \alpha \sqrt{x}$,where $\alpha$ is a constant and $x$ is the distance covered. The correct graph of the magnitude of its tangential acceleration $(a_t)$ and centripetal acceleration $(a_c)$ versus $t$ will be:

$A$ particle of mass $M$ moves with constant speed along a circular path of radius $r$ under the action of a force $F$. Its speed is

In a conical pendulum,the bob of mass $m$ moves in a horizontal circle of radius $r$ with uniform speed $V$. The string of length $L$ describes a cone of semi-vertical angle $\theta$. The centripetal force acting on the bob is $(g = \text{acceleration due to gravity})$

If mass,speed,and radius of the circular path of a particle are increased by $100 \%$,then the necessary force required to maintain the circular path will have to be increased by (in $\%$)