What should be the speed of a car in $m/s$ to cross a bridge with a radius of curvature of $20 \; m$ without losing contact with the bridge? $(g = 9.8 \; m/s^2)$

$A$ conical pendulum of length $1\,m$ makes an angle $\theta = 45^\circ$ with respect to the $Z$-axis and moves in a circle in the $XY$ plane. The radius of the circle is $0.4\,m$ and its centre is vertically below $O$. The speed of the pendulum,in its circular path,will be ..... $m/s$ (Take $g = 10\,ms^{-2}$)

One end of a string of length $l$ is connected to a particle of mass $m$ and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $v$,the net force on the particle (directed towards the centre) is :
$(i) \; T$
$(ii) \; T - \frac{m v^{2}}{l}$
$(iii) \; T + \frac{m v^{2}}{l}$
$(iv) \; 0$
$T$ is the tension in the string. [Choose the correct alternative].

If the radius of curvature of the path of two particles of same masses are in the ratio $1 : 2$,then in order to have constant centripetal force,their velocity,should be in the ratio of

$A$ mass is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed centre at an angular velocity $\omega_0$. If the length of the string and angular velocity are doubled,the tension in the string which was initially $T_0$ is now

$A$ particle of mass $m$ is suspended from a ceiling through a string of length $L$. If the particle moves in a horizontal circle of radius $r$ as shown in the figure,then the speed of the particle is