A sphere of mass $m$ is tied to end of a string of length $l$ and rotated through the other end along a horizontal circular path with speed $v$. The work done in full horizontal circle is
$0$
$\left( {\frac{{m{v^2}}}{l}} \right)\,.\,2\pi l$
$mg\,.\,2\pi l$
$\left( {\frac{{m{v^2}}}{l}} \right)\,.\,(l)$
When an object is shot from the bottom of a long smooth inclined plane kept at an angle $60^{\circ}$ with horizontal. it can travel a distance $\mathrm{x}_{1}$ along the plane. But when the inclination is decreased to $30^{\circ}$ and the same object the shot with the same velocity, it can travel $x_{2}$ distance. Then $x_{1}: x_{2}$ will be
For a particle in a uniformly accelerated circular motion
A particle moves with constant angular velocity in circular path of certain radius and is acted upon by a certain centripetal force $F$. if the angular velocity is kept same but the radius of the path is halved, the new force will be
A smooth wire of length $2\pi r$ is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed $\omega$ about the vertical diameter $AB$, as shown in figure, the bead is at rest with respect to the circular ring at position $P$ as shown. Then the value of $\omega^2$ is equal to
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 particle will be: