$A$ door $1.6 \,m$ wide requires a force of $1 \,N$ to be applied at the free end to open or close it. The force that is required at a point $0.4 \,m$ distance from the hinges for opening or closing the door is (in $\,N$)

$A$ thin ring of radius $R$ is rotated about its center. Then its radius:

The instantaneous velocity of point $B$ of a rod of length $L = 0.5 \ m$ is $v = 3 \ m/s$ at an angle of $30^{\circ}$ with the rod as shown. The angular velocity of the rod for the minimum velocity of end $A$ is:

In the figure shown,a mass $m$ is attached to a light string which is wrapped around a solid cylinder of mass $M$ and radius $R$. The system starts from rest at $t = 0$. If friction is negligible,what will be the angular velocity at time $t$?

$A$ flywheel having mass $3 \ kg$ and radius $5 \ m$ is free to rotate about a horizontal axis. $A$ string having negligible mass is wound around the wheel and the loose end of the string is connected to a $3 \ kg$ mass. The mass is kept at rest initially and released. The kinetic energy of the flywheel when the mass descends by $3 \ m$ is . . . . . . $J$. $(g = 10 \ m/s^{2})$

$A$ uniform cylinder of mass $m$ can rotate freely about its own axis,which is horizontal. $A$ particle of mass $m_0$ hangs from the end of a light string wound around the cylinder,which does not slip over it. When the system is allowed to move,the acceleration of the descending mass will be