Torque Questions

A uniform rod of mass $20\text{kg}$ and length $5\text{m}$ leans against a smooth vertical wall making an angle of $60^{\circ }$ with it. The other end rests on a rough horizontal floor. The friction force that the floor exerts on the rod is (take $g=10{\text{m/s}}^2$):

A coil in the shape of an equilateral triangle of side l is suspended between the pole pieces of a permanent magnet such that is in plane of the coil. If due to a current i in the triangle a torque τ acts on it, the side l of the triangle is :

If $\vec{F}$ is the force acting on a particle having position vector $\vec{r}$ and $\vec{\tau }$ be the torque of this force about the origin, then:

A magnetic needle suspended parallel to a magnetic field requires $\sqrt{3}\text{J}$ of work to turn it through $60^{\circ }$. The torque needed to maintain the needle in this position will be:

$ABC$ is an equilateral triangle with $O$ as its centre. ${\vec{F}}_1,{\vec{F}}_2$ and ${\vec{F}}_3$ represent three forces acting along the sides $AB,BC$ and $AC$ respectively. If the total torque about $O$ is zero then the magnitude of $F_3$ is:

A current loop in a magnetic field:

A solid cylinder of mass $50\text{kg}$ and radius $0.5\text{m}$ is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of $2{\text{revolutions/s}}^2$ is: