A uniform thin rod of length $L$ is hinged about one of its ends and is free to rotate about the hinge without friction. Neglect the effect of gravity.A force$\mathit{ }F\mathit{ }$is applied at a distance $x$ from the hinge on the rod such that force is always perpendicular to the rod. Find the normal reaction at the hinge as function of $x\mathit{,}$ at the initial instant when the angular velocity of rod is zero.

  

A solid cylinder of mass $M=1 \mathrm{kg}$ & radius $R=0.5 \mathrm{m}$ is pivoted at its centre & has three particles of mass $m=0.1\mathrm{kg}$ mounted at its perimeter in the vertical plane as shown in the figure. The system is initially at rest. Find the angular speed of the cylinder, when it has swung through ${90}^0$ in anticlockwise direction. $[\mathrm{Take} g=10 \mathrm{m} {\mathrm{s}}^{-2}]$

A rigid body is made of three identical uniform thin rods each of length $L$ fastened together in the form of letter $H.$The body is free to rotate about a fixed horizontal axis $\mathrm{AB}$that passes through one of the legs of the$H.$ The body is allowed to fall from rest from a position in which the plane of $H$ is horizontal. What is the angular speed of the body, when the plane of $H$ is vertical. 

A uniform rod of mass m and length $L$ lies radially on a disc rotating with angular speed $\mathrm{\omega }$ in a horizontal plane about its axis. The rod does not slip on the disc and the centre of the rod is at a distance $R$ from the centre of the disc. Find out the kinetic energy of the rod.

The moment of inertia of the pulley system as shown in figure is $3 {\mathrm{kgm}}^2 .$ Its radius is $1\mathrm{m}.$ The system is released from rest ,find the linear velocity of the block when it has descended through $40 \mathrm{cm}.$(Assume that there is no slipping between string & pulley and string is light)$[\mathrm{Take} g=10 \mathrm{m} {\mathrm{s}}^{-2} ]$

Two beads (each of mass $m$) can move freely in a frictionless wire whose rotational inertia with respect to the vertical axis is $I$. The system is rotated with an angular velocity ${\mathrm{\omega }}_0$ when the beads are at a distance $\frac{r}{2}$ from the axis. What is the angular velocity of the system when the beads are at a distance$\mathit{ }r$ from the axis ?