A uniform metre stick having mass $400 \mathrm{g}$ is suspended from the fixed supports through two vertical light strings of equal lengths fixed at the ends. A small object of mass $100 \mathrm{g}$ is put on the stick at a distance of $60 \mathrm{cm}$ from the left end. Calculate the tensions in the two strings.$(g=10 \mathrm{m} {\mathrm{s}}^{-2})$

Assuming frictionless contacts, determine the magnitude of external horizontal force $P$ applied at the lower end for equilibrium of the rod as shown in figure. The rod is uniform and its mass is $m\mathit{.}$

In the system as shown in figure, $\mathrm{AB}$ is a uniform rod of mass $10 \mathrm{kg}$ and $\mathrm{BC}$is a light string which is connected between wall and rod, in vertical plane. There is block of mass $15 \mathrm{kg}$ connected at $\mathrm{B}$with a light string. $[\text{Take }g=10 \mathrm{m} {\mathrm{s}}^{-2}]$ ($\mathrm{BC}$ and $\mathrm{BD}$are two different strings)

$\left(\mathrm{i}\right)$ Tension in the string $\mathrm{BC}$

$\left(\mathrm{ii}\right)$Hinge force exerted on beam at point $\mathrm{A}$

A rod of negligible mass having length $l=2 \mathrm{m}$is pivoted at its centre and two masses of $m_1=6 \mathrm{kg}$ and $m_2=3 \mathrm{kg}$ are hung from the ends as shown in figure.

$\left(\mathrm{a}\right)$Find the initial angular acceleration of the rod if it is horizontal initially.

$\left(\mathrm{b}\right)$ If the rod is uniform and has a mass of $m_3=3 \mathrm{kg}.$

$\left(\mathrm{i}\right)$ Find the initial angular acceleration of the rod.

$\left(\mathrm{ii}\right)$ Find the tension in the supports to the blocks of mass $3 \mathrm{kg}$ and $6 \mathrm{kg}$ $(g=10 {\mathrm{ms}}^{-2}).$

The uniform rod $AB$of mass m is released from rest when $\mathrm{\beta }= 60º.$ Assuming that the friction force between end $A$ and the surface is large enough to prevent sliding, determine (for the instant just after release)

$\left(\mathrm{a}\right)$The angular acceleration of the rod

$\left(\mathrm{b}\right)$ The normal reaction and the friction force at $A.$

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.