A uniform rectangular plate has side length $l$ and $2l$. The plate lies in $x$- $y$ plane with its centre at origin and sides parallel to $x$- and $y$-axes. The moment of inertia of the plate about an axis passing through a vertex (say $A$) perpendicular to the plane of the figure is $I_0$. Now the axis is shifted parallel to itself so that moment of inertia about it remains $I_0$. Write the locus of point of intersection of the axis with $x$- $y$ plane.

A beam of weight $W$ supports a block of weight $W.$ The length of the beam is $L$ and weight is at a distance $L/4$ from the left end of the beam. The beam rests on two rigid supports at its ends. Find the reactions of the supports.

A uniform rod of length $l$ and mass $m$ is hung from two strings of equal length from a ceiling as shown in figure. Determine the tensions in the strings.

A uniform ladder of length $L$ and mass $m_1$ rests against a frictionless wall. The ladder makes an angle $\theta$ with the horizontal.
(a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when a firefighter of mass $m_2$ is at a distance $x$ from the bottom.

(b) If the ladder is just on the verge of slipping when the firefighter is a distance $d$ from the bottom, what is the coefficient of static friction between ladder and ground?

A uniform beam of mass $m$ is inclined at an angle $\theta$ to the horizontal. Its upper end produces a ninety degree bend in a very rough rope tied to a wall, and its lower end rests on a rough floor (figure).
(a) If the coefficient of static friction between beam and floor is ${\mu }_s,$ determine an expression for the maximum mass $M$ that can be suspended from the top before the beam slips.

(b) Determine the magnitude of the reaction force at the floor and the magnitude of the force exerted by the beam on the rope at $P$ in terms of $m, M,$ and ${\mu }_s.$

The ladder shown in figure has negligible mass and rests on a frictionless floor. The crossbar connects the two legs of the ladder at the middle. The angle between the two legs is ${60}^{\mathrm{o}}$. The fat person sitting on the ladder has a mass of $80\mathrm{}\mathrm{k}\mathrm{g}$. Find the contact forces exerted by the floor on each leg and the tension in the crossbar.