Test on Mechanical Properties of Solids

If S is stress and Y is Young's modulus of material of a wire, the energy stored in the wire per unit volume is

A wooden wheel of radius R is made of two semicircular parts (see figure); The two parts are held together by a ring made of a metal strip of cross sectional area A and length L. L is slightly less than 2πR. To fit the ring on the wheel, it is heated so that its temperature rises by ΔT and it just steps over the wheel. As it cools down to surrounding temperature, it pressure the semicircular parts together. If the coefficient of linear expansion of the metal is α, and its Young’s modulus is Y, the force that one of the wheel applies on the other part is:

Two wires are made of the same material and have the same volume. However wire 1 has cross–section area A and wire 2 has cross–sectional area 3A. If the length of wire 1 increases by Δx on applying force F, how much force is needed to stretch wire 2 by the same amount?

The pressure that has to be applied to the ends of a steel wire of length 10 cm to keep its length constant when its temperature is raised by 100 °C is :
(For steel Young's modulus is 2x 10¹¹ Nm^-2 and coefficient of thermal expansion is 1.1 × 10^-5 K^-1)

A thin rod of length L is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/length) varies with x as  where n can be zero or any positive number. If the position x_CM of the centre of mass of the rod is plotted against n, which of the following graphs best approximates the dependence of x_CM on n ?