A string of area of cross-section $4\,mm ^{2}$ and length $0.5$ is connected with a rigid body of mass $2\,kg$. The body is rotated in a vertical circular path of radius $0.5\,m$. The body acquires a speed of $5\,m / s$ at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is $\ldots . . \times 10^{-5}$. (Use Young's modulus $10^{11}\,N / m ^{2}$ and $g =10\,m / s ^{2}$ )

A wooden wheel of radius $R$ is made of two semicircular part (see figure). The two parts are held together by a ring made of a metal strip of cross section area $S$ and length $L$. $L$ is slighly less than $2\pi R$. To fit the ring on the wheel, it is heated so that its temperature rises by $\Delta T$ and it just steps over the wheel.As it cools down to surronding temperature, it presses the semicircular parts together. If the coefficint of linear expansion of the metal is $\alpha$, and its young's modulus is $Y$, the force that one part of wheel applies on the other part is 

Density of rubber is $​d$​. $​ A$​ thick rubber cord of length $​L$​ and cross-section area $​A$​ undergoes elongation under its own weight on suspending it. This elongation is proportional to

A rigid massless rod of length $6\ L$ is suspended horizontally by means of two elasticrods $PQ$ and $RS$ as given figure. Their area of cross section, young's modulus and lengths are mentioned in figure. Find deflection of end $S$ in equilibrium state. Free end of rigid rod is pushed down by a constant force . $A$ is area of cross section, $Y$ is young's modulus of elasticity

A wire of length $L$ and radius $r$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $F$, its length increases by $5\,cm$. Another wire of the same material of length $4 L$ and radius $4\,r$ is pulled by a force $4\,F$ under same conditions. The increase in length of this wire is $....cm$.

Column$-II$ is related to Column$-I$. Join them appropriately :