$A$ rod of length $1000\, mm$ and coefficient of linear expansion $\alpha = 10^{-4} / ^\circ C$ is placed symmetrically between fixed walls separated by $1001\, mm$. The Young's modulus of the rod is $10^{11} N/m^2$. If the temperature is increased by $20^\circ C$,then the stress developed in the rod is ........... $MPa$.

If the ratio of lengths,radii,and Young's moduli of steel and brass wires in the figure are $a, b,$ and $c$ respectively,then the corresponding ratio of increase in their lengths is

$A$ sphere of mass $2 \,kg$ and diameter $4.5 \,cm$ is attached to the lower end of a steel wire of $2 \,m$ length and area of cross-section $0.24 \times 10^{-6} \,m^2$. The wire is suspended from a $205 \,cm$ high ceiling of a room. When the system is made to oscillate as a simple pendulum, the sphere just grazes the floor at its lowest position. Find the velocity of the sphere at the lowest position. (Young's modulus of steel $= 2 \times 10^{11} \,Nm^{-2}$ and acceleration due to gravity $= 10 \,ms^{-2}$) (in $\,ms^{-1}$)

$A$ ring of a thin wire of cross-sectional area $a$ and length $l$ is dipped into a liquid of surface tension $\sigma$ and taken out so that a film of liquid is formed in the ring. If Young's modulus of the material of the wire is $Y$,then the longitudinal strain developed in the ring will be:

$A$ steel rod of radius $20 \ mm$ and length $2 \ m$ is acted upon by a force of $400 \ kN$ along its length. The values of stress and strain are respectively $(Y_{\text{steel}} = 2 \times 10^{11} \ N \ m^{-2})$.

The lengths of four wires $A, B, C$ and $D$ made of the same material are $1 \,m, 2 \,m, 3 \,m$ and $4 \,m$ respectively. The radii of the wires $A, B, C$ and $D$ are $0.2 \,mm, 0.4 \,mm, 0.6 \,mm$ and $0.8 \,mm$ respectively. For the same applied tension, the elongation is more in which wire?