Consider the situation shown in the figure. The force $F$ is equal to $m_2g/2$. If the area of cross-section of the string is $A$ and its Young's modulus is $Y$,find the strain developed in it. The string is light and there is no friction anywhere.

$A$ brass wire of length $2 \ m$ and radius $1 \ mm$ at $27 ^\circ C$ is held taut between two rigid supports. Initially, it was cooled to a temperature of $-43 ^\circ C$, creating a tension $T$ in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to $1.4 \ T$ is . . . . . . $^\circ C$.

$A$ force $F$ is applied on a wire of radius $r$ and length $L$,and the change in the length of the wire is $l$. If the same force $F$ is applied on a wire of the same material with radius $2r$ and length $2L$,then what is the change in the length of the second wire?

$A$ 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 $\,ms^{-1}$)

$A$ student performs an experiment to determine the Young's modulus of a wire of length $2 \, m$ using Searle's method. In an observation,for a load of $10 \, kg$,the extension of the wire is measured as $0.88 \, mm$ with an uncertainty of $\pm 0.05 \, mm$. The student also measures the diameter of the wire as $0.4 \, mm$ with an uncertainty of $\pm 0.01 \, mm$. Take $g = 9.8 \, m/s^2$ (exact). Calculate the Young's modulus of the wire.

$A$ wire of length $L$ and radius $r$ is rigidly fixed at one end. On stretching the other end of the wire with a force $F$,the increase in its length is $l$. If another wire of same material but of length $2L$ and radius $2r$ is stretched with a force of $2F$,the increase in its length will be