The area of cross-section of the rope used to lift a load by a crane is $2.5 \times 10^{-4} \, m^2$. The maximum lifting capacity of the crane is $10$ metric tons. To increase the lifting capacity of the crane to $25$ metric tons,the required area of cross-section of the rope should be $......... \times 10^{-4} \, m^2$ (take $g = 10 \, m/s^2$).

When a uniform wire of radius $r$ is stretched by a $2 \, kg$ weight,the increase in its length is $2.00 \, mm$. If the radius of the wire is $r/2$ and other conditions remain the same,the increase in its length is .......... $mm$. (in $.00$)

$A$ thin rod having a length of $1\;m$ and area of cross-section $3 \times 10^{-6}\;m^2$ is suspended vertically from one end. The rod is cooled from $210^{\circ}C$ to $160^{\circ}C$. After cooling,a mass $M$ is attached at the lower end of the rod such that the length of the rod again becomes $1\;m$. Young's modulus and coefficient of linear expansion of the rod are $2 \times 10^{11}\;Nm^{-2}$ and $2 \times 10^{-5}\;K^{-1}$,respectively. The value of $M$ is $.......kg$. (Take $g=10\;ms^{-2}$)

$A$ steel rod has a radius of $10 \,mm$ and a length of $1 \,m$. $A$ $80 \,kN$ force stretches it along its length. If the Young's modulus of the rod is $2 \times 10^{11} \,N/m^2$, then the change in length is

Two wires of the same material have lengths in the ratio $1:2$ and diameters in the ratio $2:1$. If they are stretched by forces $F_A$ and $F_B$ respectively to produce the same extension,then the ratio $\frac{F_A}{F_B}$ is:

$A$ rod of length $L$ and radius $r$ is held between two rigid walls so that it is not allowed to expand. If its temperature is increased,then the force developed in it is proportional to .........