A material has Poisson's ratio of \(0.5\). If a uniform rod made of it suffers a longitudinal strain of \(2\times 10^{-3}\), what is the percentage increase in volume?
1. \(2\%\)
2. \(4\%\)
3. \(0\%\)
4. \(5\%\)

If the ratio of lengths, radii, and Young's modulus of steel and brass wires in the figure are \(a,\) \(b\) and \(c\) respectively, then the corresponding ratio of increase in their lengths will be:

                                    
1. \(\dfrac{2 a^{2} c}{b}\)

2. \(\dfrac{3 a}{2 b^{2} c}\)

3. \(\dfrac{2 a c}{b^{2}}\)

4. \(\dfrac{3 c}{2 a b^{2}}\)

The density of metal at normal pressure is \(\rho\)$\mathrm{}$. lts density when it is subjected to an excess pressure \(P\) is \(\rho'\). lf \(B\) is the bulk modulus of the metal, the ratio \(\frac{ρ'}{\rho }\) is:
1. \(\frac{1}{1-\frac{p}{B}} \)
2. \(1+\frac{B}{P} \)
3. \(\frac{1}{1-\frac{B}{P}} \)
4. \(2+\frac{P}{B}\)

A uniform cylinder rod of length \(L\), cross-sectional area \(A\) and Young's modulus \(Y\) is acted upon by the forces, as shown in the figure. The elongation of the rod is:

           
1. \(\frac{3FL}{5AY}\)

2. \(\frac{2FL}{5AY}\)

3. \(\frac{2FL}{8AY}\)

4. \(\frac{8FL}{3AY}\)

An elastic material of Young's modulus \(Y\) is subjected to a stress \(S\). The elastic energy stored per unit volume of the material is:
1. \(\frac{SY}{2}\)
2. \(\frac{S^2}{2Y}\)
3. \(\frac{S}{2Y}\)
4. \(\frac{2S}{Y}\)

lf \(\rho\) is the density of the material of a wire and \(\sigma\) is the breaking stress, the greatest length of the wire that can hang freely without breaking is:
1. \(\dfrac{2}{\rho g}\)

2. \(\dfrac{\rho}{\sigma g}\)

3. \(\dfrac{\rho g}{2 \sigma}\)

4. \(\dfrac{\sigma}{\rho g}\)

The work done per unit volume to stretch the length of a wire by \(1\%\) with a constant cross-sectional area will be:
\((Y = 9\times10^{11}~\text{N/m}^2)\)
1. \(9\times 10^{11}~\text{J}\)
2. \(4.5\times 10^{7}~\text{J}\)
3. \(9\times 10^{7}~\text{J}\)
4. \(4.5\times 10^{11}~\text{J}\)

A wire of length \(L\) and cross-sectional area \(A\) is made of a material of Young's modulus \(Y.\) It is stretched by an amount \(x.\) The work done is:

1. \(\dfrac{Y x A}{2 L}\)

2. \(\dfrac{Y x^{2} A}{L}\)

3. \(\dfrac{Y x^{2} A}{2 L}\)

4. \(\dfrac{2 Y x^{2} A}{L}\)

The Young's modulus of a wire is \(Y.\) If the energy per unit volume is \(E,\) then the strain will be:
1. \(\sqrt{\frac{2E}{Y}}\)
2. \(\sqrt{2EY}\)
3. \(EY\)
4. \(\frac{E}{Y}\)