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. \(\frac{2S}{Y}\)

If \(E\) is the energy stored per unit volume in a wire having \(Y\) as Young's modulus of the material, then the stress applied is:
1. \(\sqrt{2EY}\)
2. \(2\sqrt{EY}\)
3. \(\frac{1}{2}\sqrt{EY}\)
4. \(\frac{3}{2}\sqrt{EY}\)

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. \(\frac{E}{Y}\)

A \(5~\text{m}\) long wire is fixed to the ceiling. A weight of \(10~\text{kg}\) is hung at the lower end and is \(1~\text{m}\) above the floor. The wire was elongated by \(1~\text{mm}.\) The energy stored in the wire due to stretching is:
1. zero                        
2. \(0.05~\text J\) 
3. \(100~\text J\)                          
4. \(500~\text J\)

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 ratio of Young's modulus of the material of two wires is \(2:3\). If the same stress is applied on both, then the ratio of elastic energy per unit volume will be:
1. \(3:2\)
2. \(2:3\)
3. \(3:4\)
4. \(4:3\)

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. \(4.5\times 10^{11}~\text{J}\)

If the force constant of a wire is \(K\), the work done in increasing the length of the wire by \(l\) is:
1. \(\frac{Kl}{2}\)
2. \(Kl\)
3. \(\frac{Kl^2}{2}\)
4. \(Kl^2\)