: A hanging wire made of an alloy of iron with diameter 0.09cm is initially 2.2m long. When a 66kg mass is hung from it, the wire stretches an amount of 1.12cm. A mole of iron has a mass of 56g, and its density is 7.87 g/cm. Based on these experimental measurement, what is Young’s modulus for this alloy iron

The given data can be listed below,

• The diameter of wire is,$D=0.09cm\left(\frac{1m}{100cm}\right)=0.09\times {10}^{-2}m$
• The length of the wire is,$L=2.2m$
• Mass of the hanging object is,$m=66kg$ .
• The elongation is wire is,$∆L=1.12cm$
• Molecular mass of the iron is,$M_{Fe}=56g$
• The density of iron is,${\rho }_{Fe}=7.87g/c{m}^3\left(\frac{1000kg/m^3}{1g/c{m}^3}\right)=7.87\times {10}^{3}kg/m^3$

The elastic modulus of a material is defined of its stiffness that is constant throughout a wide range of stresses in most materials.

Young's modulus is defined as the proportion of longitudinal strain to longitudinal stress.

The force on the hanging object is given by,

$F=mg$

Here, m is the mass of the object and g is the acceleration of gravity

Substitute all the values in the above,

$$\begin{gathered} F=66kg\times 9.8m/s^2 \\ =646.8N \end{gathered}$$

Cross-sectional area of the wire is given by,

$A=\frac{{\mathrm{\pi D}}^2}{4}$

Here, D is the diameter of wire.

Substitute the value in the above,

$$\begin{gathered} A=\frac{\mathrm{\pi }{\left(0.09\times {10}^{-2}\mathrm{m}\right)}^2}{4} \\ =6.36\times {10}^{-7}m \end{gathered}$$

Young modulus of the wire is given by,

$Y=\frac{FL}{A∆L}$

Here, F is the force on object, L is the length of the wire, A is the elongation in wire, and$∆L$ is the elongation in wire.

Substitute all the values in the above expression.

$$\begin{gathered} Y=\frac{\left(646.8N\right)\left(2.2m\right)}{\left(6.36\times {10}^{-7}{m}^2\right)\left(1.12cm\right)\left({\displaystyle \frac{1m}{100cm}}\right)} \\ =2.0\times {10}^{11}N/m^2 \end{gathered}$$

Thus, the young modulus of the iron alloy is $2.0\times {10}^{11}N/m^2$.