Chapter 9, Problem 76GP

To determine

To Find: The value of mass which is suspended on a steel wire.

Answer to Problem 76GP

The value of mass which is suspended on a steel wire is $\text{19}\text{.2 kg}$ .

Explanation of Solution

Given:

Diameter of steel wire, $\text{d = 2 mm = 2}\times {\text{10}}^{\text{-3}}\text{ m}$

Stretch in the length of steel wire $\text{= 0}\text{.030 \%}$

Young modulus of steel wire, $\text{= 200 GPa = 200}\times {\text{10}}^9\text{ Pa}$

Formula used:

The relation between the Young modulus, stress and strain is

  $\begin{array}{l}\text{Young modulus Y = }\frac{\text{Stress}}{\text{Strain}}\\ \text{Stress = }\frac{\text{Force}}{\text{Area}}\text{ = }\frac{\text{m}\times \text{g}}{\text{A}}\\ \text{Strain = }\frac{\text{Stretch in length}}{\text{Original length}}\text{ = }\frac{\text{ΔL}}{\text{L}}\end{array}$

Calculation:

Substitute the given values, we get

  $\begin{array}{c}\text{Stress = }\frac{\text{Force}}{\text{Area}}\\ \text{= }\frac{\text{m}\times \text{g}}{\text{A}}\\ \text{= }\frac{\text{m}\times \text{9}\text{.8}}{\left(\frac{\text{Π}}{\text{4}}\times {\text{d}}^{\text{2}}\right)}\\ \text{= }\frac{\text{9}\text{.8}\times \text{m}}{\text{3}\text{.14}\times {\text{10}}^{\text{-6}}}\\ \text{= 3}\text{.12}\times {\text{10}}^{\text{6}}\text{m}\\ \text{Strain = }\frac{\text{Stretch in length}}{\text{Original length}}\text{ = }\frac{\text{ΔL}}{\text{L}}\\ \text{= }\frac{\text{0}\text{.030}}{\text{100}}\text{ = }\\ \text{0}\text{.030}\times {\text{10}}^{\text{-2}}\end{array}$

  $\begin{array}{c}\text{Young modulus Y = }\frac{\text{Stress}}{\text{Strain}}\\ \text{200}\times {\text{10}}^{\text{9}}\text{ = }\frac{\text{3}\text{.12}\times {\text{10}}^{\text{6}}\times \text{m}}{\text{0}\text{.030}\times {\text{10}}^{\text{-2}}}\\ \text{m = }\frac{\text{200}\times {\text{10}}^{\text{9}}\times \text{0}\text{.030}\times {\text{10}}^{\text{-2}}}{\text{3}\text{.12}\times {\text{10}}^{\text{6}}}\\ \text{m = 19}\text{.2 kg}\end{array}$

Conclusion:

Thus, the value of mass which is suspended on a steel wire of $2mm$ diameter and $0.030\%$ stretches is $\text{19}\text{.2 kg}$ .