Question

prove that the young modulus is the ratio of stress to the ratio of strain.

Answer 1

According to Hooke's law,
$$\begin{gathered} \mathrm{Stress}\mathrm{\alpha }\mathrm{Strain} \\ \frac{\mathrm{Stress}}{\mathrm{Strain}}=\mathrm{Constant}=\mathrm{Modulus}\mathrm{of}\mathrm{elasticity} \end{gathered}$$
Young's modulus of elasticity is defined as the ratio of normal stress to the longitudinal strain within the elastic limit.
$$\begin{gathered} \mathrm{Y}=\frac{\mathrm{F}/\mathrm{A}}{∆\mathrm{L}/\mathrm{L}} \\ =\frac{\mathrm{FL}}{\mathrm{A}∆\mathrm{L}} \\ \mathrm{Where},\mathrm{F}\mathrm{is}\mathrm{the}\mathrm{force}\mathrm{acting}\mathrm{on}\mathrm{area}\mathrm{A}\mathrm{and}\mathrm{length}\mathrm{L}\mathrm{which}\mathrm{causes}\mathrm{a}\mathrm{change}\mathrm{in}\mathrm{length}∆\mathrm{L}. \end{gathered}$$