Stress And Strain

According to NCERT Class 11, Chapter 9 (Mechanical Properties of Solids), within the proportional limit stress is directly proportional to strain, a relationship known as Hooke's law. Stress is force per unit area and strain is the fractional change in length, so the proportionality constant is Young's modulus $Y$. Here the graph plots elongation $\Delta L$ against applied stress $\sigma =F/A$. Since $\sigma =Y(\Delta L/L)$, rearranging gives $\Delta L=\frac{L}{Y}\sigma$. The slope of $\Delta L$ versus $\sigma$ is therefore $L/Y$, which can also be written as $\frac{1}{Y}\times L$, the reciprocal of Young's modulus scaled by length. The option 'Young's modulus directly' is wrong because $Y$ is the slope only when stress is plotted against strain, not elongation against stress. The option 'breaking stress' is incorrect since that is a single point at fracture, not a slope. Poisson's ratio relates lateral to longitudinal strain and is unrelated to this slope. The key insight is that the slope of a graph always carries the units of the vertical axis quantity divided by the horizontal axis quantity, so one must read the axes carefully rather than memorising that the slope equals Young's modulus. Dimensionally, $L/Y$ has units of $\text{m}/({\text{N/m}}^2)$, consistent with elongation per unit stress, confirming the choice. As a final check, a longer or less stiff wire stretches more for the same stress, so the slope should grow with $L$ and shrink with $Y$, exactly as $L/Y$ predicts.