The maximum elongation of a steel wire of $1 ext{ m}$ length if the elastic limit of steel and its Young's m

Question

The maximum elongation of a steel wire of $1 	ext{ m}$ length if the elastic limit of steel and its Young's modulus, respectively, are $8 	imes 10^8 	ext{ N m}^{-2}$ and $2 	imes 10^{11} 	ext{ N m}^{-2}$, is:

Accepted Answer

1.  **Identify Given Values:**
    *   Length ($L$) = $1 	ext{ m}$
    *   Elastic Limit (Maximum Stress, $\sigma$) = $8 	imes 10^8 	ext{ N m}^{-2}$
    *   Young's Modulus ($Y$) = $2 	imes 10^{11} 	ext{ N m}^{-2}$

2.  **Formula Application:**
    Young's Modulus is defined as the ratio of longitudinal stress to longitudinal strain within the elastic limit.
    $$Y = \frac{	ext{Stress} (\sigma)}{	ext{Strain} (\varepsilon)} = \frac{\sigma}{\Delta L / L}$$
    Rearranging the formula to solve for elongation ($\Delta L$):
    $$\Delta L = \frac{\sigma \cdot L}{Y}$$

3.  **Calculation:**
    Substitute the values into the equation:
    $$\Delta L = \frac{(8 	imes 10^8) 	imes 1}{2 	imes 10^{11}}$$
    $$\Delta L = 4 	imes 10^{8-11} 	ext{ m}$$
    $$\Delta L = 4 	imes 10^{-3} 	ext{ m}$$

4.  **Unit Conversion:**
    Since $10^{-3} 	ext{ m} = 1 	ext{ mm}$:
    $$\Delta L = 4 	ext{ mm}$$

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