A wire of length $L$, area of cross section $A$ is hanging from a fixed support. The length of the wire change

Question

A wire of length $L$, area of cross section $A$ is hanging from a fixed support. The length of the wire changes to $L_1$ when mass $M$ is suspended from its free end. The expression for Young's modulus is:

Accepted Answer

1.  **Definition of Young's Modulus ($Y$):** Young's modulus is defined as the ratio of longitudinal stress ($\sigma$) to longitudinal strain ($\varepsilon$) within the elastic limit.
    $$Y = \frac{\sigma}{\varepsilon}$$
2.  **Longitudinal Stress ($\sigma$):** Stress is the restoring force per unit area. Here, the force ($F$) is the weight of the suspended mass $M$.
    $$F = Mg$$
    $$\sigma = \frac{F}{A} = \frac{Mg}{A}$$
3.  **Longitudinal Strain ($\varepsilon$):** Strain is the ratio of the change in length ($\Delta L$) to the original length ($L$).
    Original Length = $L$
    Final Length = $L_1$
    Change in Length ($\Delta L$) = $L_1 - L$
    $$\varepsilon = \frac{\Delta L}{L} = \frac{L_1 - L}{L}$$
4.  **Substitution:** Substituting the values of stress and strain into the Young's modulus formula:
    $$Y = \frac{\frac{Mg}{A}}{\frac{L_1 - L}{L}}$$
    $$Y = \frac{Mg}{A} 	imes \frac{L}{L_1 - L} = \frac{MgL}{A(L_1 - L)}$$

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