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:

• A. $\frac {MgL_1}{AL}$
• B. $\frac {Mg(L_1-L)}{AL}$
• C. $\frac {MgL}{AL_1}$
• D. $\frac {MgL}{A(L_1-L)}$

Answer 1

The Correct Option is D

To determine the expression for Young's modulus \(Y\) when a wire of length \(L\) and cross-sectional area \(A\) stretches to a new length \(L_1\) due to a suspended mass \(M\), we need to understand the concept of Young's modulus and its application to stretching a wire.

Young's modulus is defined as the ratio of tensile stress to tensile strain:

\[ Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} \]

Step 1: Calculate Tensile Stress

Tensile stress is the force applied per unit area. Here, the force applied is the weight of the mass \(M\), which is \(Mg\) (where \(g\) is the acceleration due to gravity). Thus, the tensile stress is given by:

\[ \text{Tensile Stress} = \frac{Mg}{A} \]

Step 2: Calculate Tensile Strain

Tensile strain is the change in length divided by the original length of the wire. The change in length \(\Delta L\) is given by:

\[ \Delta L = L_1 - L \]

Therefore, the tensile strain is:

\[ \text{Tensile Strain} = \frac{\Delta L}{L} = \frac{L_1 - L}{L} \]

Step 3: Derive Young's Modulus

Substitute the expressions for tensile stress and tensile strain into the formula for Young's modulus:

\[ Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{\frac{Mg}{A}}{\frac{L_1 - L}{L}} = \frac{MgL}{A(L_1 - L)} \]

The correct expression for Young’s modulus is therefore:

\[ \frac{MgL}{A(L_1 - L)} \]

This matches the answer option: $\frac{MgL}{A(L_1-L)}$, which is the correct choice.

The other options can be ruled out because they do not correctly represent the relationship between the stress and strain for the given scenario.