Question

Two wires $A$ and $B$ of the same length are made of the same material. The Load ($F$) vs. elongation ($x$) graph for these two wires is shown in the figure. Then which of the following statement(s) is/are true?

• A. The cross-section area of A is greater than that of B
• B. Young's modulus of A is greater than Young's modulus of B.
• C. The cross-sectional area of B is greater than that of A.
• D. Young's modulus of both A and B are same.

Answer 1

The Correct Option is A, D

We need to analyze the relationship between load, elongation, Young's modulus, length, and cross-sectional area to determine the correct statements.

Step 1: Recall the Formula for Young's Modulus

Young's modulus \(Y\) is defined as: \[Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L_0}\] where:

• \(F\) is the applied force (Load)
• \(A\) is the cross-sectional area
• \(\Delta L\) is the elongation (change in length) which we denote as \(x\)
• \(L_0\) is the original length

We can rearrange this formula to express the elongation \(x\) as: \[x = \frac{F L_0}{A Y}\] or, equivalently, \[F = \frac{AY}{L_0} x\]

Step 2: Analyze the Graph

The graph plots \(F\) (Load) vs. \(x\) (elongation). This represents a linear relationship, and the slope of the graph is given by: \[\text{Slope} = \frac{F}{x} = \frac{AY}{L_0}\]

From the graph, we can observe that the slope of line \(A\) is greater than the slope of line \(B\). Therefore: \[\frac{A_A Y_A}{L_A} > \frac{A_B Y_B}{L_B}\] where \(A_A\) and \(A_B\) are the cross-sectional areas of wires \(A\) and \(B\), and \(Y_A\) and \(Y_B\) are their respective Young's moduli.

Step 3: Apply Given Conditions

We are given that the wires are made of the same material, which means \(Y_A = Y_B\). We are also given that the wires have the same length, so \(L_A = L_B\). Therefore, the inequality simplifies to: \[A_A > A_B\]

Step 4: Determine the Correct Statements

1. The cross-section area of A is greater than that of B: This statement is TRUE, as we derived that \(A_A > A_B\).
2. Young's modulus of A is greater than Young's modulus of B: This statement is FALSE because the wires are made of the same material, implying \(Y_A = Y_B\).
3. The cross-sectional area of B is greater than that of A: This statement is FALSE as \(A_A > A_B\).
4. Young's modulus of both A and B are same: This statement is TRUE, since it is mentioned they are made of the same material.

Conclusion

The correct statements are:

• The cross-section area of A is greater than that of B
• Young's modulus of both A and B are the same.