Question

When a tension of \(F_1\) is applied on a metal wire its length is \(L_1\), if the tension is \(F_2\), length becomes \(L_2\). Then the original length of the wire is:

• A. 2\( \frac{F_2 - F_1}{F_1 + F_2}L_2 \)
• B. \( \frac{F_2 L_1 - F_1 L_2}{F_2 - F_1} \)
• C. \( F_1 L_1 - F_2 L_2 \)
• D. \( (F_1 - F_2) L_2 \)

Hint:

In problems involving elasticity, the original length can often be deduced by considering changes under different forces and applying principles from material science.

Answer 1

The Correct Option is B

Step 1: Apply the principle of superposition of forces. Assuming linear elasticity, the change in length due to each force can be superimposed. Step 2: Calculate the original length \(L\). Using the relationship between force and elongation: \[ \Delta L = \frac{F \cdot L}{EA} \] Solving for \(L\) with the known conditions gives: \[ L = \frac{F_2 L_1 - F_1 L_2}{F_2 - F_1} \]

Answer 2

To solve the problem, we need to determine the original length of a metal wire based on its change in length when subjected to different tensions.

1. Understanding the Problem:
When a tension \( F_1 \) is applied to the wire, its length is \( L_1 \). When the tension is increased to \( F_2 \), the length becomes \( L_2 \). We are tasked with finding the original length of the wire, which is the length when no force is applied (denoted as \( L_0 \)).

2. Concept of Hooke's Law:
According to Hooke's Law, the elongation or compression of a wire is directly proportional to the applied force. The change in length, \( \Delta L \), is related to the applied force \( F \) and the original length \( L_0 \) by the equation: \[ \Delta L = \frac{F}{k} \] where \( k \) is the spring constant. In this case, the difference in length is due to the change in tension, so we have the following two relations:
\( L_1 = L_0 + \Delta L_1 = L_0 + \frac{F_1}{k} \)
\( L_2 = L_0 + \Delta L_2 = L_0 + \frac{F_2}{k} \)

Now, solving for \( L_0 \) (the original length), we can rearrange the equations and find that the original length is: \[ L_0 = \frac{F_2 L_1 - F_1 L_2}{F_2 - F_1} \]

3. Conclusion:
Based on the above calculation, the correct formula for the original length of the wire is:

Final Answer:
The correct option is (B) \( \frac{F_2 L_1 - F_1 L_2}{F_2 - F_1} \).