Question

The length of a metal wire is $l_{1}$ when the tension is $T_{1}$ and $l_{2}$ when the tension is $T_{2}$. The unstretched length of the wire is ________.

• A. $\frac{T_{2}l_{1}-T_{1}l_{2}}{T_{2}-T_{1}}$
• B. $\frac{T_{2}l_{1}+T_{1}l_{2}}{T_{2}-T_{1}}$
• C. $\frac{T_{2}l_{1}-T_{1}l_{2}}{T_{2}+T_{1}}$
• D. $\frac{T_{2}l_{1}+T_{1}l_{2}}{T_{2}+T_{1}}$

Hint:

This formula effectively uses a weighted average logic to find the zero-tension point.

Answer 1

The Correct Option is A

Step 1: Concept
Hooke's Law: $T = k(l - L)$, where $L$ is the unstretched length.
Step 2: Analysis
$T_1 = k(l_1 - L)$ ...(1) $T_2 = k(l_2 - L)$ ...(2)
Step 3: Calculation
Divide (1) by (2): $T_1 / T_2 = (l_1 - L) / (l_2 - L)$. $T_1 l_2 - T_1 L = T_2 l_1 - T_2 L$. $L(T_2 - T_1) = T_2 l_1 - T_1 l_2$. $L = \frac{T_2 l_1 - T_1 l_2}{T_2 - T_1}$.
Step 4: Conclusion
Hence, the unstretched length is $\frac{T_{2}l_{1}-T_{1}l_{2}}{T_{2}-T_{1}}$.
Final Answer:(A)