The following four wires are made of the same material. Which of these will have the largest extension when th

Question

The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?

Accepted Answer

1. **Formula:** The extension ($\Delta l$) of a wire under tension ($F$) is derived from Young's Modulus ($Y = \frac{FL}{A\Delta l}$), giving $\Delta l = \frac{FL}{AY}$ .
2. **Dependencies:** Since the material (same $Y$) and tension ($F$) are constant, the extension is proportional to the ratio of length to cross-sectional area: $\Delta l \propto \frac{L}{A}$.
3. **Area:** The cross-sectional area $A = \pi r^2 = \pi (d/2)^2 \propto d^2$. Therefore, $\Delta l \propto \frac{L}{d^2}$.
4. **Comparison:** We calculate the ratio $\frac{L}{d^2}$ for each option (keeping units consistent is not strictly necessary for comparison if we treat cm and mm consistently as relative units, but converting to standard units prevents errors. Here we simply compare the numerical values of $L_{cm}/d_{mm}^2$):
   - Option A: $\frac{100}{1^2} = 100$
   - Option B: $\frac{200}{2^2} = \frac{200}{4} = 50$
   - Option C: $\frac{300}{3^2} = \frac{300}{9} \approx 33.3$
   - Option D: $\frac{50}{(0.5)^2} = \frac{50}{0.25} = 200$
5. **Conclusion:** The wire in Option D has the highest ratio, meaning it will experience the largest extension.

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