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

Question

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

Accepted Answer

1. **Formula:** The extension ($\Delta L$) in a wire is given by the formula for Young's Modulus ($Y$): $\Delta L = \frac{F L}{A Y}$, where $F$ is the tension, $L$ is the length, and $A$ is the cross-sectional area .
2. **Area:** The cross-sectional area $A = \pi r^2 = \pi (d/2)^2 = \frac{\pi d^2}{4}$, where $d$ is the diameter.
3. **Proportionality:** Since the material ($Y$) and tension ($F$) are the same for all wires, the extension is directly proportional to length and inversely proportional to the square of the diameter:
   $$ \Delta L \propto \frac{L}{d^2} $$
4. **Comparison:** Calculating the ratio $\frac{L}{d^2}$ for each option (keeping units consistent relative to each other):
   - **Option A:** $\frac{50}{(0.5)^2} = \frac{50}{0.25} = 200$
   - **Option B:** $\frac{100}{(1)^2} = 100$
   - **Option C:** $\frac{200}{(2)^2} = \frac{200}{4} = 50$
   - **Option D:** $\frac{300}{(3)^2} = \frac{300}{9} \approx 33.3$
5. **Conclusion:** The wire with length 50 cm and diameter 0.5 mm has the highest ratio, and thus the largest extension.

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