Question

Let L be the length and d be the diameter of cross-section of a wire. Wires of the same material with different L and d are subjected to the same tension along the length of the wire. In which of the following cases, the extension of wire will be the maximum?

• A. $L=200cm,d=0.5mm$
• B. $L=300cm,d=1.0mm$
• C. $L=50cm,d=0.05mm$
• D. $L=100cm,d=0.2mm$

Answer 1

Answer: (C)

$L=50cm,d=0.05mm$

Since material is same, Young's modulus is also same.

We have: $Y\frac{\Delta l}{l}=\frac{F}{A}\Rightarrow \Delta l=\frac{Fl}{YA}\propto \frac{l}{d^2}$

where, $Y$ is the Young's Modulus, $F$ is the tension which is same in all cases, $A$ is the area of cross section of the wire and $l$ is the length of the wire.

Calculating the value of $\frac{l}{d^2}$ in all cases, we get:

A: $\frac{l}{d^2}=\frac{2000}{0.5\times 0.5}=8000$

B: $\frac{l}{d^2}=\frac{3000}{1.0\times 1.0}=3000$

C: $\frac{l}{d^2}=\frac{50}{0.05\times 0.05}=20000$

D: $\frac{l}{d^2}=\frac{100}{0.2\times 0.2}=2500$

So maximum elongation will be in case C.