Question

Two copper wires have the lengths in the ratio $1 : 2$ and their radii are in the ratio $3 : 1$. If they are stretched by the same force, the ratio of the respective longitudinal strains in the two wires is

• A. $1 : 9$
• B. $9 : 1$
• C. $1 : 27$
• D. $27 : 1$
• E. $1 : 3$

Hint:

Often in elasticity problems, unnecessary data like the "length ratio" is given to distract you. Strain specifically depends on radius and force, but not on initial length if the material is identical.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Longitudinal strain is defined as the change in length per unit original length. According to Hooke's Law for wires, the strain depends on the stress (force per unit area) and the material's Young's Modulus.
Key Formula or Approach:
Young's Modulus \( Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\text{Strain}} \)
So, \( \text{Strain} = \frac{F}{A \cdot Y} = \frac{F}{\pi r^2 \cdot Y} \).

Step 2: Detailed Explanation:
1. Both wires are made of copper, so their Young's Modulus (\( Y \)) is the same.
2. Both are stretched by the same force \( F \).
3. Therefore, strain \( \epsilon \propto \frac{1}{r^2} \). Note that strain is independent of the original length \( L \) in this context.
Given radii ratio \( \frac{r_1}{r_2} = \frac{3}{1} \).
The ratio of strains is:
\[ \frac{\epsilon_1}{\epsilon_2} = \left( \frac{r_2}{r_1} \right)^2 \]
\[ \frac{\epsilon_1}{\epsilon_2} = \left( \frac{1}{3} \right)^2 = \frac{1}{9} \]

Step 3: Final Answer:
The ratio of the longitudinal strains is $1 : 9$.