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 the ratio of change in length to original length. It can be found from Young's Modulus, which relates stress and strain. Since both wires are made of copper, their Young's modulus is identical.
Step 2: Key Formula or Approach:
Young's Modulus \(Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\text{Strain}}\).
Rearranging for strain: \(\text{Strain} = \frac{F}{A \cdot Y}\).
Since \(F\) and \(Y\) are constant for both wires, \(\text{Strain} \propto \frac{1}{A} \propto \frac{1}{r^2}\).
Step 3: Detailed Explanation:
Notice that the length ratio is extra information and is not needed to find strain, since strain fundamentally depends only on the cross-sectional stress when the material is identical.
Let the strain in wire 1 be \(S_1\) and in wire 2 be \(S_2\).
\[ \frac{S_1}{S_2} = \frac{A_2}{A_1} = \frac{\pi r_2^2}{\pi r_1^2} = \left(\frac{r_2}{r_1}\right)^2 \] We are given \(\frac{r_1}{r_2} = \frac{3}{1}\), so \(\frac{r_2}{r_1} = \frac{1}{3}\).
Substitute this into the ratio equation:
\[ \frac{S_1}{S_2} = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \] Step 4: Final Answer:
The ratio of longitudinal strains is 1 : 9.