Question

Two wires \(W_1\) and \(W_2\) of the same material and equal radius are stretched by equal forces well within their elastic limit. If the lengths of wires \(W_1\) and \(W_2\) are in the ratio \(1:3\), then what will be the ratio of strains produced in wires \(W_1\) and \(W_2\)?

• A. \(1:1\)
• B. \(1:3\)
• C. \(3:1\)
• D. \(1:6\)

Hint:

For wires of the same material and same cross-sectional area subjected to equal forces: Stress is the same and therefore \[ \text{Strain is also the same}. \] The original lengths do not affect the strain in this case.

Answer 1

The Correct Option is A

Step 1: Connect strain to stress.
Young's modulus says $Y = \frac{\text{stress}}{\text{strain}}$, so rearranging, $\text{strain} = \frac{\text{stress}}{Y}$, where stress $= \frac{F}{A}$.
Step 2: List what is equal for the two wires.
Both wires are the same material (so same $Y$), have equal radius (so same area $A$), and are stretched by equal forces $F$. Their lengths differ in the ratio $1:3$, but length does not appear in the stress.
Step 3: Compare the stresses.
Since $F$ and $A$ are the same for both, the stress $\frac{F}{A}$ is identical in the two wires.
Step 4: Compare the strains.
Strain $= \frac{\text{stress}}{Y}$. With equal stress and equal $Y$, the strains are equal too.
Step 5: Note why length does not matter.
Length would matter for the actual extension, but strain is a ratio (extension divided by length), so it depends only on stress and $Y$, not on how long the wire is.
Step 6: Write the ratio.
Strain of $W_1$ to strain of $W_2$ is $1:1$.
\[ \boxed{\text{Strain ratio} = 1:1} \]