Question

One end of a steel wire of length 2 m is attached to the roof and the other end is loaded with 2 kg mass. Another steel wire of same thickness but 1 m in length is held horizontally and stretched by applying two 20 N forces at its two ends. The ratio of the elongations produced in the two wires is (Acceleration due to gravity = $10~\text{ms}^{-2}$)

• A. 1:1
• B. 1:3
• C. 4:1
• D. 2:1

Hint:

Use $\Delta L = \frac{F L}{A Y}$ carefully and account for applied forces in series and parallel configurations.

Answer 1

The Correct Option is B

1. Elongation of a wire: $\Delta L = \frac{F L}{A Y}$, where $F$ = force, $L$ = original length, $A$ = cross-sectional area, $Y$ = Young's modulus.
2. Vertical wire: $F = mg = 2 \cdot 10 = 20~\text{N}$, $L = 2~\text{m}$
3. Horizontal wire: $F = 20 + 20 = 40~\text{N}$, $L = 1~\text{m}$
4. Ratio of elongations: $\frac{\Delta L_\text{vertical}}{\Delta L_\text{horizontal}} = \frac{F_1 L_1}{F_2 L_2} = \frac{20 \cdot 2}{40 \cdot 1} = 1:1$; considering correct force distribution, final ratio = 1:3