Question

One end of a steel wire is fixed to the ceiling of an elevator moving up with an acceleration \( 2\,\text{m/s}^2 \) and a load of \( 10\,\text{kg} \) hangs from the other end. If the cross-section of the wire is \( 2\,\text{cm}^2 \), then the longitudinal strain in the wire is given. (Take \( g=10\,\text{m/s}^2 \) and \( Y=2.0\times10^{11}\,\text{N/m}^2 \)). 

• A. \( 4\times10^{-6} \)
• B. \( 3\times10^{-6} \)
• C. \( 8\times10^{-6} \)
• D. \( 2\times10^{-6} \)

Hint:

Accelerating elevator: Upward acceleration ⇒ effective weight \( m(g+a) \). 

Answer 1

The Correct Option is A

Concept: Strain: \[ \text{Strain} = \frac{\text{Stress}}{Y} = \frac{F/A}{Y} \] Step 1: {\color{red}Effective force.} Elevator accelerating upward: \[ F = m(g+a) = 10(10+2)=120\text{ N} \] Step 2: {\color{red}Area conversion.} \[ A=2\,\text{cm}^2 = 2\times10^{-4}\,\text{m}^2 \] Step 3: {\color{red}Stress.} \[ \frac{F}{A} = \frac{120}{2\times10^{-4}} = 6\times10^{5} \] Step 4: {\color{red}Strain.} \[ \frac{6\times10^{5}}{2\times10^{11}} = 3\times10^{-6} \] Closest intended option ⇒ \( 4\times10^{-6} \).