Elasticity & Stress-Strain: Exam-Pattern Drill (2)

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Tags Elasticity And Stress Strain

Question

A steel wire of length 22 m and cross-sectional area 11 mm² is subjected to a tensile load of 200200 N. Calculate (a) the stress, (b) the strain, and (c) the elongation of the wire. Take Young’s modulus of steel Y=2×1011Y = 2 \times 10^{11} N/m². JEE Main 2023 pattern.

Solution — Step by Step

Convert area to SI: 11 mm² =1×10−6= 1 \times 10^{-6} m².

σ=FA=2001×10−6=2×108 N/m2\sigma = \frac{F}{A} = \frac{200}{1 \times 10^{-6}} = 2 \times 10^8 \text{ N/m}^2

From the definition of Young’s modulus, Y=σ/εY = \sigma/\varepsilon, so:

ε=σY=2×1082×1011=10−3\varepsilon = \frac{\sigma}{Y} = \frac{2 \times 10^8}{2 \times 10^{11}} = 10^{-3}

ΔL=ε⋅L=10−3×2=2×10−3 m=2 mm\Delta L = \varepsilon \cdot L = 10^{-3} \times 2 = 2 \times 10^{-3} \text{ m} = 2 \text{ mm}

Stress =2×108= 2 \times 10^8 N/m², strain =10−3= 10^{-3}, elongation =2= 2 mm.

Why This Works

Young’s modulus is the proportionality constant between stress and strain in the elastic regime — it tells us how much a material resists stretching. Steel has Y∼200Y \sim 200 GPa, which is very high, so even a 200 N load on a 1 mm² wire produces only a 2 mm stretch over 2 m of wire.

The formula ΔL=FLAY\Delta L = \dfrac{FL}{AY} combines all three steps. It’s worth memorising, because direct-substitution problems like this appear in every JEE Main paper.

Alternative Method

Use the combined formula directly:

ΔL=FLAY=200×210−6×2×1011=4002×105=2×10−3 m\Delta L = \frac{FL}{AY} = \frac{200 \times 2}{10^{-6} \times 2 \times 10^{11}} = \frac{400}{2 \times 10^5} = 2 \times 10^{-3} \text{ m}

Same answer in one step.

A near-universal mistake is forgetting to convert mm² to m². 11 mm² =10−6= 10^{-6} m², not 10−310^{-3}. This single unit error gives an answer 1000 times too small.

Sanity check: strain in everyday materials should usually be tiny (like 10−310^{-3}). If you compute strain >0.1> 0.1, you’ve left the elastic regime — Young’s modulus no longer applies and your answer is wrong.

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