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What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under a tension of 940 N? Take Young's modulus of steel as 20 × 1010 Pa
  • a)
    3.2 mm
  • b)
    3.0 mm
  • c)
    3.4 mm
  • d)
    3.6 mm
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
What diameter should a 10-m-long steel wire have if we do not want it ...
Y=F x l/A x Δ l
Δ l=0.5cm=0.5x10-2m, l=10M, F=940N
Y=20x1010pa
20x1010=940x10/πr2x0.5x10-10
πr2=94x100/5x10-3x2x1011=94x102/10x108
r2=94/π x 10-7 =2.99 x 10-6
r2 ≅3x10-6
r=1.13x10-10 m
diameter=2r=3.6mm
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What diameter should a 10-m-long steel wire have if we do not want it ...
Given:
- Length of steel wire = 10 m
- Tension applied = 940 N
- Maximum elongation allowed = 0.5 cm = 0.005 m
- Young's modulus of steel = 20 x 10^10 Pa

To find: Diameter of the steel wire

Formula used:

- Tension (T) = (π/4) x (d^2) x σ
- Elongation (ε) = (TL) / (A x E)

where,
- T = Tension applied
- d = Diameter of the wire
- σ = Stress
- ε = Elongation
- L = Length of the wire
- A = Cross-sectional area of the wire
- E = Young's modulus of the material

Solution:

1. Finding the stress applied on the wire:

From the formula, Tension (T) = (π/4) x (d^2) x σ, we can rearrange to find the stress (σ) as:

σ = (4 x T) / (π x d^2)

Substituting the given values, we get:

σ = (4 x 940) / (π x d^2) = 298.77 / d^2

2. Finding the cross-sectional area of the wire:

We know that the cross-sectional area (A) of a wire is given by:

A = (π/4) x d^2

3. Finding the elongation of the wire:

From the formula, Elongation (ε) = (TL) / (A x E), we can rearrange to find the diameter (d) as:

d = sqrt((4 x T x L) / (π x ε x E))

Substituting the given values, we get:

d = sqrt((4 x 940 x 10) / (π x 0.005 x 20 x 10^10)) = 0.0034 m = 3.4 mm

Therefore, the diameter of the steel wire should be 3.4 mm to ensure that it does not stretch more than 0.5 cm under a tension of 940 N.
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What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under a tension of 940 N? Take Young's modulus of steel as 20 × 1010 Paa)3.2 mmb)3.0 mmc)3.4 mmd)3.6 mmCorrect answer is option 'C'. Can you explain this answer?
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