Stress in a steel rod at -20 degrees Celsius

Given data:
- Cross-sectional area of the steel rod, A = 150 mm²
- Tensile load at 20 degrees Celsius, F = 5000 N
- Coefficient of linear expansion, α = 11.7 * 10^(-6) per degree Celsius
- Young's modulus, Y = 200 * 10^11 N/m²

To calculate the stress in the steel rod at -20 degrees Celsius, we need to consider the change in dimensions due to temperature variation and apply Hooke's Law.

1. Calculate the change in length due to temperature variation:
- The change in temperature, ΔT = -20 - 20 = -40 degrees Celsius
- The original length of the rod, L0, can be determined using the formula:
L0 = F * L / A * Y, where L is the original length of the rod.
- The change in length, ΔL = α * L0 * ΔT

2. Calculate the new length of the rod at -20 degrees Celsius:
- The new length, L' = L0 + ΔL

3. Calculate the stress in the rod at -20 degrees Celsius:
- Stress, σ = F / A

Let's calculate each step in detail:

1. Calculate the change in length due to temperature variation:
- Given that α = 11.7 * 10^(-6) per degree Celsius and ΔT = -40 degrees Celsius.
- The original length of the rod, L0, can be determined using the formula:
L0 = F * L / A * Y
Rearranging the formula, L = L0 * A * Y / F
Substituting the values, we get:
L = (5000 * L) / (150 * 200 * 10^11)
Solving for L, we find L ≈ 1.67 m
- The change in length, ΔL = α * L0 * ΔT
Substituting the values, ΔL = (11.7 * 10^(-6)) * (1.67) * (-40)
Solving for ΔL, we find ΔL ≈ -9.34 * 10^(-6) m

2. Calculate the new length of the rod at -20 degrees Celsius:
- The new length, L' = L0 + ΔL
Substituting the values, L' = 1.67 + (-9.34 * 10^(-6))
Solving for L', we find L' ≈ 1.66999066 m

3. Calculate the stress in the rod at -20 degrees Celsius:
- Given that A = 150 mm² and F = 5000 N.
- Stress, σ = F / A
Substituting the values, σ = 5000 / (150 * 10^(-6))
Solving for σ, we find σ ≈ 33.33 * 10^6 N/m²

Therefore, the stress in the steel rod at -20 degrees Celsius is approximately 33.33 * 10^6 N/m².