A 90 cm length of steel wire with a diameter of 0.5 mm is stretched between the inside walls of an oven the wire is just taut with no tension when the temperature in the oven is 250 °C. What is the the tension force in the wire when the oven cools to a temperature of 150 °C? The distance between the oven walls does not change as the oven cools.
To calculate the tension force in the wire, we need to consider the thermal expansion of the wire material. The change in temperature causes the wire to expand or contract, which in turn affects the tension in the wire.
To solve this question, we need to determine the change in length of the wire due to the temperature difference between 250 °C and 150 °C. We can then use this change in length to calculate the tension force in the wire.
The formula that relates the change in length of a wire to the temperature change is given by:
ΔL = α * L * ΔT
Where:
ΔL is the change in length of the wire,
α is the coefficient of linear expansion for the material (steel in this case),
L is the original length of the wire, and
ΔT is the temperature difference.
The coefficient of linear expansion for steel is approximately 12 × 10^-6 / °C.
First, let's calculate the change in length of the wire:
ΔT = 250 °C - 150 °C = 100 °C
L = 90 cm = 0.9 m
α = 12 × 10^-6 / °C
ΔL = (12 × 10^-6 / °C) * (0.9 m) * (100 °C) = 0.0108 m
Now, we can calculate the tension force in the wire. To do this, we need to consider the change in length as well as the initial length of the wire and its cross-sectional area.
The formula for tension force in a wire is given by:
T = (E * A * ΔL) / L
Where:
T is the tension force,
E is the Young's modulus of the material (steel in this case),
A is the cross-sectional area of the wire, and
L is the original length of the wire.
The Young's modulus for steel is approximately 200 GPa (200 × 10^9 Pa).
To calculate the cross-sectional area of the wire, we can use the formula for the area of a circle:
A = π * r^2
Where:
A is the cross-sectional area,
π is approximately 3.14, and
r is the radius of the wire.
Given that the diameter of the wire is 0.5 mm, the radius would be half of that:
r = 0.5 mm / 2 = 0.25 mm = 0.00025 m
Now, we can calculate the cross-sectional area:
A = 3.14 * (0.00025 m)^2 = 1.9635 × 10^-7 m^2
Finally, we can calculate the tension force in the wire:
T = (200 × 10^9 Pa) * (1.9635 × 10^-7 m^2) * (0.0108 m) / (0.9 m)
T ≈ 237,333 N
Therefore, the tension force in the wire, when the oven cools to a temperature of 150 °C, is approximately 237,333 N.