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.

The length of the wire IF UNSTRETCHED decreases by an amount

delta L = 90 cm*alpha*100C,
where alpha is the coefficient of thermal expansion of steel. Look that up.

But since the wire does not stretch, it is under tension T that can be calculated using Young's modulus of steel, E. Look that one up, too.

Stress = strain * E
= (deltaL/L) *E
= E*alpha*deltaT

(L cancels out)

The tension force is the stress times the wire cross-sectional area.

To find the tension force in the wire when the oven cools to a temperature of 150 °C, we can use the coefficient of linear expansion for steel and the equation for thermal expansion.

First, let's calculate the change in length of the wire due to the change in temperature. The formula for thermal expansion is given by:

ΔL = α * L * ΔT

Where:
ΔL is the change in length
α is the coefficient of linear expansion
L is the original length
ΔT is the change in temperature

The coefficient of linear expansion for steel is approximately 12 x 10^(-6) per °C.

Given:
L = 90 cm
ΔT = 250 °C - 150 °C = 100 °C
α = 12 x 10^(-6) per °C

Now we can calculate the change in length:

ΔL = (12 x 10^(-6) per °C) * (90 cm) * (100 °C)
ΔL = 0.00108 cm

The change in length is 0.00108 cm. However, this value is so small that it can be considered negligible for the purpose of this calculation.

Since the distance between the oven walls does not change, the wire will remain taut even with the temperature difference. This means that the tension force in the wire remains the same.

Therefore, the tension force in the wire when the oven cools to a temperature of 150 °C is the same as when the temperature was 250 °C, with no tension.