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.
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To find the tension force in the wire when the oven cools to a temperature of 150 °C, we need to use the concept of thermal expansion.
When the wire is heated to 250 °C, it expands due to thermal expansion, which causes an increase in its length. Similarly, when the wire cools to 150 °C, it contracts, leading to a decrease in its length.
To start, let's find the initial length of the wire at 250 °C. We know that the length of the wire is 90 cm at this temperature.
Next, we need to find the final length of the wire at 150 °C with no tension. Since the wire is just taut with no tension when the temperature is 250 °C, it means that at 150 °C, the wire will be shorter than its initial length.
To calculate the final length, we can use the formula for linear thermal expansion:
Δ𝐿 = 𝐿₀ * 𝛼 * Δ𝑇
where:
Δ𝐿 is the change in length,
𝐿₀ is the initial length,
𝛼 is the coefficient of linear expansion of steel, and
Δ𝑇 is the change in temperature.
The coefficient of linear expansion of steel is approximately 12 × 10^(-6) per degree Celsius (12 × 10^(-6) °C^(-1)).
Using the formula, we can calculate the change in length for the wire:
Δ𝐿 = 90 cm * (12 × 10^(-6) °C^(-1)) * (250 °C - 150 °C)
Simplifying the equation gives us:
Δ𝐿 ≈ 90 cm * (12 × 10^(-6) °C^(-1)) * 100 °C
Δ𝐿 ≈ 0.108 cm
Therefore, the final length of the wire at 150 °C is approximately:
𝐿_f = 𝐿₀ - Δ𝐿
𝐿_f ≈ 90 cm - 0.108 cm
𝐿_f ≈ 89.892 cm
Since the distance between the oven walls does not change as the oven cools, the tension force in the wire can be calculated using Hooke's law:
𝐹 = 𝑘 * 𝛥𝐿
where:
𝐹 is the tension force,
𝑘 is the spring constant, and
𝛥𝐿 is the change in length.
Assuming the wire behaves linearly with a constant spring constant, we can rearrange the equation as follows:
𝐹 = 𝑘 * (𝐿_f - 𝐿₀)
To find the spring constant, we need more information about the wire's properties. Without this information, we cannot determine the exact tension force in the wire when the oven cools to 150 °C.
However, if the spring constant is provided, you can substitute the values into the equation to find the tension force.