Suppose a bimetallic strip is constructed of copper and steel strips of thickness 1.33 mm and length 23.5 mm, and the temperature of the strip is reduced by 4.50 K. If the strip is 23.5 mm long, how far is the maximum deviation of the strip from the straight orientation?
To calculate the maximum deviation of the bimetallic strip, we can use the formula for thermal expansion:
ΔL = α * L * ΔT
where:
ΔL is the change in length of the strip,
α is the coefficient of linear expansion,
L is the original length of the strip, and
ΔT is the change in temperature.
In this case, we have a copper and steel strip, so we need to calculate the individual contributions of each material and then add them together.
First, let's calculate the change in length for the copper strip:
ΔL_copper = α_copper * L * ΔT
The coefficient of linear expansion for copper is approximately 16.6 × 10^(-6) K^(-1).
ΔL_copper = (16.6 × 10^(-6) K^(-1)) * (23.5 mm) * (4.50 K)
Next, let's calculate the change in length for the steel strip:
ΔL_steel = α_steel * L * ΔT
The coefficient of linear expansion for steel is approximately 12 × 10^(-6) K^(-1).
ΔL_steel = (12 × 10^(-6) K^(-1)) * (23.5 mm) * (4.50 K)
Finally, we can sum up the individual contributions to get the total change in length of the bimetallic strip:
ΔL_total = ΔL_copper + ΔL_steel
Now, the maximum deviation from the straight orientation occurs at the end of the strip, so the maximum deviation is equal to half of the total change in length:
maximum deviation = 0.5 * ΔL_total
Substituting the values, we can now calculate the maximum deviation:
maximum deviation = 0.5 * (ΔL_copper + ΔL_steel)
To determine the maximum deviation of the bimetallic strip from its straight orientation, we need to consider the difference in thermal expansion between the copper and steel strips. The bimetallic strip bends due to the varying expansion and contraction of the two different metals.
To calculate the maximum deviation, we can use the formula:
δ = (T * L^2) / (2 * D * d)
Where:
δ = maximum deviation
T = change in temperature
L = length of the strip
D = difference in the coefficients of linear expansion between the two metals
d = thickness of the strip
Let's calculate it step by step:
1. Convert the length of the strip to meters:
Length (L) = 23.5 mm = 23.5 / 1000 meters = 0.0235 meters
2. Convert the thickness of the strip to meters:
Thickness (d) = 1.33 mm = 1.33 / 1000 meters = 0.00133 meters
3. Determine the difference in coefficients of linear expansion between copper and steel. Let's assume that the coefficient of linear expansion for copper is α_copper = 1.7 × 10^-5 K^-1 and for steel is α_steel = 1.2 × 10^-5 K^-1.
D = α_copper - α_steel
D = (1.7 × 10^-5 K^-1) - (1.2 × 10^-5 K^-1)
D = 0.5 × 10^-5 K^-1
D = 5 × 10^-6 K^-1
4. Substitute the values into the formula:
δ = (T * L^2) / (2 * D * d)
δ = (4.50 K * (0.0235 meters)^2) / (2 * (5 × 10^-6 K^-1) * 0.00133 meters)
δ = (4.50 * 0.0235^2) / (2 * 5 × 10^-6 * 0.00133)
δ ≈ 0.233 meters (rounded to three decimal places)
Therefore, the maximum deviation of the bimetallic strip from the straight orientation is approximately 0.233 meters.