As the drawing shows, two thin strips of metal are bolted together at one end and have the same temperature. One is steel, and the other is aluminum. The steel strip is 0.062% longer than the aluminum strip. By how much should the temperature of the strips be increased, so that the strips have the same length?
Let's denote the original length of the steel strip at temperature T as L_s and the original length of the aluminum strip as L_a. From the problem, we know that:
L_s = 1.00062 * L_a
When the temperature of both strips is increased by ΔT, their lengths will change according to their linear expansion coefficients. The coefficient of linear expansion for steel (α_s) is approximately 12 x 10^-6 K^-1, and for aluminum (α_a), it's about 24 x 10^-6 K^-1.
So, the change in length for each strip would be:
ΔL_s = α_s * L_s * ΔT
ΔL_a = α_a * L_a * ΔT
The final lengths of the strips after the temperature increase would be:
L_s' = L_s + ΔL_s
L_a' = L_a + ΔL_a
We are looking for a temperature increase, ΔT, that makes L_s' equal to L_a', so:
L_s + ΔL_s = L_a + ΔL_a
Substitute the expressions of ΔL_s and ΔL_a:
L_s + α_s * L_s * ΔT = L_a + α_a * L_a * ΔT
Now substitute the original relationship between L_s and L_a:
1.00062 * L_a + α_s * 1.00062 * L_a * ΔT = L_a + α_a * L_a * ΔT
Now we can factor out L_a and rearrange the terms:
1.00062 + α_s * 1.00062 * ΔT = 1 + α_a * ΔT
Divide both sides by L_a:
1.00062 = 1 + (α_a - α_s * 1.00062) * ΔT
Now solve for ΔT:
ΔT = (1.00062 - 1) / (α_a - α_s * 1.00062)
ΔT = (0.00062) / (24 x 10^-6 - 12 x 10^-6 * 1.00062)
ΔT ≈ (0.00062) / (12 x 10^-6)
ΔT ≈ 51.67 K
So, the temperature of the strips should be increased by about 51.67 K for the strips to have the same length.
To find the temperature increase needed for the steel and aluminum strips to have the same length, we can use the concept of thermal expansion.
The thermal expansion coefficient is a measure of how much a material expands or contracts with changes in temperature. Let's denote the thermal expansion coefficient of steel as α_s and the thermal expansion coefficient of aluminum as α_a.
Given that the steel strip is 0.062% longer than the aluminum strip, we can write the following equation:
α_s * L_s - α_a * L_a = 0.062% * L_a
Here, L_s represents the initial length of the steel strip, and L_a represents the initial length of the aluminum strip.
Since both strips have the same initial temperature, we can assume that the change in temperature is the same for both materials. Let's denote this change in temperature as ΔT.
The change in length for each strip can be calculated as:
ΔL_s = α_s * L_s * ΔT
ΔL_a = α_a * L_a * ΔT
By setting the two change in lengths equal to each other, we can solve for ΔT:
α_s * L_s * ΔT = α_a * L_a * ΔT
α_s * L_s = α_a * L_a
(α_s - α_a) * L_s = α_a * L_a
ΔT = (α_a * L_a) / (α_s - α_a)
Thus, to make the steel and aluminum strips have the same length, the temperature needs to be increased by ΔT = (α_a * L_a) / (α_s - α_a).
To find the temperature increase needed for the steel and aluminum strips to have the same length, we need to use the concept of thermal expansion. The change in length of a material due to temperature change can be calculated using the linear expansion coefficient.
The linear expansion coefficient (α) is a measure of how much a material expands or contracts per unit length for a given change in temperature.
Given that the steel strip is 0.062% longer than the aluminum strip, we can assume that the steel strip has expanded by 0.062% more than the aluminum strip. Let's call the change in length for the aluminum strip ΔL_aluminum and the change in length for the steel strip ΔL_steel.
We have the equation: ΔL_steel = ΔL_aluminum + 0.062% of ΔL_aluminum
To simplify the calculation, let's assume the initial length of both strips is L, and the change in length due to temperature change is ΔL.
Thus, ΔL_steel = L * α_steel * ΔT (1)
ΔL_aluminum = L * α_aluminum * ΔT (2)
Where α_steel and α_aluminum are the linear expansion coefficients for steel and aluminum, respectively, and ΔT is the temperature increase.
Given that the temperature increase needed for the strips to have the same length is ΔT, we can rearrange equations (1) and (2):
ΔL_steel = ΔL_aluminum + 0.062% * ΔL_aluminum
L * α_steel * ΔT = L * α_aluminum * ΔT + 0.062% * (L * α_aluminum * ΔT)
α_steel * ΔT = α_aluminum * ΔT + 0.062% * α_aluminum * ΔT
Simplifying further:
α_steel * ΔT - α_aluminum * ΔT = 0.062% * α_aluminum * ΔT
(α_steel - α_aluminum) * ΔT = 0.062% * α_aluminum * ΔT
Canceling ΔT from both sides:
α_steel - α_aluminum = 0.062% * α_aluminum
Now we can solve for ΔT to find the temperature increase needed for the strips to have the same length:
ΔT = (0.062% * α_aluminum) / (α_steel - α_aluminum)
To calculate the exact value, we need the specific linear expansion coefficients for steel and aluminum. These coefficients can be found in reference materials or online databases for material properties.