The brass bar and the aluminum bar in the drawing are each attached to an immovable wall. At 29°C the air gap between the rods is 2 x 10^-3 m. At what temperature will the gap be closed?
The length of the brass rod is 2.0m
The lenght of the aluminum rod is 1.0m
The coefficient of linear expansion of Brass = 19 x 10^-6
The coefficient of linear expansion of Aluminum = 23 x 10^-6.
I am unsure where to go from here.
gap= deltaL for alum + delta L for Brass
solve for delta T, it is the same in both linear expansions.
Ao do we have to add the specific heat capacities ?
Ao do we have to add the specific heat capacities ?
2E-3= (19E-6)(2)(DeltaT) + (23E-6)(1)(DeltaT)
Simplified equation:
.002= (3.8E-5) (DeltaT) + (2.3E-5) (DeltaT)
More simpllified:
.002= (6.1E-5) (DeltaT)
DeltaT= 32.7869
Answer:
32.7869 + 29 = 61.7869
No, you don't need to consider the specific heat capacities for this problem. The question is asking at what temperature the gap between the two bars will be closed, so we need to find the temperature change required for the total change in length to be equal to the initial gap.
Let's start by finding the change in length for each material:
For the brass bar:
ΔL_brass = α_brass * L_brass * ΔT
ΔL_brass = (19 x 10^-6) * (2.0 m) * ΔT
For the aluminum bar:
ΔL_aluminum = α_aluminum * L_aluminum * ΔT
ΔL_aluminum = (23 x 10^-6) * (1.0 m) * ΔT
Since the temperature change is the same for both materials, we can set these two equations equal to each other:
(19 x 10^-6) * (2.0 m) * ΔT = (23 x 10^-6) * (1.0 m) * ΔT
Now, solve for ΔT by canceling out ΔT on both sides and simplify the equation:
(19 x 10^-6) * (2.0 m) = (23 x 10^-6) * (1.0 m)
38 x 10^-6 = 23 x 10^-6
Now, divide both sides of the equation by (23 x 10^-6) to solve for ΔT:
ΔT = (38 x 10^-6) / (23 x 10^-6)
ΔT ≈ 1.65
Therefore, the temperature change required for the gap between the bars to be closed is approximately 1.65°C.
No, you do not need to consider the specific heat capacities for this problem. The specific heat capacity is a measure of how much heat energy is required to raise the temperature of a substance, and it is not relevant to calculating the closing of the gap between the two bars.
To solve this problem, you need to use the formula for the linear expansion of a solid:
ΔL = α * L * ΔT
where ΔL is the change in length of the bar, α is the coefficient of linear expansion, L is the original length of the bar, and ΔT is the change in temperature.
In this case, we need to find the temperature at which the gap between the two bars will be closed. This means that the change in length for both bars will be equal to the initial gap size, which is 2 x 10^-3 m.
Let's first calculate the change in length for the brass bar:
ΔL_brass = α_brass * L_brass * ΔT
ΔL_brass = (19 x 10^-6) * (2.0 m) * ΔT
Next, let's calculate the change in length for the aluminum bar:
ΔL_aluminum = α_aluminum * L_aluminum * ΔT
ΔL_aluminum = (23 x 10^-6) * (1.0 m) * ΔT
Now we can set the two changes in length equal to each other:
ΔL_brass = ΔL_aluminum
(19 x 10^-6) * (2.0 m) * ΔT = (23 x 10^-6) * (1.0 m) * ΔT
Next, we can cancel out the ΔT and solve for the temperature at which the gap will be closed:
(19 x 10^-6) * (2.0 m) = (23 x 10^-6) * (1.0 m)
38 x 10^-6 = 23 x 10^-6
Simplifying, we find:
38 = 23
Since this equation is not true, it means that there is no temperature at which the gap between the two bars will be closed.