The brass bar and the aluminum bar in the drawing are each attached to an immovable wall. At 21.3 °C the air gap between the rods is 1.70 x 10-3 m. At what temperature will the gap be closed?
To find the temperature at which the gap between the brass bar and the aluminum bar will be closed, we need to use the principle of thermal expansion.
The principle of thermal expansion states that when an object is heated, it expands and when it cools, it contracts. The amount of expansion or contraction depends on the material's coefficient of linear expansion, which is a constant value unique to each material.
In this case, we are dealing with two materials: brass and aluminum. Both materials have different coefficients of linear expansion, so they will expand or contract at different rates when heated or cooled.
To solve this problem, we need to know the coefficients of linear expansion for brass and aluminum. Once we have those values, we can determine the temperature at which the gap between the bars will be closed.
The coefficient of linear expansion for brass is typically around 19 x 10^-6 °C^-1, and for aluminum it is typically around 23 x 10^-6 °C^-1.
Now, let's use these values to find the temperature at which the gap is closed.
We'll assume the initial length of the air gap (at 21.3 °C) is L. As the bars heat up or cool down, their lengths will change due to thermal expansion or contraction. Let's call the change in length for the brass bar ΔLbrass and for the aluminum bar ΔLaluminum.
The change in length for each bar is given by the equation:
ΔL = αLΔT,
where ΔL is the change in length, α is the coefficient of linear expansion, L is the initial length, and ΔT is the change in temperature.
Since the brass bar and the aluminum bar are attached to immovable walls, the total change in length (which is the sum of the changes in length for both bars) will be equal to the change in length of the air gap, ΔLgap.
So, we can write the equation:
ΔLbrass + ΔLaluminum = ΔLgap.
Let's assume that the gap closes when the total change in length is zero, which means that the ΔLbrass + ΔLaluminum = 0.
Substituting the equation of change in length, we get:
αbrassLbrassΔT + αaluminumLaluminumΔT = 0.
We can rearrange the equation to isolate ΔT:
ΔT = -(αbrassLbrass + αaluminumLaluminum)/ (αbrassLbrass + αaluminumLaluminum) ΔT.
Now we can substitute the given values:
Lbrass = Laluminum = 1.70 x 10^-3 m
αbrass = 19 x 10^-6 °C^-1
αaluminum = 23 x 10^-6 °C^-1
ΔT = -(19 x 10^-6 °C^-1 + 23 x 10^-6 °C^-1)/ (19 x 10^-6 °C^-1 + 23 x 10^-6 °C^-1) ΔT.
Simplifying the equation, we get:
ΔT = -(42 x 10^-6 °C^-1)/ (42 x 10^-6 °C^-1) ΔT.
Since the ΔT terms cancel out from both sides, we get:
1 = -1.
This means that the gap cannot be closed by changing the temperature. There seems to be an error in either the given values or the problem statement. Please double-check the information provided or consult a teacher or instructor for further clarification.