Two cubes with
sides of length 121.0 mm fit in a space that is 243.3 mm
wide, as shown in the figure. One cube is made of aluminum,
and the other is made of brass. What temperature increase is
required for the cubes to completely fill the gap? Give answer
in kelvin, do not enter unit.
To find the temperature increase required for the cubes to completely fill the gap, we need to consider the thermal expansion of the aluminum and brass cubes.
The thermal expansion of a material can be calculated using the formula:
ΔL = α * L0 * ΔT
where:
ΔL is the change in length,
α is the coefficient of linear expansion,
L0 is the initial length, and
ΔT is the change in temperature.
We know that the cubes have sides of length 121.0 mm and that the width of the gap is 243.3 mm. Since the cubes need to completely fill the gap, the change in length (ΔL) for both cubes combined should be equal to the width of the gap.
Let's first calculate the change in length for the aluminum cube:
ΔL_aluminum = α_aluminum * L0_aluminum * ΔT
We can rearrange the formula to solve for ΔT:
ΔT = ΔL_aluminum / (α_aluminum * L0_aluminum)
Similarly, we can calculate the change in length for the brass cube:
ΔL_brass = α_brass * L0_brass * ΔT
Again, rearranging the formula to solve for ΔT:
ΔT = ΔL_brass / (α_brass * L0_brass)
Since ΔT should be the same for both cubes, we can set the two equations equal to each other and solve for ΔT:
ΔL_aluminum / (α_aluminum * L0_aluminum) = ΔL_brass / (α_brass * L0_brass)
Now, we can substitute the given values:
ΔL_aluminum = 243.3 mm
L0_aluminum = 121.0 mm
ΔL_brass = 243.3 mm
L0_brass = 121.0 mm
This gives us:
243.3 / (α_aluminum * 121.0) = 243.3 / (α_brass * 121.0)
Simplifying the equation:
1 / (α_aluminum * 121.0) = 1 / (α_brass * 121.0)
Since the length of the cube is the same for both materials, L0_aluminum = L0_brass = 121.0 mm, we can cancel it out from both sides of the equation:
1 / α_aluminum = 1 / α_brass
This means that the coefficients of linear expansion for aluminum and brass must be equal for the cubes to completely fill the gap. Therefore, the temperature increase required for the cubes to completely fill the gap is 0 Kelvin.