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 0.10% longer than the aluminum strip. By how much should the temperature of the strips be increased, so that the strips have the same length?
To find the change in temperature needed for the strips to have the same length, we can use the concept of thermal expansion.
The formula for linear expansion is given by:
ΔL = α * L0 * ΔT
where:
ΔL is the change in length of the strip,
α is the coefficient of linear expansion for the material,
L0 is the original length of the strip, and
ΔT is the change in temperature.
Let's denote the change in length for the steel strip as ΔLs and for the aluminum strip as ΔLa.
We are given that the steel strip is 0.10% longer than the aluminum strip. This can be written as:
ΔLs = 0.10% * La
We want to find the change in temperature ΔT, so that the strips have the same length. This means:
ΔLs + La = 0
Substituting the value of ΔLs from the earlier equation:
0.10% * La + La = 0
Combining like terms:
1.10% * La = 0
Dividing both sides by 1.10%:
La = 0 / 1.10% = 0
This means that the aluminum strip has to contract by 0 to have the same length as the steel strip. Therefore, no change in temperature is needed for the strips to have the same length.
To find out by how much the temperature of the strips should be increased, we can use the concept of thermal expansion.
Thermal expansion is a property of matter where its size or volume changes with temperature. Different materials have different rates of thermal expansion, so when two materials with different rates of expansion are subjected to the same temperature change, they will not change length by the same amount.
To solve this problem, we need to use the relation between the change in length (ΔL) and the original length (L) of the material. The equation is given by:
ΔL = L * α * ΔT
Where:
ΔL is the change in length
L is the original length
α is the coefficient of linear expansion
ΔT is the change in temperature
In this case, we know that one strip is steel and the other is aluminum. The steel strip is 0.10% longer than the aluminum strip, which means the change in length ratio is 0.10%.
Let's assume the original length of the aluminum strip is L, then the original length of the steel strip will be L + 0.10% of L.
Now let's find out the change in temperature (ΔT) required to make the steel strip the same length as the aluminum strip.
ΔL_aluminum = L * α_aluminum * ΔT
ΔL_steel = (L + 0.10% of L) * α_steel * ΔT
Since we want the two strips to have the same length after the temperature change, we can set the changes in length equal to each other:
L * α_aluminum * ΔT = (L + 0.10% of L) * α_steel * ΔT
We can cancel out ΔT on both sides of the equation:
L * α_aluminum = (L + 0.10% of L) * α_steel
Now we can solve for the change in temperature by rearranging the equation:
α_aluminum / α_steel = (L + 0.10% of L) / L
α_aluminum / α_steel = 1 + 0.10%
This equation gives us the ratio of the coefficients of linear expansion. To find the change in temperature, we just need to multiply this ratio by the change in length, ΔL.
ΔT = (α_aluminum / α_steel) * ΔL
By plugging in the appropriate values for the coefficients of linear expansion and the change in length, you can find the change in temperature required to equalize the lengths of the two strips.