Final answer:
The student's question about finding the ratio of the length of an aluminium rod to the combined length of an aluminium and steel rod, both of which expand the same amount upon temperature change, can be answered using the formula for linear thermal expansion. By equating the expansions and cancelling out the common factors, the ratio is found to be the coefficient of linear expansion of steel divided by the sum of the coefficients of linear expansion for aluminium and steel.
Step-by-step explanation:
Given that two rods, one of aluminium and one of steel, are joined end to end and have the same expansion upon a change in temperature, we can find the ratio of the length of the aluminium rod to the total length of both rods combined by using the formula for linear thermal expansion ΔL = αLΔT, where ΔL is the change in length, L is the initial length, α is the coefficient of linear expansion, and ΔT is the change in temperature.
Since the rods are joined end to end and the expansion in both rods is the same, we can equate the expansion expressions for both materials: αaL1ΔT = αsL2ΔT. We can simplify this equation since the change in temperature ΔT is the same for both rods and therefore cancels out, yielding αaL1 = αsL2. To find the desired ratio, we solve for L1/(L1 + L2):
This can be expressed as L1/(L1 + L2) = αs / (αa + αs) after rearranging the terms and doing the algebra.