Question

A brick is initially 15.11 cm high and an aluminum can is initially 15.12 cm high. By how much must their temperatures be raised in order for the brick and the can to have exactly the same height?

Answer 1

Coefficients α of linear expansion for common materials are often given in terms of the relative change in length in ppm (parts per million) for each degree Celsius change in temperature. This may be expressed in the form

ΔL/L = α.ΔT

where the value for α includes the factor 10^-6 to correct the ppm figure so that ΔL and L may be expressed in the same units. ΔT is the total change in temperature for both materials.

Typical values for α for these two materials are 5.5 for brick and 23 for aluminium. That for brick is more variable, since the composition and manufacturing methods for this material can vary widely. However, we note immediately that this gives rise to a problem with the question as presented. The initial height of the aluminium can is larger than that of the brick - by 0.01 cm - and, as the temperature is raised, it will increase more rapidly than the brick as a result of its greater coefficient of expansion. So the two objects cannot be brought to the same vertical height by heating. That result may however be achieved by cooling, and we may alternatively calculate the temperature decrease ΔT required for that.

Using the equation above for aluminium (Al) and brick (Br)

ΔL(Al)/L(Al) - ΔL(Br)/L(Br) = [α(Al) - α(Br)].ΔT

To a good approximation L(Al) = L(Br) = 15.10cm, so this simplifies to

ΔL(Al) - ΔL(Br) = -0.01 = 15.10 x [(23 - 5.5) x 10^-6] x ΔT

which gives ΔT = - 37.8°C

That is, you would need to cool the aluminium can and the brick by 37.8°C to get them to the same vertical size. Assuming that the initial temperature was 20°C, this would involve taking them both down to -17.8°C - not an easy task