For a solid, the linear thermal expansion coefficient α measures the fractional increase in length per degree:
α ≡ (∆L/L)/∆T
where ∆L is a change in length, L is the length, and ∆T is the change in temperature.
A similar coefficient may be derived for the volume V , which changes an amount ∆V when we change the temperature by ∆T:
β = (∆V/V)/∆T
Such a coefficient is especially useful for describing the thermal properties of fluids. (Note: You may take the changes to be small. Indeed, we’ll eventually be interested in infinitesimal changes so that things like ∆L turn into derivatives.)
(a) Derive the relationship between α and β for an isotropic solid (i.e., one for which α does not depend on the direction in the material which we measure it).
(b) For a piece of concrete, the linear thermal expansion coefficient is about α = 1 × 10−5 K−1. Imagine now a concrete bridge that is 1 kilometer long. What is the variation in length ∆L for this concrete bridge between a freezing cold temperature of 32◦F and a hot, summer temperature of 100◦F? Is thermal expansion a relevant consideration for bridge design?