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?