Example: Expansion

Question:

Suppose the railroad used a single piece of steel from Lansing to Chicago (400 km.) How much would it expand from the coldest winter day (-40^$\circ$F) to the warmest summer day (110^$\circ$ F)?

Answer:

Because it is the change in temperature that counts, we only need to convert $\Delta$T = 150 $\cdot$ (5/9) = 83.3 ^$\circ$C
$\Delta$L = (12$\cdot$ 10^-6 m/^$\circ$C)(400,000 m)(83.3 ^$\circ$C) = 400 m

This is why using a single piece of steel for this project would be a very bad idea. Instead the railroad tracks consist of many individual pieces with expansion joints between them.

Question:

How much force does it take to keep a 5 m piece of track 25 cm² in cross section from expanding? Here is a case in which we are given the strain and asked to find the stress.

Answer:

strain = $\Delta$L/L = $\alpha\Delta$T = (12$\cdot$ 10^-6 ^$\circ$C^-1)(83.3^$\circ$C) = 1.00$\cdot$ 10^-3
stress = F/A = strain $\cdot$ elastic modulus

F = strain $\cdot$ Y $\cdot$ A = (1.00$\cdot$ 10^-3)$\cdot$ (20$\cdot$ 10¹⁰)$\cdot$ (25$\cdot$ 10^-4) = 500,000 N, independent of length!