In the (pre-metric) ’60s, a mining company in Venezuela laid a flat stretch of straight railway track from its main entrance. The track was 5000 feet long.
The consulting engineer calculated that in the hottest weather the track would expand only two feet. Since the relatively small increase would be spread over the whole track, he proposed to the company (as a cost saving measure) that the track be fixed at both ends, and that a little slack be provided in the fixing of the rails to allow for the expansion.
Now, assuming that the track bends symmetrically, how high do you think the bulge rises above the ground in the middle? Half an inch? An inch? Two feet?
And why was the engineer soon unemployed?
Solution
This is one of those not infrequent situations where it is dangerous to trust our intuition. And while it is hard to determine exactly what will happen, a simple approximation will give us a good idea.
Let us assume that the expanded track is approximated by the two sloping sides of the triangle shown in this diagram. This is obviously unrealistic, but it will be easy to determine x.
We calculate x by applying Pythagoras theorem to the right angled triangle on the left. This gives
x2 = (2501)2 (2500)2 = 5001,
using the difference of two squares:
(2501)2 (2500)2 = (2501 + 2500)(2501 – 2500).
But now the value of x is approximately 70 feet!
Extensions
1. The amount of lift caused by a small expansion is quite surprising. Investigate the effect of an expansion of 1 cm on a straight stretch of model railway 3 metres long.
2. A similar problem concerns a metal hoop placed around the earth’s equator. Suppose that the hoop expands by 1 metre. If it lifts evenly off the surface of the earth all the way round, would you be able to crawl under it? walk under it? A curious aspect of this problem is that you do not even need to know the length of the hoop!
35. THE 
LINE 
