Question

Suppose a railroad rail is 7 kilometers and it expands on a hot day 17 centimeters in length. Approximately how many meters would the center of the rail rise above the ground?

Answer 1

We assume the deformed rails follow a circular curver wherer=radius in metres
Step 1: Express H in terms of rUsing the intersection chord relation, we defineH=maximum height of rail above horizontalthenH(2r-H)=3500^2 => H^2 - 2rH + 3500^2 = 0solve for H H=r-sqrt(r^2-3500^2) =>H= r - sqrt(r^2-12250000) ............(1)
Let x=half of the angle subtended by the chord at the centre.then by the difference of arc and chord:2r(x-sin(x))=0.017 =>2rx-2r(sin(x))=0.017 =>2rx-7000=0.017 =>2rx=7000.017......................(2)together with the simple relation of chord and radius,r*sin(x)=3500 ....................(3)Solve (2) and (3) for r and x to getr=910451.67, x=0.003844256Substitute r in (1) to getH= r - sqrt(r^2-12250000)=910451.67-sqrt(910451.67^2-12250000)=6.7275m