A steel rail `30m` long is firmly attached to the road be only at its ends. The sun raises the temperature of the rail by `50^(0)C`, causing the rail to buckle. Assuming that the buckled rail consitst of two straight parts metting in the centre, calculate how much the centre of the rail rise? Given, `alpha_(steel) = 12 xx 10^(-6//@)C.

Explanation

Let the initial length be `2x` and the final total length be `2(x + Delta x)` as shown.
Let `y` be the height of the centre of the buckled rail Clearly, `Deltax = alphax DeltaT` and
`y = sqrt((x + Deltax)^(2) - x^(2)) = sqrt(2x(Deltax)) = sqrt(2x^(2)alphaDeltat)`
`y = xsqrt(2alphaDeltaT) ["neglecting" (Deltax)^(2)]`
Thus, `y = [15 sqrt(2(12 xx 10^(-6)))50]cm = 0.52m`