Steel expands 11 parts in a million for each 1 ?Cchange. Consider a 40,000-km st

Question

Steel expands 11 parts in a million for each 1 ?Cchange. Consider a 40,000-km steel pipe that forms a ring to fit snugly all around the circumference of the Earth. Suppose people along its length breathe on it so as to raise its temperature 1.5?C. The pipe gets longer. It also is no longer snug.

How high does it stand above ground level? (To simplify, consider only the expansion of its radial distance from the center of Earth, and apply the geometry formula that relates circumference Cand radius r, C = 2?r. The result is surprising!)?

Explanation / Answer

Given Data

temperature, Delta T = 1.5 degree Celsius

C = 40,000 km

We have coefficient of expansion for steel pipe, alpha = 12 x 10-6 /0 C

To Find

Delta r = ?

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it stand above the ground level at a height which is given as ::

geometry formula that relates circumference C and radius r,

circumference of steel pipe, delta C = 2 *Pi* delta r                           { eq. 1 }

or r = C / 2 *Pi

where, C = 40,000 km

r = radius of the pipe

inserting the values in above eq,

r = (40,000 km) / 2 (Pi) = 6366.2 km

delta C = 2 *Pi* delta r = 2 *Pi* r *alpha*Delta T

Due to the thermal expansion increase in radius is given as ::

Delta r = r *alpha*Delta T                                   { eq. 2 }

inserting the values in eq.2

Delta r = (6366.2 km) (12 x 10-6 /o C) (1.5 oC)

Delta r = 0.1146 km = 114.6 m

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(or)

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delta C = 2*pi*delta r = 2*pi*r*alpha*deltaT

The distance the piper rises is delta r.

delta r = r*alpha*deltaT
          = [4*10*7 m/(2pi)]*11*10^-6*1.5 = 105 m

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