How do i work this out?

Question

A railway line is laid with 20m lengths of rail on a day when the temperature is 15C. Gaps of 2cm are left between the rails. At what temperature will the gaps close ? ( linear expansion coefficient of steel = 1.2 x 10^-5K ^-1)

Answer 1

First, you have to understand the constant.  The constant is 1.2x10^-5/°K, or 0.000012/°K

So, the thing is 20m long, and you want it to be 20.02m long.  That means you're looking for an expansion of 2cm÷20m = 0.02m÷20m = 0.001.  You're also looking for an answer in °K or °C (same thing).  So, you've got this:

0.001 <what operation> 0.000012/°K = ___°K

So, the operation must be ÷

0.001÷0.000012°K ≈ 83°K

So, you get a temperature of 15+83=98°C.  Since we're not boiling the rails, something's not reasonable in the problem description.

Answer 2

For every degree hotter it gets, steel expands by 0.000012 times its original length.  So a 20m. steel rail will expand by 0.00024m. per degree C.

By the time the rail has grown to 20.02m., it will have half-filled the gap in front of it and half-filled the gap behind it; and assuming all the rails are at the same temperature, then this is when the gaps will close up.

Now do you see how to work it out?