To ensure a tight fit, cylindrical metal elements are sometimes made the same si

Question

To ensure a tight fit, cylindrical metal elements are sometimes made the same size (or even slightly larger) than the holes they will be placed into, and are cooled significantly just before being inserted into them. Suppose that on the UNR Zebra z-pinch, at 20 degrees Celsius, a cylindrical aluminum alloy cathode of diameter 6.9 cm needs to be cooled to fit into a hole in a holder. Suppose that the cathode needs to made 117 microns smaller (a micron, or micrometer, is 0.001 mm), in order to be inserted. To what temperature (in degrees Celsius) must the aluminum alloy cathode be cooled? The expansion coefficient of the aluminum alloy is 2.4E-5.

Explanation / Answer

here ,

a = 2.4 *10^-5 C^-1

diameter = 6.9 cm

for change in length , l = -117 microns

l = -117 *10^-4 cm

Now,

as change in length = a*L*(Tf - T)

-117 *10^-4 = 6.9*2.4 *10^-5 (Tf - 20)

Tf = -50.7 degree

the aluminum alloy cathode should be cooled to -50.7 degree

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