The Liquid Paraboloid
The process of
grinding, polishing and testing is very long and tedious. Surface area is
roughly proportional to the square of the radius, so the bigger the mirror’s
radius, the more work there is to do. Fortunately, modern telescope makers have
a new technique that eliminates the need for grinding. Glass can be heated
until it is a liquid and then spun in a huge container. When spun, the surface
becomes a paraboloid (see section on parabolas), leaving a lot less work to do
once it is cooled and solid. The following informal mathematical proof shows
why this is so.
The three
dimensional paraboloid is the same as the rotation of a parabola around its
axis, so the problem can be reduced to two dimensions. The main concept in
proving that the surface is parabolic is that the net force due to both gravity
and the centrifugal force* must be perpendicular to the surface of the water.
From here, we can obtain the slope of the surface (using the opposite inverse
rule for perpendicular lines).
* The centrifugal
force, Fc, is not a real force. It is really a reactionary force
caused by the container, but from the rotating cylinder’s reference frame,
there is an apparent force that may be called the centrifugal force.
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This equation was
loosely verified with a simple setup of a spinning water bottle being filmed
with a camera (to find frequency). Professor Chris Martin of Oberlin College
verified the math.