john boyce said:
Does anyone know the formula for converting RPM into focal length in the spin casting of mirrors? Any help would be welcome as always thank you.
I suspect that these forums aren't very good at formatting math, but I'll do what I can to make this readable. In case it doesn't go through, the character ² below is the superscript 2, for squaring a quantity. It
should go through, but it might not.
The short answer is that the focal length of the mirror, in meters, is 450
g/(pi*
n)², where
g is the acceleration due to gravity (about 9.8 m/s²
, pi is the usual 3.14159..., and
n is the number of revolutions per minute. Thus, for example, for
n = 20 (RPM), we get 450*9.8/(3.14*20)² = 1.1 m.
This strange looking formula comes ultimately from the centrifugal force field set up by the mirror's rotation. As a result of this field, each point on the mirror's surface is subject to two accelerations: it's mostly pulled downward by
g, but it also has an outward component
w²
r, due to the rotation, where
w is the angular velocity of rotation, and
r is the distance of that point from the mirror's center. The slope of this force is therefore
g/(
w²
r). The mirror's surface must be perpendicular to that, so its slope is one over that, or
w²
r/
g.
Notice that the slope of the mirror at any point is proportional to the point's distance from the mirror's center. Such a shape must be a paraboloid, with the height
y of the mirror above its bottom given by
y =
w²
r²/(2
g). The focal length of any such paraboloid is equal to 4 over the coefficient of
r², so it must be
g/(2
w²
.
Now we're almost there, but
w, unfortunately, is given in radians per second, which most people aren't comfortable with. So instead, we give it as rotations per minute. There are 2pi radians in a rotation, and 60 seconds in a minute, so the conversion factor from
n RPM to
w radians per second is pi/30. That's what accounts for the final form, given above.
