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[Phys-L] Re: telescope mirrors



Anthony Lapinski wrote:
I tell my (high school) astronomy students
that reflecting telescopes are made by melting glass and then rotating it
at high speed while the glass solidifies.

Sometimes they spin 'em, sometimes they don't.

The magnificent 200-inch mirror for the Hale telescope wasn't spun;
they spent years grinding away 5 tons of excess glass to reach the
desired figure.

How can one show, in simple terms (no calculus), that the surface
of a spinning liquid is a parabola?

It's the usual situation in high-school physics: how to find
circumlocutions so kids can do simple calculus problems without
_calling_ it calculus.

We know from the kinematics of throwing things that the position
is parabolic when the velocity is changing with time and the
acceleration is constant. Don't tell anybody the big secret:
this is calculus. It is finding the slope of a function when
the slope is not constant.

Anyway, for a mirror, we know what the slope has to be at each
point. It is easy to work backward and show that a paraboloid
has the required slope. The average slope over the interval
[a,b] is (b^2-a^2)/(b-a) which -- by algebra -- we can show is
just (b+a) which is twice the average position (b+a)/2. To
repeat: the average slope is proportional to the average position.
(The constant of proportionality is 2.)

As Feynman was fond of saying: the same equations have the same
solutions. The algebraic pseudo-calculus tricks that work for
throwing things work just as well for mirror-shapes.

As a separate problem, it is easy to show that the paraboloid
is the solution to the hydrostatic equilibrium question (static
means static in the rotating frame). Consider a tiny marble
free to roll on a paraboloidal surface, and show that the
normal component of the gravitational force is balanced by the
centrifugal force.
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