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Re: [Phys-l] time to bottom of ramp



For purpose of comparison, please specify the beta function, or whatever it is you get. These are tabulated functions and are no worse that sines, cosines, or Bessel Functions.

1. The ram calculation gives 5/sqrt(2) s (I agree) = 3.54 s (not 5.53), which is shorter than your arc time. I'll recheck my arc calcuolation.
The Beta function B(x,y)= Gamma(x)Gamma(y)/Gamma(x+y) is also known as Euler's Integral of the first kind.

On Fri, 27 Oct 2006, Krishna Chowdary wrote:

On 10/27/06, Jack Uretsky <jlu@hep.anl.gov> wrote:

Hi all-
For a frictionless quarter-circle arc of radius h, the time to
travel the arc is (barring arithmetic errors) about 2.6 times to slide
down an inclined ramp between the same 2 points. The answer can be
expressed as a beta-function.
Regards,
Jack


I get a very different result for a specific example of the case above. For
a frictionless quarter circle of radius h = 100 feet, I find the travel time
to be 3.27757 seconds (I needed to solve an integral numerically in
Mathematica - if someone can direct me to a closed form solution that
doesn't have elliptic integrals in it, I'd be most appreciative). I assume
the quarter circle is oriented so that the initial part of the arc is
vertical (aligned with the uniform gravitational field) and that the
end of
the arc is horizontal.

For an inclined plane of slope 1 with a vertical drop and horizontal
traverse each of 100 feet (ie an inclined plane between the initial point
and the final point of the quarter circle arc above), I find the travel time
to be 5/Sqrt(2) = 5.5355 seconds.

Is the beta-function the Euler beta function?

It seems unreasonable to my physical intuition that the ramp beats out a
quarter-circle arc. Of course the cycloid beats out both.



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