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

["Krishna Chowdary" <professor.head@gmail.com>]

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.

--
regards
-Krishna

Krishna Chowdary
Department of Physics & Astronomy
Bucknell University