This phenomenon is well-known to anyone who has attempted golf. A putt aimed at the hole but travelling sufficiently fast disappears briefly therein, only to re-emerge cheerily at a substantial angle to its original line of travel --- to the frustration and bafflement of its propellor, and the grim amusement of any mathematicians happening to lurk in the bushes nearby. WFL On 12/5/11, Allan Wechsler <acwacw@gmail.com> wrote:
With the problem idealized exactly as stated, I'm pretty sure this wrapped-parabola is correct. On the other hand, about four years ago at the Gathering for Gardner, we watched a juggler roll rubber balls along the inside of a plexiglass cylinder. This departs from the given idealization in a couple of important ways: the ball has a high coefficient of friction with the cylinder surface, and rolls pretty much without slipping; and the ball's moment of inertia is not negligible. The resulting behavior is surprising. The best video I was able to find illustrating this is http://www.youtube.com/watch?v=1t1grbgT5pE.
On Mon, Dec 5, 2011 at 1:43 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Here's a related problem:
A point particle of mass M (i.e., zero moment of inertia) is sliding frictionlessly around inside a vertical cylinder; gravity is downwards of strength G. What are the paths?
Imagine drawing a parabolic arc on a sheet of transparent paper, and wrapping it around the inside of the cylinder. This is parametrised by:
x = r cos(wt) y = r sin(wt) z = -½gt²
(w, r and g are constants, namely the angular speed, radius and gravitational field strength, respectively.)
Sincerely,
Adam P. Goucher
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