Re: [math-fun] spherical pendulums?
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? Ditto with a vertical cone instead of a cylinder. At 12:44 AM 12/5/2011, Bill Gosper wrote:
Are they chaotic or just hairy? The Wolfram demonstration seems quasiperiodic.? Gene once disabused me of the folly of trying to resolve the motion into x and y. But suppose we hung the string from the inside of an upward cusp of a cycloid of revolution. Would the pendulum simply describe an ellipse? --rwg
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
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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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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Hmmmm.... Does it matter if we wrap the transparent paper around a cylinder with circular cross section, or could the cross section be another shape -- e.g., an ellipse or something else? Obviously, if the shape included some sort of cusp, it wouldn't stay on the shape. Aren't we assuming that any accelerations are perpendicular to the shape of the cross section? But if the moving object is accelerating/decelerating along other dimensions, wouldn't that change things? At 10:43 AM 12/5/2011, Adam P. Goucher 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
The curved cylinder imposes a normal force on the point mass, and this does not affect the motion parallel to the surface. So indeed the particle moves in a parabola wrapped around the cylinder, no matter what the cross section, at least for smooth surfaces. -- Gene
________________________________ From: Henry Baker <hbaker1@pipeline.com> To: Adam P Goucher <apgoucher@gmx.com> Cc: math-fun@mailman.xmission.com Sent: Tuesday, December 6, 2011 8:01 AM Subject: Re: [math-fun] spherical pendulums?
Hmmmm....
Does it matter if we wrap the transparent paper around a cylinder with circular cross section, or could the cross section be another shape -- e.g., an ellipse or something else?
Obviously, if the shape included some sort of cusp, it wouldn't stay on the shape.
Aren't we assuming that any accelerations are perpendicular to the shape of the cross section? But if the moving object is accelerating/decelerating along other dimensions, wouldn't that change things?
At 10:43 AM 12/5/2011, Adam P. Goucher 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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For the cone, if the particle is in a stable circle, I would expect that a small impulse (down or up) would give it an oscillatory behavior, similar to being in an orbit with constant angular momentum, with the resulting ellipse being viewed as an oscillation about the circular orbit. Harmonic is not clear. -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker Sent: Monday, December 05, 2011 8:28 AM To: Bill Gosper Cc: math-fun@mailman.xmission.com Subject: [EXTERNAL] Re: [math-fun] spherical pendulums? 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? Ditto with a vertical cone instead of a cylinder. At 12:44 AM 12/5/2011, Bill Gosper wrote:
Are they chaotic or just hairy? The Wolfram demonstration seems quasiperiodic.? Gene once disabused me of the folly of trying to resolve the motion into x and y. But suppose we hung the string from the inside of an upward cusp of a cycloid of revolution. Would the pendulum simply describe an ellipse? --rwg
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It's not clear to me that the resulting orbit is an ellipse. I would expect the radius to be a periodic function of time, indeed, but I think that in general the periapse would precess. For particular values the orbit would close after tracing a p/q rosette, but for general initial conditions the orbit would never close. On Wed, Dec 7, 2011 at 10:03 AM, Cordwell, William R <wrcordw@sandia.gov>wrote:
For the cone, if the particle is in a stable circle, I would expect that a small impulse (down or up) would give it an oscillatory behavior, similar to being in an orbit with constant angular momentum, with the resulting ellipse being viewed as an oscillation about the circular orbit. Harmonic is not clear.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker Sent: Monday, December 05, 2011 8:28 AM To: Bill Gosper Cc: math-fun@mailman.xmission.com Subject: [EXTERNAL] Re: [math-fun] spherical pendulums?
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?
Ditto with a vertical cone instead of a cylinder.
At 12:44 AM 12/5/2011, Bill Gosper wrote:
Are they chaotic or just hairy? The Wolfram demonstration seems quasiperiodic.? Gene once disabused me of the folly of trying to resolve the motion into x and y. But suppose we hung the string from the inside of an upward cusp of a cycloid of revolution. Would the pendulum simply describe an ellipse? --rwg
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Sorry for my awkward wording; if a particle is in a circular orbit around the earth (the "similar"), and it has a small perturbation, the new orbit is necessarily an ellipse, but may also be viewed as an oscillation about the circular orbit. It is not clear at all that a small perturbation to the particle moving in a stable circle in the cone would give an ellipse. -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Wednesday, December 07, 2011 8:36 AM To: math-fun Subject: Re: [math-fun] [EXTERNAL] Re: spherical pendulums? It's not clear to me that the resulting orbit is an ellipse. I would expect the radius to be a periodic function of time, indeed, but I think that in general the periapse would precess. For particular values the orbit would close after tracing a p/q rosette, but for general initial conditions the orbit would never close. On Wed, Dec 7, 2011 at 10:03 AM, Cordwell, William R <wrcordw@sandia.gov>wrote:
For the cone, if the particle is in a stable circle, I would expect that a small impulse (down or up) would give it an oscillatory behavior, similar to being in an orbit with constant angular momentum, with the resulting ellipse being viewed as an oscillation about the circular orbit. Harmonic is not clear.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto: math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker Sent: Monday, December 05, 2011 8:28 AM To: Bill Gosper Cc: math-fun@mailman.xmission.com Subject: [EXTERNAL] Re: [math-fun] spherical pendulums?
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?
Ditto with a vertical cone instead of a cylinder.
At 12:44 AM 12/5/2011, Bill Gosper wrote:
Are they chaotic or just hairy? The Wolfram demonstration seems quasiperiodic.? Gene once disabused me of the folly of trying to resolve the motion into x and y. But suppose we hung the string from the inside of an upward cusp of a cycloid of revolution. Would the pendulum simply describe an ellipse? --rwg
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participants (6)
-
Adam P. Goucher -
Allan Wechsler -
Cordwell, William R -
Eugene Salamin -
Fred lunnon -
Henry Baker