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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