Earl

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13 years, 162 days

MaplePrimes Activity


These are replies submitted by Earl

@Joe Riel Please see my reply to Dr. Lopez above. Am I correctly interpreting the hint you have given me?

@rlopez Thank you for  your reply above and your email (I'll reply to that as well)

Joe Riel's hint to consider the cone tangent to the unit sphere at the path of transport lit a light for me. I imagined the following:

On a rectangular sheet of paper (i.e. a 2d Euclidean space) draw successive images of a vector parallel transported along a horizontal line. Then draw on the paper the shape which will form the tangent cone. While still flat, translate the vector images without changing their orientation, from the roots on the horizontal line to roots on the arc which will be the line on the cone tangent to the unit sphere.

The translated vector images will appear to rotate horizonally vis-a-vis the cone's tangent line, as they will when the cone is formed from the paper cutout and placed on the unit sphere.

In this sense, the vectors on the unit sphere maintain the parallelism they had before Euclidean space was transformed into the tangent cone.

Does this sound correct?

 

@Carl Love The ode is the first equation in the output of EulerLagrange(L, t, [theta(t)]) where

L = KE - PE and 

KE := (1/2)*BeadMass*((RingRadius*(diff(theta(t), t)))^2+(RingRadius*cos(theta(t)-(1/2)*Pi)*`Dφ0`)^2)

and 

PE = BeadMass*g*RingRadius*(1+sin(theta(t)-(1/2)*Pi)).

BeadMass = 1, RingRadius = 1, the ring's angular velocity `Dφ0` is 1, g is 9.81.

The ring is initially in the xz plane and centred at the origin. Theta is the angle a radius to the bead makes with the negative z axis and phi is the azimuth angle the ring makes with the positive x axis as its diameter rotates in the xy plane.

@Kitonum Thank you for showing me that lowering the order of my original ODE permits a closed solution which can then be probed for the bracketing values of DthetaZero.

But, why are your values for DthetaZero neither the under/over values I found nor the exact value which Carl Love found above?

@Preben Alsholm The techniques in your reply are beyond my level of knowledge, however they give me the opportunity to learn, so I appreciate your answer. 

@Carl Love Thanks for finding the true value of DthetaZero at which the bead stops at the top of the ring (and presumably remains there as the ring continues to rotate).

I have only a rudimentary knowledge of differential equations, but would like to learn more. Please send me the form of the dsolve command which you used to treat the problem as a BVP.

@Carl Love I apologize Carl. The number 4 in the ODE should be 1. I have provide a link to a worksheet (correctly I hope) showing the physical situation from which the ODE is derived.

Rotating_Pendulum_explained.mw.

@Kitonum You have been a help to me on a number of occasions and I truly appreciate this. I hope you continue to provide cogent replies to my inquiries.

@Kitonum Does your solution assume the top is a hollow shell with all its mass in its surface?

If the top is solid with uniform density of one, is the following the correct calculation of its Moment of Inertia (MI) about the z axis?

int((1/2)*Pi*TopSC[1]^2*(TopSC[1]^2), z = 0 .. Pi)

The logic of the above integral is:

Integrate the MI of a cylinder (MI = (m*r^(2))/(2)),;
centred on the z axis (the top's axis of revolution), of height dz, radius = TopSC[1], ;mass = Pi*(TopSC[1])^(2)*dz, ;
from the top's bottom (z = 0) to its top (z = π).

 

 For anyone else answering, the top has a uniform density of one.

@acer Your suggestion works well and is more intuitive to me. I would like to praise your answer but I don't see a thumbs up or trophy icon in its header.

@Carl Love As usual, your answer is informative and to the point.

@Carl Love Hello Carl, I've examined all the ListTool commands but cannot find help text for operator ?[] which you used above. Is help available and if so how can I find it?

@Carl Love Thank you, Carl for the insight into the light ray vector and for several list manipulations in Maple that I was not aware were possible.

Sorry, plot3d parameter Plot3Daxes from my archive displays boxed, colored axes.

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