@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.

@Kitonum I have gratefully learned several coding techniques from your examples. Thank you.

On examining an open umbrella, it appears that the surface between a pair of ribs in not a portion of a sphere but a minimal surface.

Is this true? Does any Maple resource (Maple Primes, Maple Application Centre, Maple package) demonstrate or provide the code for calculating and displaying such a surface?

@Markiyan Hirnyk Thanks for pointing me to the appropriate help page, I don't know how I overlooked it.

@Jarekkk I hope you receive this question to an old answer.

The solve commands in your answer above work in Maple 2016.0 but I can't find an explanation of the 'parametric' or 'parameters' options in Maple 2016.0 help.

Where can I find help text explaining these options?