@Rouben Rostamian  Thank you for your reply. The uploaded worksheet below shows a chain hanging from two gears and several questions regarding this situation. I'd appreciate any answers you can give me.

ChainOnTwoGears.mw

@Carl Love Thanks. I'll look that up and see if I can continue from there.

@Carl Love Let's assume that the gear faces are all oriented vertically i.e. that gravity's direction is perpendicular to the axes of all gears, and from top to bottom in the image parallel to the image's vertical side lines.

@Kitonum I was stymied by the ode's apparent mix of spherical and cartesian coordinates. Your answer cleared this up by defining the radial coordinate in cartesian terms, for which I thank you

@nm 

 diff(v(t), t) := 2/7*(Omega &x v(t)) + 5/7*g*sin(theta)*e__y

 where r is the radial coordinate, v(t) = diff(r(t) ,t), Omega = <0,0,1>, g = 9.81, theta; = Pi/8 and e__y is the Cartesian y axis;

@vv I thought you might be interested in this application of the time delay technique you have shown me.

The worksheet requires a connection to the DirectSearch package.

Carom.mw

@vv I'll try to use your delay technique in my application.

@vv Thank you for showing me how to delay processing using the Thread package.

Can the time delays demonstrated in the Explore display be achieved through any other coding technique, for example via an animate command?

@Kitonum Thank you for showing me the ODE technique and how to code an animation of a procedure.

@Kitonum I appreciate the beauty in Rouben Rostamian's answer, however I have chosen yours as the best answer because you have shown me an important technique of which I was unaware. Thanks again! 

@Rouben Rostamian  What a lovely display! Especially the shading produced when the two colors overlap. Thank you greatly!

@Kitonum I have tried many ways to apply your technique to the website's last equation containing differentials and cannot produce a differential equation, which, when solved, yields the slope lines portrayed in the website's accompanying diagram.

How can y*cos(x)*dy -sin(x)*dx be converted to a solvable differential equation?

@Kitonum I am grateful to you for this lesson. After studying its implications I will try to apply it to several of such equations in the cited website

@Carl Love My searches will benefit from your advice.

@Rouben Rostamian  You have given me two gifts! First, I will carefully study your answer and try to absorb this lovely technique.

Secondly, you have opened up a new field of exploration for me as I attempt to apply this to a variety of other interesting surfaces.

You likely already are aware of the almost endless variety of these on https://mathcurve.com/surfaces.gb/surfaces.shtml