Ok, and that must mean the period of the steel ball in a cycloidal track is NOT \( 4\pi \sqrt {\frac {R} {g}} \). The problem here is that, according to the physics text (which was used in undergrad physics for several years at a recognised university), the period is the same as for a cycloidal pendulum. So, you must be wrong. Oh dear. Furthermore, the steel ball follows a tautochrone curve and this can be used to analyse its motion: Please Register or Log in to view the hidden image!
How did you infer that? That is for you to find out by solving the equations of motion. Also this time you have been wasting u could have gotten started on the equations. I realize that it is much tougher to put the equations together than to post stuff and wait for the solutions to be given to you.
I think you'll find that is referring to the disc rolling on a flat surface, not the bead on a cycloidal track.
I see that Tach has now diverged into his usual state of denial, that he could possibly be in error. I predict that further correspondence with Tach on this thread subject will be a monumental waste of bandwidth.
I can see that you have been wasting your time trolling instead of attempting to put together the equations of motion and solving them. It has been about 21 hours since you have been given the method and you are still waiting for someone else to solve your homework.
If I'm "trolling", there's a purpose to it: namely to demonstrate that Tach, although self-assured and confident about what he knows, is capable of making mistakes. So, that's what this thread has really been about (for the conspiracists out there)--yeah, that's you, sunshine.
I don't know what fuels your delusion, probably the fact that I did not give you the solution to your homework ready made. If you waste more time trolling you might not have any homework to hand in besides the first half done for you by AN.
Ah yes, the sweet smell of a conspiracist's self-delusion. That you have to believe I was looking for a solution to some "homework" rather than leading the likes of you down the garden path of "Ooops, I made a mistake" is telling, my son. I know what the solution is, I know what my physics text has to say and why a cycloidal pendulum's period is independent of amplitude unlike a 'circular' pendulum. So stop trying to make me look like the same kind of idiot you are, please?
Ok, good for you. Then why do you keep asking how to solve the second half of the problem? (because you obviously don't). You made yourself look like you look through your posts. I wasn't trying to make you look like anything, I was giving you the tools how to solve the problem. Sorry , I failed, I won't try again.
I predict that Tach will not be able to explain how the period of an "ordinary" cycloidal pendulum and the period of a steel ball rolling in an inverted cycloidal track are the same, but the equations of motion are different. Nor will anyone else for one very good reason: the second premiss is demonstrably false.
Making friends again, Tach? Perhaps just a slight change in the color of your delivery would go a long way. My opinion is that your condescending tone brings up emotion-laden and defensive responses, which makes for a non-constructive atmosphere if we're all seeking "truth". There shouldn't be a feeling of dread in making a mistake in a science forum, for you or anyone else posting. Yes that's why I made such a production in the other thread. Just making a point that hyper-criticism isn't fun, and ideas should be able to be discussed in a mature, emotionally detached and respectful manner.
Hint 1: the max kinetic energy of the cycloidal pendulum is \(\frac{mv^2}{2}\) Hint 2: the max kinetic energy of the rolling ball is \(\ne \frac{mv^2}{2}\) Basic physics says that different energies trigger different equations of motion. Different equations of motion produce different periods.
So then, the only difference between the velocities of a rolling ball and a sliding ball (bead on a wire) is a constant factor? The constant factor depends only on the shape of the rolling body and hence its radius of gyration? That all means: if the curve is isochronous and the weight (or particle) moves without friction (slides or slips), then the shape of the moving body is irrelevant; if the curve is isochronous and the weight rolls along the track (which implies it has a smoothly curved boundary) then its radius of gyration becomes a constant factor in the equations of motion: a rolling body has kinetic energy of rotation so it will move more slowly than a sliding body. So nya nya.
So far, you got nothing, just the result from the book. How about you try doing the calculations all by yourself?
'Yawn' Hmm. This is what Tach posted originally: Given that the total energy of the moving body depends on its mass and height above the lowest part of the cycloid, the velocity must vary according to the two formulas if the first is for a body with no kinetic energy of rotation, and the second is for a rolling sphere. In fact, this principle--that a rolling body moves more slowly than a sliding one with the same weight--is independent of the curve it follows. It's also true for spheres, cylinders, and disks rolling down a flat plane. The difference in speed depends only on the shape of the rolling body (its radius of gyration). It's because a sliding body doesn't convert any potential energy into rotation, therefore it moves faster than a rolling body which does.
So, you still can't do any of the calculations to find the two different periods? Your book should have a "Solutions" section at the back. You'll get an "incomplete" for your homework.
Still unable to explain why the period of a cycloidal pendulum is independent of the amplitude, which you said was not true of a ball rolling along a cycloidal track? Still unable to explain why a constant geometric factor, namely the radius of gyration, will make the equations of motion "different"? Still unable to figure out that I'm not asking you or anyone else to do "my homework"? Is that because you prefer to stick to whatever occurred to you first? It's just the way your brain works, you have major problems with letting an idea go?
Looks like you are. You only have the book answer for one case, you don't have any derivation and you are desperately trying to elicit it from the others. It's been more than one day since you have been give the tools to put together the equations of motion and you haven't managed even one line. Why aren't you rolling up your sleeves and start solving the problem by yourself?
