btm said:

sibeen said:

A tautochrone curve is another name for an isochrone curve. I didn’t kow that until today.

The brachistochrone problem was first posed by one of the Bernoulli brothers (Jakob, IIRC, but I may be wrong); it led directly to the development of the calculus of variations. The solution of the problem turns out to also be the solution to the tautochrone problem. The names mean “shortest time” (brachistochrone) and “same time” (tautochrone). “Isochrone” is a recent appelation, but, as noted, means the same tautochrone.

The brachistochrone problem:
Consider a frictionless wire connecting two points on a vertical plane, and a bead moving on that wire under the influence of gravity. Find the equation of the curve the wire must trace to minimise the time it takes the bead to move from the top of the wire to the bottom.

It turns out that the solution curve set doesn’t include a time parameter. The solution curve is a cycloid.

sibeen said:

mollwollfumble said:

I had a similar problem to this recently, but it wasn’t the same and I never figured out how to solve it.

Worth a thread?

https://www.youtube.com/watch?v=eBc827pwKf0

ROFL

A tobacco tin was used on a kids show. That’d get you crucified now days. :)

BTW, I use the term ‘tobacco’ loosely. Dr Pat shudders

sibeen said:

btm said:

Do you know the calculus of variations? If so, did you try that on it? Are you familiar with the Euler-Legrange equation? (I assume so on all three counts, but they were not part of my BEng, even with the extra maths and physics I did.)

Mate, I have trouble remembering even the basics of calculus, let alone some esoteric variations :)

Have done calculus of variations. A bit of a black art to me, but I did try it.

The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can only use up to the first half rotation, and always ends at the horizontal.

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That’s handy to know.

But what happens if you take into account the energy required for rotation?

If a vehicle is rolling down a cycloid then the energy of rotation contained in the wheels isn’t handled correctly. Not a bad approximation for a car or scateboard, but bad for a bicycle or motorcycle.

To put it more in perspective, a yo-yo does not fall down the string at an acceleration of 9.8 m/s² but rather somewhat slower because of the rotational inertia.

mollwollfumble said:

But what happens if you take into account the energy required for rotation?

If a vehicle is rolling down a cycloid then the energy of rotation contained in the wheels isn’t handled correctly. Not a bad approximation for a car or scateboard, but bad for a bicycle or motorcycle.

To put it more in perspective, a yo-yo does not fall down the string at an acceleration of 9.8 m/s² but rather somewhat slower because of the rotational inertia.

So, how does the rotational moment of inertia modify the cycloid tautochrone?

An object’s moment of inertia I determines how much it resists rotational motion. In this simulation, four objects are placed on a ramp and left to roll without slipping. Starting from rest, each will experience an angular acceleration based on their moment of inertia.

(Warning: Devil’s advocate mode) I hope you all realise that this means that Galileo was wrong. He used an inclined plane to measure the acceleration due to gravity.

mollwollfumble said:

mollwollfumble said:

But what happens if you take into account the energy required for rotation?

If a vehicle is rolling down a cycloid then the energy of rotation contained in the wheels isn’t handled correctly. Not a bad approximation for a car or scateboard, but bad for a bicycle or motorcycle.

To put it more in perspective, a yo-yo does not fall down the string at an acceleration of 9.8 m/s² but rather somewhat slower because of the rotational inertia.

So, how does the rotational moment of inertia modify the cycloid tautochrone?

An object’s moment of inertia I determines how much it resists rotational motion. In this simulation, four objects are placed on a ramp and left to roll without slipping. Starting from rest, each will experience an angular acceleration based on their moment of inertia.

(Warning: Devil’s advocate mode) I hope you all realise that this means that Galileo was wrong. He used an inclined plane to measure the acceleration due to gravity.

That was very interesting, molly.

sibeen said:

mollwollfumble said:

mollwollfumble said:

But what happens if you take into account the energy required for rotation?

If a vehicle is rolling down a cycloid then the energy of rotation contained in the wheels isn’t handled correctly. Not a bad approximation for a car or scateboard, but bad for a bicycle or motorcycle.

To put it more in perspective, a yo-yo does not fall down the string at an acceleration of 9.8 m/s² but rather somewhat slower because of the rotational inertia.

So, how does the rotational moment of inertia modify the cycloid tautochrone?

An object’s moment of inertia I determines how much it resists rotational motion. In this simulation, four objects are placed on a ramp and left to roll without slipping. Starting from rest, each will experience an angular acceleration based on their moment of inertia.

(Warning: Devil’s advocate mode) I hope you all realise that this means that Galileo was wrong. He used an inclined plane to measure the acceleration due to gravity.

That was very interesting, molly.

Agreed.

I didn’t know Galileo used an inclined ramp.

But I know he was wrong about many things.

He should have used sliders rather than rollers.

Though of course, that would increase the friction…

should have used very very smooth sliders

dv said:

He should have used sliders rather than rollers.

Though of course, that would increase the friction…

should have used very very smooth sliders

Ice with dieseline thrown over it.

Michael V said:

dv said:

He should have used sliders rather than rollers.

Though of course, that would increase the friction…

should have used very very smooth sliders

Ice with dieseline thrown over it.

LOL. Does that have low friction?

mollwollfumble said:

Michael V said:

dv said:

He should have used sliders rather than rollers.

Though of course, that would increase the friction…

should have used very very smooth sliders

Ice with dieseline thrown over it.

LOL. Does that have low friction?

I’d imagine so. Ice does. Dieseline does.

I don’t know whether the combination does, but it has always sounded like it should – so I started using notion to describe a low friction surface in 1973.

I’d like to see the result of that on a conveyor belt.

sibeen said:

mollwollfumble said:

mollwollfumble said:

But what happens if you take into account the energy required for rotation?

If a vehicle is rolling down a cycloid then the energy of rotation contained in the wheels isn’t handled correctly. Not a bad approximation for a car or scateboard, but bad for a bicycle or motorcycle.

To put it more in perspective, a yo-yo does not fall down the string at an acceleration of 9.8 m/s² but rather somewhat slower because of the rotational inertia.

So, how does the rotational moment of inertia modify the cycloid tautochrone?

An object’s moment of inertia I determines how much it resists rotational motion. In this simulation, four objects are placed on a ramp and left to roll without slipping. Starting from rest, each will experience an angular acceleration based on their moment of inertia.

(Warning: Devil’s advocate mode) I hope you all realise that this means that Galileo was wrong. He used an inclined plane to measure the acceleration due to gravity.

That was very interesting, molly.

A funny aside.In all my engineering profession I have never had to use the moment of inertia for anything and the above was a curious tidbit that I found interesting a few weeks ago. So I’ve got a current gig where I’m helping a client out with their protection settings on their incoming supply and they are having quite a barny with the electrical supplier. So I’m going through all original documentation for the site and trying to determine how the original engineers made some of their design decisions.

I came to a page where it showed how the ROCOF (rate of change of frequency) and Vector Shift settings were determined. A heap of equations that I’d never come across before that made no rhythm nor reason to me, so off I went in search of the papers the thinking was based on. A few hours later I’m performing calculations based upon the moment of inertia of some generators and the associated rotational energy stored within their rotors. Managed to get the same figures out as the original engineer, which made me feel chuffed with myself, and then discovered that the original engineer had made a boo boo when interpreting the equations and had stuffed up some site settings.