# Energy to increase the radius of a circle composed of several disks

• B
Hi,

I take a big number of disks to composed a circle of a radius of 1 m, the blue curved line is in fact several very small disks:

I take a big number of disks to simplify the calculations, and I take the size of the disks very small in comparison of the radius of the circle. The center A1 of the circle is fixed to the ground. I take each disk from the center of the circle A1 with a segment. The disks keep constant their orientation in the space. There is friction between the disks, I count the energy from the friction. I insert a disk to rotate clockwise all the disks, the disk is inserted before the dot A, like each disk is taken from the center and like there is friction between the disks and like the segment that taken the disks rotate clockwise the forces are like that:

It is the dot A that blocks the force F1. The dot A will move in horizontal translation to the right.

Like I increase the radius of the circle, I need to increase the lengths of the segments that taken the disks.

I drew a limited number of disks.
Black arrows: forces (same value F) from friction between the disks.
Red arrows on the center of each disk: the sum of forces of friction on each center of the disk (c0..c10), F1 is the algebric sum of all these forces.
The dot A receives all the sum of the red forces (F1=2piF)
The violet segments can rotate freely around A1, only the dot A blocks the rotation of the violet segments around A1.
Each disk can rotate around itself (from c0 to c10) but keep constant their orientation.
The orange arrows are the forces due to the lateral forces from the last disks (it is like pressure).

I suppose the value of the force of friction between the disks like a constant to simplify the calculations. So, F1 is constant in value.

For the example of a radius of 1 m, the energy needed to insert the new disk is : ##2*r*2*\pi*F##. The angle of rotation of the disks is ##2r##, so the energy from friction is ##2*\pi*F*2*r## the same energy I need to enter the disk. I win an energy to increase the length of the segments that taken the disks, the perimeter is increased of ##2*r##, so the radius is increased of ##2*r/(2*\pi)##, so the energy recovered is ##2*r*\pi*F##. I don't find the sum at 0. Maybe when I increase the radius of the circle the friction is not so much and the angle of rotation lower than I calculate ?

I took a pure geometric example to simplify the calculations and to keep constant the orientation of the disks in a fixed referential there are a lot of method.

Have you an idea where my mistake is ?

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Dale
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Well, the red arrows don't make any sense to me. I don't see any physical reason why the net forces would do that. I must say, however, that I don't really understand your description.

What is keeping the disks from just flying off everywhere? Why are you doing friction instead of frictionless? With high enough friction I am not convinced that you can add another disk. So to add a disk you need sufficiently low friction, so why not frictionless for a first analysis?

Well, the red arrows don't make any sense to me. I don't see any physical reason why the net forces would do that.
The red forces is the sum of the 2 black arrows (forces of friction) on each disk.

What is keeping the disks from just flying off everywhere?
There is one segment for each disk (violet color). Each segment is attached on the center of the circle and take a disk. I can increase the length of the segments because the circle increases its radius.

Why are you doing friction instead of frictionless?
Because, I want to understand with friction. Without any friction, there is no force.

With high enough friction I am not convinced that you can add another disk. So to add a disk you need sufficiently low friction, so why not frictionless for a first analysis?
There is finite friction not an infinite friction, the force of friction could be 1 N for example.

Dale
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The red forces is the sum of the 2 black arrows (forces of friction) on each disk.
That doesn't work by symmetry, clearly.

That doesn't work by symmetry, clearly.
I don't understand. What do you mean ?

Dale
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2020 Award
I don't understand. What do you mean ?
You have a circle. Circles are symmetric if you look at them from the front or the back. Therefore by symmetry you can look at this arrangement from the top or from the bottom and everything must be identical. So you cannot have the red arrows since they will flip directions if you look at them from the top or from the bottom.

You also obviously cannot have the orange arrows for the same reason.

nasu
The disks rotates CW around A1 and don't rotate around themselves so I don't understand why the forces are not like I drew.

I understood my mistake, the angular rotation of the disks around A1 is not 2r it is r because I need to take the mean.

Thanks

Dale
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The disks rotates CW around A1
Why would they rotate clockwise around A1? Are you not pushing in the new disk radially?

I am sorry, your description is so complicated and confusing that I am spending my effort trying to figure out what you mean rather than help you on the physics. I am done. Good luck, but until you can communicate more clearly I think this is something you will have to figure out on your own. You cannot expect people to help figure out the physics if they are spending all of their time trying to figure out what you are saying.

In the future spend some time to SIMPLIFY, SIMPLIFY, SIMPLIFY your problem to its very minimal bare essence and then write your description of the simplest system with utmost CLARITY so that it is unambiguous and easy to understand. Then people can help you with the physics instead of wasting time trying to decode your meaning. This is very frustrating

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Dale
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2020 Award
@Gh778 sorry about that. I got irritated and overreacted. I am not closing your thread, but I am leaving it myself. I don't have time to try to figure out what you mean and then help with the physics, but maybe someone else does. I am not optimistic, but I will let other people decide for themselves.

No problem. Have a good day !

jbriggs444
Homework Helper
I will try my hand. But, like @Dale, I do not understand the arrangement.

We seem to have a wheel on a frictionless axle anchored to the ground. We seem to have an array of disks somehow bound to the wheel. Perhaps there are spokes holding them in.

There is no friction between wheel and disks.

The disks are not free to rotate about their own centers. Possibly the spokes have some kind of chain drive so that they allow the spoke to rotate without allowing the disk on the end of the spoke to rotate.

The disks are not free to move inward. The wheel prevents it. The presence of the other disks prevents it.
The disks are not free to move outward. The spokes prevent it.
The disks are only permitted to orbit about the wheel if they do so in lock step.

We assume that everything is confined to a plane. We are working in two dimensions.

Now the confusing part...

Somehow a new disk is inserted. There is no place for it to go. And yet it is inserted. Somehow the spokes are lengthened and the wheel is expanded. How does that work? Who is pushing on what to make it happen? Who is cranking out the spokes to make them longer? Is there any tension in the spokes? Any compression in the ring of disks?

I took time to reply because yesterday I said I found my mistake because the angle is half for the friction so I have well the correct result I think. I don't want to be boring, don't reply if you think the question is nonsense. I reply, only, if I have question.

We assume that everything is confined to a plane. We are working in two dimensions.
Yes, in 2 dimensions is enough.

We seem to have a wheel on a frictionless axle anchored to the ground.
Yes, there is no friction except between the disks.

We seem to have an array of disks somehow bound to the wheel.
There is no wheel, the disks are arranged to draw a circle, that's all. I redrew without the circle:

Violet color: spoke
Blue: disks
Black arrows: forces of friction
Red arrows: sum of the 2 forces of friction on each disk
Orange arrows: the spoke are free to rotate around A1, except the last spoke (at the dot A) which it is blocked.

The disks are not free to rotate about their own centers. Possibly the spokes have some kind of chain drive so that they allow the spoke to rotate without allowing the disk on the end of the spoke to rotate.
I cancel all the torques from the black arrows (the friction) from the ground, I think (maybe I'm wrong here) it costs/gives no energy to cancel a torque on a disk which doesn't rotate in a fixed referential.

Perhaps there are spokes holding them in.
Yes, there is a spoke for each disk. I called it segment, but yes it is a spoke.

The disks are not free to move inward. The wheel prevents it. The presence of the other disks prevents it.
There is no wheel, the disks are arranged like a circle. It is the spoke that does the job, I can increase the length of the spokes (the circle increases its radius so I need to increase the length of the spokes).

Somehow a new disk is inserted. There is no place for it to go. And yet it is inserted. Somehow the spokes are lengthened and the wheel is expanded. How does that work?
I increase the radius of the "circle" (the circle is composed of N disks), I increase the length of each segment that takes a disk (the violet segments).

The disks are not free to move outward. The spokes prevent it.
Yes.

Somehow the spokes are lengthened
Yes.

How does that work?
I think in theory with lengths and forces, to build it, it is possible to use some mechanical devices (pneumatic cylinders, etc).

Who is pushing on what to make it happen?
An external device, but I count the energy I need to give to enter a new disk.

Is there any tension in the spokes?
Yes, because I can recover an energy to increase the lengths of the spokes.

Any compression in the ring of disks?
The lateral forces of friction on each disk (black arrows) give the red force on each center on each disk. I stop at the dot A the last disk, the dot A receives the force F1. The spokes are free to rotate around A1, so there are the orange forces on the center of the disks.

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jbriggs444
Homework Helper
Nope. I am out as well. I asked for an explanation and got nothing but a regurgitation.
Somehow a new disk is inserted. There is no place for it to go. And yet it is inserted. Somehow the spokes are lengthened and the wheel is expanded. How does that work? Who is pushing on what to make it happen? Who is cranking out the spokes to make them longer? Is there any tension in the spokes? Any compression in the ring of disks?
Your response to "how does that work" was:?
I think in theory with lengths and forces, to build it, it is possible to use some mechanical devices (pneumatic cylinders, etc).
That is not an answer. That is an assertion of possibility. But I do not understand what you are even describing. I am asking you to describe it.

Your response to "what is pushing on what?" was:
An external device, but I count the energy I need to give to enter a new disk.
That tells me what was pushing (answer: something). But I asked what it was pushing on.

vanhees71