B Why does a coin take 2 full rotations around another coin?

[Devin-M]

But the general idea, that the circle rotates more, because it is travelling further, is on the right track.

Another intuitive way to think about it, in the lego gears case, for me, is suppose you “unwind” the outer geared surface of a lego gear so that it becomes a straight path— watch the tips of the neighboring gear teeth get closer together.

 

[DaveC426913]

Another intuitive way to think about it, in the lego gears case, for me, is suppose you “unwind” the outer geared surface of a lego gear so that it becomes a straight path— watch the tips of the neighboring gear teeth get closer together.

Hm. That's only the case for teeth of inordinate length. There is no reason why the height of the teeth can't approach zero, which means the amount by which they get closer together likewise approaches zero.

In fact, the scenario works by friction alone; teeth are superfluous.

Since we can eliminate the effect without altering the scenario, it's got to be a spurious factor.

 

[Devin-M]

But if the teeth are lengthened, then the tips of the neighboring teeth can be considerably closer when the path is unwound & straightened.

 

[DaveC426913]

But if the teeth are lengthened, then the tips of the neighboring teeth can be considerably closer when the path is unwound & straightened.

Teeth are not relevant to the problem or solution; they are a distraction (one that you added late in the game). And, after all, coins don't have interlocking teeth.

Witness the fact that the teeth can (and should) be removed completely and absolutely nothing about the problem will change.

 

[Devin-M]

Teeth are not relevant to the problem or solution; they are a distraction (one that you added late in the game). And, after all, coins don't have interlocking teeth.

Witness the fact that the teeth can (and should) be removed completely and absolutely nothing about the problem will change.

Here’s why I believe they are relevant and illustrative of the problem.

In the case of a radius 1 stationary coin and the radius 10 moving coin, the factor of change of # of rotations between the straight line case versus the circular case is directly proportional to the change in area swept by the tangent that extends through the radius 10 coin in straight vs circular cases.

In other words the area swept by the tangent is 11 times greater in the circular case & the coin rotates more times by a factor of 11 times greater.

The line forming center of base to tip of a gear tooth is also a tangent to the inner coin. In fact if the gear teeth on the radius 1 coin were length 20, if we rotate the radius 1 coin 1 rotation, then a length 20 tooth will sweep the same area as the tangent of the radius 10 coin.

So essentially the changing distance between the tips of the teeth in the straight line vs circular case can visually illustrate the changing area swept by the tangent of the radius 10 coin around the radius 1 coin… a change in area in which the factor of change is directly proportional to the factor of change in number of rotations.

 

[Lnewqban]

... I'm simply musing out loud how difficult it is to intuit the offered answer.

-Suppose we have a radius 1 coin and a radius 10 coin.

 

[A.T.]

In the case of a radius 1 stationary coin and the radius 10 moving coin, the factor of change of # of rotations between the straight line case versus the circular case is directly proportional to the change in area swept by the tangent that extends through the radius 10 coin in straight vs circular cases.

In other words the area swept by the tangent is 11 times greater in the circular case & the coin rotates more times by a factor of 11 times greater.

The line forming center of base to tip of a gear tooth is also a tangent to the inner coin. In fact if the gear teeth on the radius 1 coin were length 20, if we rotate the radius 1 coin 1 rotation, then a length 20 tooth will sweep the same area as the tangent of the radius 10 coin.

So essentially the changing distance between the tips of the teeth in the straight line vs circular case can visually illustrate the changing area swept by the tangent of the radius 10 coin around the radius 1 coin… a change in area in which the factor of change is directly proportional to the factor of change in number of rotations.

I don't see how your construction that involves areas is more intuitive, even if it was correct. It seems that this is just a unnecessarily complicated attempt to get to the distance traveled by the center of the coin, which is indeed directly related to the number of rations, as explained in the Veritasium video at 9:45.

What the "teeth" can offer, is a visualization of the local normal vectors, which can be used to define a local frame of reference moving along with the contact point. Relative to the local normal vector, the rolling coin has the same number of rotations as for straight rolling. But if it rolls around a coin, the normal vector itself rotates along the path in the same direction, so it adds one rotation. For rolling inside a circle the normal vector rotates opposite to the rolling rotation, so it subtracts one rotation.

 

[Averagesupernova]

We start with the above.

Rolling it around so the dull coin is on the bottom gives us an inverted dull coin relative to the first pic.

This last image is of course wrong. The part of the coin with Washington's nose would have to continually contact the shiny coin on its trip around. This action is not rolling.
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When intuition gets you in trouble as it often does, work out a path to the answer by showing how it cannot work as your intuition says it would.

 

[davenn]

Have you watched the video?

yeah, saw it about a week ago
I love Destin's videos, they are incredibly informative :)

 

[Herman Trivilino]

on the circular path than the circumference “unwound” into a straight path.

The difference is that following the circular path causes it to make one additional rotation.

Note that the moon makes one rotation when it makes one revolution around Earth, because the same side of the moon always faces Earth.

 

[A.T.]

Note that the moon makes one rotation when it makes one revolution around Earth, because the same side of the moon always faces Earth.

Exactly. When you take pure sliding along a straight line (no rotation), and merely roll that entire scenario so the line becomes a circle (preserving the contact point of the coin with the line), the you already have 1 rotation, like the tidally locked Moon has.

 

[sophiecentaur]

Yes, I'm simply musing out loud how difficult it is to intuit the offered answer.

I was out of the loop when this thread ran, originally. This quoted statement only goes to show how useful Maths is. Intuition is a false friend but (the appropriate) Maths will not let you down.

 

[painter]

This one baffles me, I still can’t get my head around it (no pun intended).

Ahh. May I ask you that what does"Why does a coin take 2 full rotations around another coin? "mean?
I cannot get the point .
Does coins really take 2 full rotations around another coin then it will be stationary??

 

[jbriggs444]

Ahh. May I ask you that what does"Why does a coin take 2 full rotations around another coin? "mean?
I cannot get the point .
Does coins really take 2 full rotations around another coin then it will be stationary??

Take two coins. Place them flat on a table, one above the other:

Roll the top coin around the bottom coin without slipping so that it completes a complete circle. Watch how many times the picture on the moving coin rotates [relative to the fixed orientation of the table].

 

[Averagesupernova]

Ahh. May I ask you that what does"Why does a coin take 2 full rotations around another coin? "mean?
I cannot get the point .
Does coins really take 2 full rotations around another coin then it will be stationary??

Have you actually read this entire thread?

 

[Herman Trivilino]

Does coins really take 2 full rotations around another coin then it will be stationary??

You have to apply a force to make the coin move. Anytime you stop applying the force the coin stops moving (due to friction).

When you move the coin you have to rotate it yourself in such a way that it doesn't slip against the coin in the center.

 

[sophiecentaur]

in such a way that it doesn't slip against the coin in the center.

Use gears instead; no significant friction involved.

 

[Herman Trivilino]

Use gears instead; no significant friction involved.

Some coins have ridges that act like little gear teeth. But I meant friction between the coins and the surface they rest on.

 

[DaveC426913]

You have to apply a force to make the coin move. Anytime you stop applying the force the coin stops moving (due to friction).

Some coins have ridges that act like little gear teeth. But I meant friction between the coins and the surface they rest on.

This is over-thinking the issue. The coins do not slip as they move against each other. How this happens is irrelevant and a distraction.

 

[Vanadium 50]

Before duplicating the entire thread, maybe we should give @painter an opportunity to read it.

 

[sbrothy]

I love it when you guys go all in with something like this. :)