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

[Devin-M]

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

Take 2 US quarters. Put them in contact side by side. Without slippage, roll one quarter around the circumference of the other until it returns to the starting point. It requires rotating the moving quarter 2 full times to go around the first quarter, not 1 rotation as intuition and 2R*Pi suggest. What is happening here? Is the circumference along the curved path actually 4R*Pi not 2R*Pi? Please help.

 

[russ_watters]

I don't have two quarters handy, but....Once for rotation, once for revolution? Also, the quarter isn't revolving about a radius r, it's revolving about 2r, center to center.

 

[Hill]

There is a recent thread about this:
https://www.physicsforums.com/threads/deceptively-simple-geometry-question-on-sat-test.1057821/

 

[PeroK]

I don't have two quarters handy, but....Once for rotation, once for revolution? Also, the quarter isn't revolving about a radius r, it's revolving about 2r, center to center.

You can see the "once for revolution" by sliding the quarter round, keeping the same point on the outer quarter touching the inner quarter.

 

[Ibix]

The (possibly overcomplicated) way I see it is this.

Let the central coin have radius $R$ and the orbiting coin have radius $r$.

The center of the orbiting coin moves at constant speed $v$. The linear velocity of the rim relative to the center must also be $v$ so that the point touching the central coin is stationary, not slipping. Thus its angular velocity is $v/r$.

The coin center travels distance $2\pi(r+R)$ in time $t=2\pi(r+R)/v$. In that time, the orbiting coin turns through $tv/r=2\pi(r+R)/r$ radians, or $1+R/r$ revolutions.

In this case $r=R$ and the answer is 2. In the SAT question, $R=3r$ and the answer is 4.

 

[DaveC426913]

 

[Lnewqban]

Please, see:
https://www.tec-science.com/mechani...-equation-of-planetary-gears-willis-equation/

https://en.wikipedia.org/wiki/Epicyclic_gearing

 

[Devin-M]

In this step of the example, the coin actually does travel further than it had before when simply rolling in a straight line…

 

[DaveC426913]

In this step of the example, the coin actually does travel further than it had before when simply rolling in a straight line…

Sure but that's not the point.

The demo is essentially an "exploded view" of the experiment - which means actual real-world distances are not to-scale - but only for clarification purposes, whereas relevant-to-the-problem distances are measured and labeled accurately.

The point is that there are two factors at work, and the demo merely shows that they can be teased apart.

One factor is coin B navigating the perimeter of coin A. That requires the full perimeter of coin B.

The other factor is that coin B is also revolving as it is doing so.

 

[PeroK]

In this step of the example, the coin actually does travel further than it had before when simply rolling in a straight line…

Given that you can conduct this experiment yourself with any two coins, scepticism in this case is misplaced.

 

[DaveC426913]

you can conduct this experiment yourself with any two coins

True. Any two coins - no matter the ratio of their relative sizes - have the same effect: one revolution is added to the total accounting for its revolution about the central coin.

Two coins with a ratio of 10:1 will have the smaller coin doing 11 revolutions.

 

[Devin-M]

I still don’t understand why if both perimeters are equal and there is no slippage how more than 1 revolution occurs unless the coin is indeed traveling further on the circular path than the circumference “unwound” into a straight path.

 

[Hill]

unless the coin is indeed traveling further on the circular path than the circumference “unwound” into a straight path

The center of the coin indeed is: $2 \pi (R+r) = 2 \pi R + 2 \pi r$.

 

[DaveC426913]

I still don’t understand why if both perimeters are equal and there is no slippage how more than 1 revolution occurs unless the coin is indeed traveling further on the circular path than the circumference “unwound” into a straight path.

It has nothing to do with travelling "further".

The coin follows a path that is, itself, one coin in circumerence. That results in a revolution.

But the path itself is curved, going its own 360 turn, meaning that, to follow it, the coin must also rotate 360 degrees.

Have you watched the video?

 

[Devin-M]

Think this way… take the “unwound” circumference, then fold it in half. Have the coin roll along the top, then underneath along the bottom to the starting point. The coin traveled a certain distance. Now put your finger in the middle of the folded circumference and restore it to a circle. Wouldn’t the path now followed by the coin be longer?

 

[Devin-M]

Ok I will watch the video again.

 

[Devin-M]

What about the case of A) the coin rolling along a straight unwound circumference path versus B) the case of the path being folded in half with the coin rolling along the top half, then the bottom half and back to the starting point at the top. While the coin is transitioning twice from top to bottom and from bottom to top, doesn’t it travel “further” than the coin that went in a straight line only?

 

[Kontilera]

I still don’t understand why if both perimeters are equal and there is no slippage how more than 1 revolution occurs unless the coin is indeed traveling further on the circular path than the circumference “unwound” into a straight path.

Following the perimeter of a circle implies a revolution itself.

I believe you are mixing up two concepts.
If you focus on how many revolutions the outer coin makes in the two dimensional plane, then the answer is two. If you focus on how much distance the coin travels on the perimeter of the stationary coin, then of course the answer is one perimeter, which consitutes the second revolution (given that the coins are of equal size).

Calculating how far a coin has rolled by the number of degrees it rotates is only straight forward if the line is straight. As soon as you have a curved path, the coin will also start to rotate due to it.

If you slide the outer coin along the perimeter of the inner coin without any rolling at all, you will see that outer coin will complete one full revolution even though the point touching the other coin is fixed. This is not because it rolled along the perimeter, but because the path it travelled was curved b a total of 360 degrees.

Given that the coins are of equal size the following holds:
What you see when you roll the coin along the outside of the stationary coin is an angular velocity which is the sum of two equal parts:
The rotation due to following a curved path + The rotation due to rolling along that path.
When the coin has reached halfway, it has rotated 360 degrees because the path has curved 180 degrees and the "degrees travelled" are 180.

I wrote an answer here that offers a way to visualize it:
https://www.physicsforums.com/threa...try-question-on-sat-test.1057821/post-6972604

 

[Devin-M]

But sliding the coin along a straight line equal in length to 1 perimeter, the path length taken by the top and bottom of the coin is equal, but when sliding the coin along the circular path, the path length taken by the outer portion of the coin is longer in length than the path taken by the the inner portion of the coin, so doesn’t that mean the average path length taken by various point on the coin traveling in a circle was actually longer?

 

[Kontilera]

But sliding the coin along a straight line equal in length to 1 perimeter, the path length taken by the top and bottom of the coin is equal, but when sliding the coin along the circular path, the path length taken by the outer portion of the coin is longer in length than the path taken by the the inner portion of the coin, so doesn’t that mean the path length taken by the coin traveling in a circle was actually longer?

Again, you are mixing up concepts and making it too complicated.
You where asking why a coin makes two revolutions when rolling along the perimeter of a stationary coin.
Revolutions has to do with rotation. I.e. the question is equivalent to asking: Why does the coin rotate 720 degrees when following the circular path?
And that question has been answered some times now.

Keep it simple, use coins of the same size and focus on degrees as a measure of rotation.
A question you can investigate is:
How does the point that touches the inner coin change if you drag the outer coin along its perimeter without changing its orientation in the plane?

 

[Devin-M]

Think of a car driving around the earth. If the car is tall enough (thousands of miles high), the path length taken by the top of the car can be much larger than the path of the bottom of the car.

 

[Lnewqban]

I still don’t understand why if both perimeters are equal and there is no slippage how more than 1 revolution occurs unless the coin is indeed traveling further on the circular path than the circumference “unwound” into a straight path.

If you consider the length of the circular path of the center of mass of the moving coin (rather than the perimeter of the fixed coin), it should be $2\pi2radius$, which equals the length that any wheel should cover after completing two full turns.

 

[Nugatory]

Think of a car driving around the earth. If the car is tall enough (thousands of miles high), the path length taken by the top of the car can be much larger than the path of the bottom of the car.

And you will also notice that the car rotates once as it makes the circuit: just look at which direction it is pointing as it goes around. And that’s when it isn’t rolling, equivalent to keeping the same point on the moving coin (or the underside of the car) against the edge of the stationary coin (surface of the earth).

So that’s one rotation. If the coin is rolling without slipping, that’s a second rotation.

 

[Nugatory]

@Devin-M you’ve been posting frequently enough that it is clear that you have not been stopping to think about the answers you’re being given.

You have been blocked from posting in this thread for 24 hours, so you’ll have time to carefully read and understand what’s already in it.

 

[Kontilera]

Think of a car driving around the earth. If the car is tall enough (thousands of miles high), the path length taken by the top of the car can be much larger than the path of the bottom of the car.

Yes this is true and its the reason that a cars have differential steering:


But it feels like you are taking a detour now. Either you focus on the center of the coin or you focus on the touching point. Let me see if I can lead you back from this, we start by focusing on the center of the moving coin.

When a wheel, coin or any circular object rolls around a corner its center will travel a longer path than the actual curve its travelling along, as you suggest. Even though the coins has a circumference of $2\pi r$, the center of the outer coin will trace a path of length $4\pi r$. Thus, it make sense that it needs two revolutions in order to complete its motion. This is a statement that you seem to agree upon, based on your post above.
Let us from this conclude that:
The degrees that the coin has rotated is a measure of how far the center of the coin has travelled.

Now, lets focus in the point that touches the inner coin.

What makes people confused is the fact that the coin will rotate, for example, 360 degrees but only "travel 180 degrees" on the inner coin. This confusion stems from the fact that the degrees that the outer coin has rotated is by no means a measure for the length that the inner most points traces in the plane. Thus, the amount of rotation of the moving coin doenst have to coincide with the "degrees it has travelled" on the stationary coin. It is actually quite the opposite, the inner most point went from being on the lower side of the coin to the upper side of the coin, this yields a difference of 180 degrees between degrees rotated (of the moving coin) and degrees travelled (on the stationary coin).

The degrees travelled on the stationary coin measures the path traced out by the touching point on the moving coin. Whereas the degrees rotated by the moving coin measures the distance travelled by the center of that coin. Naturally there is a ration of 2:1 between these angles.

Haha, I think this is as clear as I can make it.
Think about it. If there are any more questions Im passing over the torch to somebody else now.

Take care.

 

[OmCheeto]

While I understand the '+1' rotation, I also consider the perspective from the rotating disk to be a legitimate answer. If you were sitting in a chair on the rotating disk, rolling around a disk of the same size, focusing only on the two discs, it would only take 1 rotation to complete 1 orbit. It's only when you take notice of the stationary universe around you that you see that you actually made two revolutions.

 

[PeroK]

While I understand the '+1' rotation, I also consider the perspective from the rotating disk to be a legitimate answer. If you were sitting in a chair on the rotating disk, rolling around a disk of the same size, focusing only on the two discs, it would only take 1 rotation to complete 1 orbit. It's only when you take notice of the stationary universe around you that you see that you actually made two revolutions.

It's a legitmate answer in the same way that the Earth has 365 days per year relative to the Sun, but 366 rotations on its axis (sidereal days) per year relative to the background stars. What you describe is a direct analogy to this.

When you do the experiment, which is a critical thing to do, you clearly have to rotate the outer coin twice round the inner coin. To ignore that seems unenlightening, to say the least. Especially if you were to insist that it only rotates once, when an experiment clearly shows two rotations.

 

[DaveE]

Imagine the case where a coin is rotating around a point on its circumference, i.e. another coin with radius equal to zero.

 

[Devin-M]

Imagine the case where a coin is rotating around a point on its circumference, i.e. another coin with radius equal to zero.

Yes or imagine the case where the stationary coin is 1/10th the diameter of the rolling coin. If the stationary coin’s circumference is unwound and straightened, wouldn’t a set of points on the moving coin travel a shorter distance rolling down the unwound straightened circumference versus rolling completely around the 1/10th circumference that’s wrapped into circle?

 

[Ibix]

Yes or imagine the case where the stationary coin is 1/10th the diameter of the rolling coin. If the stationary coin’s circumference is unwound and straightened, wouldn’t a set of points on the moving coin travel a shorter distance rolling down the unwound straightened circumference versus rolling completely around the 1/10th circumference that’s wrapped into circle?

Which set of points?

Look, if the "unwrapping" argument doesn't work for you, don't use it. There are plenty of others.

Try this. Two equal sized coins. The moving one starts above the central coin with the queen's head (or whatever) upright, so the "bottom" point of the coin is in contact with the central coin. It rolls half way round. If it moves without slipping, which point must be in contact? The "top" point, half way round the coin from the initial point, right? But the moving coin has also moved half way round the central coin so it's below it. So which way up is the picture? What's the minimum it must have rotated?

 

[Devin-M]

Which set of points?

Sorry for the crude drawing but this is my argument for why the moving coin traveling in a circle "goes further."

On the bottom right, the 1/10th size coin's circumference is unwrapped and straightened. If we follow the top most point on the "moving coin" it only forms a small arc to roll the entire circumference of the smaller coin.

On the left, if we follow the same path of the top most point of the moving coin, it takes a much *longer* path as it must go all the way around the other coin on the circular path.

 

[Ibix]

Yes. So what?

 

[Devin-M]

Yes. So what?

It just makes more intuitive sense to my mind where the extra rotation comes from if we can also say the curved path followed by the coin is longer than the “unwound” straight path.

 

[Kontilera]

Devin-M, I think all that really needs to be said regarding this problem was expressed in the following post by PeroK:
https://www.physicsforums.com/threa...try-question-on-sat-test.1057821/post-6972664
Your journey to intuitively digest this phenomena is maybe too personal for us in order to guide you.

Im sure you'll find insights that will make sense and can be formulated in sentences, but as long as you dont try to understand it from a mathematical point of view, a thread like this is prone to just repeat previous statements in different wordings.

At least thats my perspective at the moment.

PS. Please read this post with the best of intentions.

 

[Devin-M]

Interestingly in the 2 US quarters case if we compare the area "swept" by the full width of the moving coin between the case where the coin rolls around another identical coin or the coin rolls along a straight line equal in length to the coin's circumference, that swept area is as far as I can tell precisely double for the coin around another coin path, versus the straight path.