How many rotations...

[Jim Bertorelli]

Take two quarters. Keep one static and roll the other one on the edge of the first one. For one such complete circle, how many times does the second coin rotate on it's own axis?

[Ashik Uzzaman]

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Muhammad Ashikuzzaman (Fahim)
Sun Certified Programmer for the Java� 2 Platform
--When you learn something, learn it by heart!

[Greg Harris]

1

[Jim Bertorelli]

Not 2? Try it, see it and then reply again

[Jim Bertorelli]

Well, now read the question again and the think about your answer

[Jim Bertorelli]

yes..."on it's own" ie. the second or the rolling one's axis.
gotcha buddy...2 is not correct

[Greg Harris]

sorry, my answer of "1" was just a quick responce without any testing or logical thought. oops.
is this one of those trick questions where the answer is something like: "none, it is rotating about the static quarter's axis..." ?
it does indeed make 2 full "rotations" as far as the direction that the head is pointing, but it is really not rotating on "it's own axis" because it is rotating around the static quarter.

[Jim Bertorelli]

Actually, it rotates only once on it's axis but the reason we see it rotate twice is because, due to the curvature of it's path, it's axis itself also rotates once.
If you straighten up the static coin's edge, you'll see the rolling coin rotate only once. The distance traveled in both the cases is same.

[Jim Bertorelli]

but it is really not rotating on "it's own axis" because it is rotating around the static quarter.

It is indeed rotating on it's own axis...but only once. It's axis, while revolving around the static coin, makes another rotation.

[Michael Ernest]

This makes sense. The cirumference of a circle is pi * diameter. If I lay out a line that is a quarter's circumference long, and roll a quarter along it, I should get one full rotation of the coin.
[This message has been edited by Michael Ernest (edited December 10, 2001).]

[Jim Yingst]

The coin makes two rotations. To see this, start the coin with the top of Washington's head pointing upward. (I'm assuming a U.S. quarter here - those of you using other coins can substitute any distinguishing feature you want to; it doesn't really matter.) Now start rolling the coin around. When you get halfway, Washington's head will again be pointing upward. That's one rotation. When you roll the coin the rest of the way around, it completes its second rotation. The fact that the axis of the rolling coin is moving around the static coin is irrelevant when discussing the angular orientation of the rolling coin.

If I lay out a line that is a quarter's circumference long, and roll a quarter along it, I should get one full rotation of the coin.

Yup. And if you now bend the line back into a circle, keeping the quarter in contact with the line, and keeping the other end of the line (your starting point) fixed in space during this process (and not rotating), you will end up pulling the quarter through another full rotation, for a total of two rotations.
[This message has been edited by Jim Yingst (edited December 10, 2001).]

[shankar vembu]

hi,
the coin makes one rotation only.
Let me generalise this,
Let the static coin radius = r1
moving coin radius = r2
In 2*pi*r2 distance , the moving coin makes 1 rotation(own axis).
Now, it moves 2*pi*r1 distance, so no. of rotations = 2*pi*r1/2*pi*r2.
Since r1=r2, answer is one.
This is a theoretical approach though one can arrive at this solution without this.
=====
shankar.

[shankar vembu]

Hi,
the coin makes one rotation only.
Let me generalise this,
let the static coin radius = r1
moving coin radius = r2
In 2*pi*r2 distance, the moving coin makes one rotation on its own axis.
Now it has to move for 2*pi*r1 distance, so no. of rotations are 2*pi*r1/2*pi*r2.
since r1=r2 here, answer is 1.
This is a theoretical approach though the solution can be arrived at without this.
=========
shankar.

[shankar vembu]

oops, i missed the catch.
Its two rotations.
The moving coin has to move a distance of (r1+r2).
Since r1=r2, the answer is two i.e. 2*pi*(r1+r2)/2*pi*r2 = 2
========
shankar

[Jim Bertorelli]

The coind indeed turns twice. But only once on it's axis. This is how you can verify it:
1. Roll the coin over the straight wire from one end (point A) to another (point B) where AB=pi*D. The coin has now rotated on it's own axis once.
2. Now, move the B end of the wire so that the wire forms a circle (B meets A).
You'll notice that the coin was stationary...it didn't rotate at all. But it sure turns again. That's because it's frame of reference moved/rotated.
It's like asking how many trips you make round the sun a year?
Ans: 1.
On your own?
Ans: 0.

[Joel McNary]

Originally posted by Justin Poggioli:
Ok guys, I have the actual answer. It all depends on HOW you are rotating the coin A around coin B.
if the static coin is flat and the moving coin is ALSO flat so that the moving coin is travelling on the static coin then the moving coin makes 2 rotations.
HOWEVER, if your moving coin is at a 90 degree angle (vertical) travelling the line AROUND the coin, then yes it indeed only spins once. Try it.

To really see what is going on here, rotate the coin both of the above suggested ways, and you will see these results. Now, start the qurter vertically and rotate it halfway, so that it is "upside down." Now lay it flat. Ta-da! It is now "right side" up.
More to the point, if the static quarter represents the earth and Washington is rolling along the earth, he starts standing at the north pole. By the time he reaches the south pole, he is standing on his head (upside down), although to us it looks like he is right sde up because our point of reference is somewhere below the static quarter. His, however, is the center of the quarter (the center of the "earth"), and, although he looks like he is right side up, he really is upside down. He finishes back at the north pole right side up.
Boy, I can't type today.
[This message has been edited by Joel McNary (edited December 11, 2001).]

[Jim Yingst]

[responding to Jim Bertorelli's post]
Well OK, it's relative to your frame of reference. From a nonrotating frame of reference, there are two rotations. From a frame of reference attached to the rolling coin itself, that coin makes no rotations - but instead the "static" coin is perceived to rotate twice in the opposite direction. Your preferred frame of reference seems to be one which isn't attached to any physical object, but instead rotates to match the revolution of the rolling coin's center of mass about the center of the static coin. I suppose this is as valid a choice as any other rotating frame of reference - which is to say, not as good as a nonrotating frame of reference for most applications. You can do it, but there's no reason to assume that it's the only way to view the problem - or even the preferred way.
Consider instead: what if you hold the moving coin with your finger so that it cannot rotate, and instead you slide the coin around the static coin in a full circle - letting the edges of the coins slip against each other. Jim B - how would you describe this system? What reference frame seems most natural for you to use? I'm curious...

[This message has been edited by Jim Yingst (edited December 11, 2001).]

[Jim Bertorelli]

Jim Y:
I did not "choose" the frame of reference. It's given in the question itself. The questions asks "...on it's own axis".
So, on it own axis, it only makes one rotation.

[Jim Bertorelli]

Originally posted by Jim Yingst:
[responding to Jim Bertorelli's post]
Well OK, it's relative to your frame of reference. From a nonrotating frame of reference, there are two rotations.

2 rotations in total.

I suppose this is as valid a choice as any other rotating frame of reference - which is to say, not as good as a nonrotating frame of reference for most applications. You can do it, but there's no reason to assume that it's the only way to view the problem - or even the preferred way.

It is not the prefered/right/worng way. It is "the" way given in the question.

Consider instead: what if you hold the moving coin with your finger so that it cannot rotate, and instead you slide the coin around the static coin in a full circle - letting the edges of the coins slip against each other. Jim B - how would you describe this system? What reference frame seems most natural for you to use? I'm curious...

In this case, the coin does not rotate at all...on it's own axis that is. Again, you cannot leave the "frame of referece" to be chosen/assumed...otherwise the answers will be different. It has to be specified in the question.

[Joel McNary]

Originally posted by Jim Yingst:
Your preferred frame of reference seems to be one which isn't attached to any physical object, but instead rotates to match the revolution of the rolling coin's center of mass about the center of the static coin. I suppose this is as valid a choice as any other rotating frame of reference - which is to say, not as good as a nonrotating frame of reference for most applications.

If I were the one rolling around the earth, my point of reference would certainly be me! In that case, I would have rolled (or flipped, or whatever) only once--I start with my head away from the earth and finish with my head away from the earth. Just because I am rotating and revolving at the same time does not increase my rotations; I still rotate only once. I just happen to also revolve once.

[Mapraputa Is]

Interesting. My first thoughts were exactly like Michael E. put it: uncurl a line and it's obvious that there is one rotation. Then Jim Y. said there are two. This idea deeply bothered me, because I believed there Is a 1-to-1 Map between points on the first and second coin circumferences. When we roll the coin on the edge of the other one, all points on both coin edges, without exception - moving is continuous, meet and there is no source of extra points, so it's 1-to-1 relationship. Then how it can be 2 rotations? No, really, how can we have two rotations?
Well, let's ask ourselves, how many points are there on the edge? The answer is: infinitely many. And with infinity, sometimes a part is equal to the whole. "Galileo paradox" can serve as an example. He was the first who figured out that there are as many squares of whole numbers, as there are whole numbers. He associated a whole number with each square: 1 -> 1, 2 -> 4, 3 -> 9, and so on. But not any whole number is a square of some other number, so there must be less squares than just numbers? We see that our intuition doesn't work well with infinity, and two rotations are not impossible.
(To better illustrate how the coin can roll twice along a line of it's circumferences length, we can modify Galileo's method and build an association between even integers and all integers: 1 -> 2, 2 -> 4, 3 -> 6, 4 -> 8... Similarly, we can show that 3, 4, 5... rotations are also possible.)
Ahhhhh... <- Map runs away very quickly...
[This message has been edited by Mapraputa Is (edited December 11, 2001).]

[Gregg Bolinger]

Man, there are a lot of people that get paid to roll quarters around each other. I wish I had time at work to do that. Wait a minute, I just read this entire forum. How silly of me, I could have been rolling quarters.
I just don't know how to utilize time I guess.

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Happy Coding,
Gregg Bolinger

[Jim Yingst]

Gregg, we here in Meaningless Drivel will be happy to offer our services in time management counseling. You have but to ask.
It seems most of us, including Jim Bertorelli, are in agreement that there are two rotations total, as seen by an outside stationary observer - but there is disagreement about whether that's what the question asked.

It is not the prefered/right/worng way. It is "the" way given in the question.

I disagree. I assume you're referring to the repeated use of the phrase "on its own axis"? That doesn't imply that we whould use a rotating frame of reference - in fact, the phrase was pretty much redundant once you said "rotate". Absent any meaningful indication of a preferred reference frame, a nonrotating frame is standard. Special relativity says your reference frame can have any velocity you want, but acceleration and rotation violate the warranty. If you really wanted something else, you should have said so.
I'm still trying to understand the mysterious properties of "rotation about an axis" as used here. Regarding the original problem you state:

Actually, it rotates only once on it's axis but the reason we see it rotate twice is because, due to the curvature of it's path, it's axis itself also rotates once.

OK. So 1 rotation about its axis + 1 rotation of the axis itself = 2 rotations as seen by us? 1 + 1 = 2 -- this seems to make sense. Can you make a similar statement about my second problem with the sliding coin and no rotation?
It seems to me that the axis of the moving coin is still making one revolution about the center coin - does this mean the axis itself rotates once? (Whatever that means.) Do we agree that the total number of rotations in this second case (as seen by an outsider) is zero? And you say that the coin also makes no rotations about its axis in this case. This seems to suggest that 1 + 0 = 0. I assume you will disagree with this assessment, but I'm curious how you'd resolve the discrepancy.
My own answer is that "rotations seen by a stationary outside observer" and "rotations about its axis" are synonymous, and the value of each is zero in the latter case (and 2 in the original problem). I have no idea what you mean by an axis "itself rotating", and so I'm trying to find out how you determine whether an axis is "itself rotating" or not (and by how much).

If I were the one rolling around the earth, my point of reference would certainly be me!

If you were fully using yourself as a reference frame, then you wouldn't refer to yourself as rolling - you'd refer to the earth rolling around you. Of course, you quite reasonably mentally detach yourself from your own rotations, and instead use a reference frame defined by the curvature of the earth. This is quite natural - gravity has a tendency to encourage us to think this way, after all, as we're apt to hurt ourselves otherwise. If the original problem had involved rolling around the earth, then there would be more reason to accept this reference frame as a natural default. But the gravity from a quarter is pretty insignificant, and I see no reason to use it to define my reference frame when I've got a perfectly good stationary one already.

[This message has been edited by Jim Yingst (edited December 12, 2001).]

[Jim Bertorelli]

Jim Y: I don't understand where is the descrip. I agree, however, that "on it's own axis" is redundant since rotate means on it's axis. Still the answer is 1. This can be verified by the experiment that I gave above. I'll repeat it here:

1. Roll the coin over the straight wire from one end (point A) to another (point B) where AB=pi*D. The coin has now rotated on it's own axis once.
2. Now, move the B end of the wire so that the wire forms a circle (B meets A).

I hope you'll agree that, if the coin rotates on a surface:
1. Either it'll move forward/backword; according to the direction of the rotation.
2. OR the coin will stay put AND there will be a slippage between the coin and the surface.
In Step 2 of the experiment that I described above, neither the coin moves nor it slips.
This shows that the coin did not rotate in step 2. It only rotated in step 1.

Now, coming your experiment. I said, "In this case, the coin does not rotate at all...on it's own axis that is." Now that "on it's axis" is redundant, the statement should be "the coin does not rotate". Which, I believe, is correct for the following reason:
Again straighten up the path and slide (do not rotate) the coin from A to B. How many times does it rotate? 0. There is a slippage between the coin and the surface BUT the coin is not stationary and that's why the coin does not rotate.

[Jim Yingst]

I hope you'll agree that, if the coin rotates on a surface:
1. Either it'll move forward/backword; according to the direction of the rotation.
2. OR the coin will stay put AND there will be a slippage between the coin and the surface.

Or the surface itself is moving. Or some combination of the above. I fail to see where that proves anything. In the two-step process, the moving surface allows another rotation to take place even with no translational motion, and no slippage.
I mean, look at a parked car with the engine running. There are a number of pulleys linked by the fan belt. These pulleys do not move forward or backward, and there is no slippage with the fan belt (assuming an ideal fan belt ). Are you saying that these pulleys do not rotate?
From your first description of this procedure:

You'll notice that the coin was stationary...it didn't rotate at all. But it sure turns again. That's because it's frame of reference moved/rotated.

There is a difference between turning and rotating? That's news to me. How does one tell the difference?
Regarding my second example, I don't see a clear response to the question about 1 + 0 = 0. That's
1 revolution of the axis about the center coin + 0 rotations about its own axis = 0 rotations total. Which term would you correct?
And what about a system like the moon, which keeps the same face pointed to the earth throughout its orbit? You can move the quarter similarly, so that as it revolves around the center coin, Washington's head keeps pointing at the center coin. How many rotations are there total? How many "about its axis"? How many rotations "of the axis itself"? For reference, my answers are "1", "1", and "this phrase is meaningless". It's because I don't understand what mean by these quantities that I'm asking this.
[This message has been edited by Jim Yingst (edited December 12, 2001).]

[Jim Bertorelli]

Originally posted by Jim Yingst:
I hope you'll agree that, if the coin rotates on a surface:
1. Either it'll move forward/backword; according to the direction of the rotation.
2. OR the coin will stay put AND there will be a slippage between the coin and the surface.
Or the surface itself is moving. Or some combination of the above. I fail to see where that proves anything. In the two-step process, the moving surface allows another rotation to take place even with no translational motion, and no slippage.
I mean, look at a parked car with the engine running. There are a number of pulleys linked by the fan belt. These pulleys do not move forward or backward, and there is no slippage with the fan belt (assuming an ideal fan belt ). Are you saying that these pulleys do not rotate?

The pulleys do rotate but if you give it a thought you'll understand that it is not much different than rolling the coin over a surface. The only difference is instead of the coin, the surface (the belt) is moving. If you stop the belt and still rotate the pulley, there will be a slippage.
Why is it so difficult to understand? If you take a big crane and rotate the car, will you say that the pulley rotated?
Moving the wire (joind B to A) is same as moving the car using a crane.

From your first description of this procedure:
You'll notice that the coin was stationary...it didn't rotate at all. But it sure turns again. That's because it's frame of reference moved/rotated.

There is a difference between turning and rotating? That's news to me. How does one tell the difference?

Now, you are just picking on me. By turning I wanted to refer the fact that although we "see" the head rotate twice but actually it rotates only once. I used the word turn only because I did not want to say rotate. I wanted to distinguish between the observation and the actual thing.

Regarding my second example, I don't see a clear response to the question about 1 + 0 = 0. That's
1 revolution of the axis about the center coin + 0 rotations about its own axis = 0 rotations total. Which term would you correct?

I couldn't reply to it because I don't understand what you mean by saying 1 + 0 = 0
But I did reply to your basic idea of preventing the coin from rotating and sliding it over the surface. In that case, again, the coin does not rotate at all.
In fact, you did not answer to what happens then we move the wire (end B) to form a circle. Does this constitue a rotation of the coin or not? I think, it does not.

And what about a system like the moon, which keeps the same face pointed to the earth throughout its orbit? You can move the quarter similarly, so that as it revolves around the center coin, Washington's head keeps pointing at the center coin.

This comparison is totally wrong. Although we see the same face of the moon and the moon does rotate, but this is because Earth also rotates at the same time.
In our case, the static coin is static. It is not rotating.

How many rotations are there total? How many "about its axis"? How many rotations "of the axis itself"? For reference, my answers are "1", "1", and "this phrase is meaningless". It's because I don't understand what mean by these quantities that I'm asking this.

We sorted out in the earlier post that "on it's axis" is redundant and "rotation" implies "on it's axis". Why are you bringing this again???

[This message has been edited by Jim Bertorelli (edited December 12, 2001).]