Solving a Rolling Coin Problem: Is it 3 or 4?

[Carbon123]

Homework Statement

A coin of radius R rolls around a coin of radius 3R which is fixed.If the coin rolls without slipping, How many times does the coin rotates around its axis ?

Homework Equations

Xpm=RΘ for non slipping

The Attempt at a Solution

The center of mass moves a distance of 2π(r+3r), which is equal to rΘ because it is non slipping,thus Θ=4x2π which is four full rotation.Would this be true ? (Some say it is 3 times)
Thanks in advance

 

[Daniel Gallimore]

The circumference of the larger coin is 2\pi(3R) or 6\pi R. The circumference of the smaller coin is just 2\pi R. If the coin rolls without slipping, then after the coin has rotated once, it should have traversed a distance of 2\pi R. Hence, after it has rotated three times, it will have traversed a distance of 6\pi R, the circumference of the larger coin.

 

[TSny]

The center of mass moves a distance of 2π(r+3r), which is equal to rΘ because it is non slipping,thus Θ=4x2π which is four full rotation.Would this be true ?

Yes, 4 rotations. I like your explanation.

Here's a visualization for when the larger coin's radius is 4 times the radius of the smaller coin. https://www.geogebra.org/m/v3a437ux

 

[Daniel Gallimore]

Yes, 4 rotations. I like your explanation.

Here's a visualization for when the larger coin's radius is 4 times the radius of the smaller coin. https://www.geogebra.org/m/v3a437ux

The radius of the larger coin in three times that of the smaller coin, not four.

 

[TSny]

The radius of the larger coin in three times that of the smaller coin, not four.

Yes. The visualization is just another example that I found on the net. It was not meant to represent the setup of the original question. In the visualization, how many revolutions does the small coin make when going once around the larger coin. Is this what you expect?

 

[ehild]

Homework Statement

A coin of radius R rolls around a coin of radius 3R which is fixed.If the coin rolls without slipping, How many times does the coin rotates around its axis ?

Homework Equations

Xpm=RΘ for non slipping

The Attempt at a Solution

The center of mass moves a distance of 2π(r+3r), which is equal to rΘ because it is non slipping,thus Θ=4x2π which is four full rotation.Would this be true ? (Some say it is 3 times)
Thanks in advance

Yes, you are right. The distance traveled by the center of mass is the same as the turning angle multiplied by the radius of the rolling coin.

 

[haruspex]

The circumference of the larger coin is 2\pi(3R) or 6\pi R. The circumference of the smaller coin is just 2\pi R. If the coin rolls without slipping, then after the coin has rotated once, it should have traversed a distance of 2\pi R. Hence, after it has rotated three times, it will have traversed a distance of 6\pi R, the circumference of the larger coin.

Imagine the smaller circle attached at point A on the larger. Cut the larger circle at A to form endpoints A, A'. Keeping A and the attached small circle fixed, unroll the circumference of the larger circle into a straight line AA' of length 6\pi R. What you say above then applies: as the smaller circle rolls along the line from A to A' it rotates three times. But now we need to roll the 6\pi R line back into being the circumference of a circle, keeping the smaller circle attached at A'. The smaller circle rotates once more in the process.

 

[Carbon123]

Thanks for the help,everyone.So it would be correct to say that the displacement of the center of mass is equal to rΘ for any non slipping motion(with respect to a non moving rotating surface )even if the surface is not a line (a curved line,for example) ?

 

[Daniel Gallimore]

Imagine the smaller circle attached at point A on the larger. Cut the larger circle at A to form endpoints A, A'. Keeping A and the attached small circle fixed, unroll the circumference of the larger circle into a straight line AA' of length 6\pi R. What you say above then applies: as the smaller circle rolls along the line from A to A' it rotates three times. But now we need to roll the 6\pi R line back into being the circumference of a circle, keeping the smaller circle attached at A'. The smaller circle rotates once more in the process.

Can you explain why the smaller circle rotates once more in the process of reattaching the endpoints? It seem this operation should have nothing to do with the smaller circle at all. The smaller circle has already rolled across the entire length of the circumference. Many thanks.

 

[TSny]

Can you explain why the smaller circle rotates once more in the process of reattaching the endpoints? It seem this operation should have nothing to do with the smaller circle at all. The smaller circle has already rolled across the entire length of the circumference. Many thanks.

See post #7 for a nice way to look at it.

Or, suppose you slide the smaller coin around the larger coin so that the smaller coin always has the same point in contact with the edge-surface of the larger coin. Thus, the smaller coin slips around the larger coin without rolling at all. Yet, the smaller coin still makes one rotation while going around the larger coin. You can think of this as the extra rotation that occurs when you include rolling without slipping.

 

[TSny]

So it would be correct to say that the displacement of the center of mass is equal to rΘ for any non slipping motion(with respect to a non moving rotating surface )even if the surface is not a line (a curved line,for example) ?

Yes, I think that's correct if by "displacement" you mean distance traveled. But I hope I'm not overlooking something.

 

[Daniel Gallimore]

See post #7 for a nice way to look at it.

Or, suppose you slide the smaller coin around the larger coin so that the smaller coin always has the same point in contact with the edge-surface of the larger coin. Thus, the smaller coin slips around the larger coin without rolling at all. Yet, the smaller coin still makes one rotation while going around the larger coin. You can think of this as the extra rotation that occurs when you include rolling without slipping.

I think I understand. It's essentially the same reason why the moon stays tidally locked with the earth: it's not the the moon isn't rotating; rather, it's rotating just as quickly as it's revolving around the earth. We can hop into the reference frame of the circle as it slides along the surface of the larger coin and perceive no rotation because we are rotating with the circle.