A Physics Puzzle: Double Circular Motion

AlexChandler

Describe a simple way to force a particle to move along two different circular trajectories at the same time. (Two circles that are not concentric) Hint: The trajectories do not have to be measured from the same frame of reference.

It need not be an elementary particle (could be even a small ball)

BobG

Or a planet?

http://faculty.fullerton.edu/cmcconnell/Planets.html#7a

AlexChandler

Haha i'm sorry but i'm afraid this is not correct.

Quite amusing though. Thank you.

Gad

A gum stuck on a tire of a car that keeps driving around a roundabout!

berkeman

Why is it not correct? Because he didn't say that the particle is sitting on the surface of the planet, and is not located at either pole?

Gad

He probably should've just said 'me' or 'you Mr. AlexChandler'.

BobG

Well, it is true that planets don't orbit the Earth via epicycles. In fact, they orbit the Sun in ellipses. In fact, it would be highly unlikely to find any particle that moved freely in a perfectly circular path.

Perfectly circular motion usually requires the particle to be fixed to the point it's circling.

But the link is an example of double circular motion, even if it doesn't reflect the real motion of the planets (nor was it intended to, since the emphasis was on the history of celestial mechanics).

And it would be simple to rig a device where the axes of a spinning gyroscope were fitted into a forked frame and the frame rotated about a fixed point.

AlexChandler

Hehe also a nice idea, but i'm afraid only the trajectory around the tire is circular. The trajectory around the roundabout is more like a cycloid.

JaredJames

Me on a merry go round spinning a ball on a string?

You're being picky. Strictly speaking you'll never get a perfect circle.

The above are all valid.

AlexChandler

But even if the epicycles were possible, this would not satisfy the puzzle. The epicycles would be circular trajectories measured in the frame of reference co-moving with the center of the epicycles, but the overall trajectory described from the frame of reference of the sun would not be circular at all.

Could you elaborate on the final idea you presented? I think it will also not work for a similar reason, but I do not fully understand what is meant by "fitted into a forked frame".

AlexChandler

I think the puzzle has been somewhat misunderstood. The point is to find a forced motion such that two distinct frames of reference could accurately describe the trajectory of the particle as circular, and the two circular trajectories must not have the same point in space as a center.

The trajectories do not have to be full circles, as long as the particle moves along two distinct circular arcs for some time.

AlexChandler

I am not being picky but the above mentioned motions do not satisfy the puzzle. I apologize if it is unclear what I am asking for, but I tried to be as exact as possible in my wording.

For example, you on the merry go round: In your frame of reference, the trajectory of the ball is absolutely a circle. But from the frame of reference of the center of the merry go round, the trajectory is very complicated, actually it is the same as thing as the epicycles described before. It is not circular. There is only one circular trajectory in each of these examples.

Gad

A motion of a dropped coin led me to this example. Imagine if there's a small body in the contact point between the coin and the surface of the floor, won't that body be moving two different circular trajectories!... Actually the body should be fixed while the two references move relatively to each other... URGH! I'd like to know the answer to this question, thanks for keeping me curious.

Jimmy Snyder

For example, if the ball is being spun about a horizontal axis even as the merry-go-round is rotating about a vertical axis. The center of the first frame of reference is the tip of of JaredJames' fingers holding the string. The center of the second frame is a point moving sinusoidally up and down through the axis of the merry-go-round in time with the rise and fall of the ball.

AlexChandler

Actually this does work. Great thinking! I thought at first that it would not work for the following reason: The oscillating frame of reference at the center of the merry go round will see that the ball moves back and forth along the circular trajectory. But actually if the merry go round is moving fast enough, the ball will only slow down in the FOR of the center of the merry go round, so the trajectory will be circular as measured from two distinct reference frames with two distinct centers. Great job! In fact I think that this answer is better than the one I had in mind. And in fact this would satisfy the puzzle even if the ball did travel back and forth along the trajectory. I am quite impressed.

JaredJames

Well done... uh... Jimmy.

AlexChandler

My solution to the problem is illustrated in the attached picture. It is just a simple machine with a roller an an elbow joint. View the small object attached to the center of the elbow joint from the frame or reference of both points O and P. Point O is held fixed with respect to the piece of paper, and point P is moving at a constant velocity to the right.

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