# How to calculate object displacement due to centripetal force of moving ball inside it?

Please note that I have little background in Physics and math and after searching for over a month I couldn't find anything that relates to what I am looking for so turning to Stackexchange for some reference material that I can study.

My questions is, I have a circular track/pipe and a ball that fits exactly in it with enough gap to allow ball to move inside pipe. Now two scenarios here, one is exactly Half circle track and another is not half but not full circle either (not sure what it is called), as in below image.

Lets say if I push ball from one end of track and ball travels through length of track to the end there has to be centripetal force applied on track itself that will result in moving/wobbling of the track. What I am trying to determine is how will the move look like, I believe it should be circular which follow path of moving ball and not linear. Also how do I calculate how much track deviates from its position. in my head I can see it move like a hula hoop around waste but I just cannot find formulas or even reference for calculations. I might be searching with incorrect terms, keywords.

I am not asking for build me up a formula, I rather prefer any way you can explain what should I do to come up with a solution. I would like to later experiment with different speed and mass of ball so its better for me to understand the process.

 As a user pointed out, here is the missing information:

• For theoretical purpose please consider track is in hypothetical environment such as vacuum of space. There is no gravity, no direction, no air so no friction etc. I want to start with learning basic motion first. As I learn more over time I will keep adding these other parameters such as friction etc.

• I didnt think I would need to be specific with mass, speed of ball etc. since I was just looking for effect on track in terms of movement. Anyway here we go. These numbers are hypothetical for me to understand the behavior of the ball and track.

The ball is made of steel, 1 inch in diameter, weight of 50g, and starting velocity of 1m/s. Track is not bolted, and is free to move with ball's movement and applied force. Track made of wood and weighs 50g. Please ignore ball slowing down while moving due to friction, because I am not looking to calculate friction loss etc. Just keeping it simple.

Thanks!

• You question only makes sense if the track is not fastened down -- but you don't specify how the track is situated. Could you edit your question to clarify this? I.e., is the track on a frictionless, level table? Is it sitting on a table but has some friction? Is it actually fastened down -- in which case it won't move after all, and your question is moot? Commented Jul 5, 2023 at 17:03
• Thanks for your comment @TimWescott. Just edited my question. Hope that makes sense.
– Rob
Commented Jul 5, 2023 at 17:39

This depends a great deal on mass:

If the track has a huge mass (perhaps thick iron pipe), and the ball has almost no mass (such as a table tennis ball), then the track will remain essentially motionless. The very small force between the track and ball is large enough to continuously change the direction of the ball along the track's path, but it cannot cause the track to begin moving enough to notice.

Now consider the reverse. Let the ball be very massive with a track that has almost no mass. The ball will move in a straight line while the track rotates to prevent the ball from breaking the track. The force cannot alter the motion of the heavy ball, but it is definitely enough to push the track out of the way.

Your question is highly hypothetical, and will definitely depend a lot on the details.

But we can state some general principles that will greatly aid any attempt at a solution. The conservation of linear and rotational momentum, and the conservation of energy, will be your guideposts. Together, they also imply that the centre of mass will only ever move inertially, if not taken to be completely fixed.

It is, however, likely that, even though those conservation laws are supposed to uniquely determine the actual trajectory, your analysis will run into problems. The collision of the ball with the walls of the pipe will be incredibly difficult to model even with a modern computer, at alone match up with reality.

I am not sure what would be the point of such a thought experiment.