Random question regarding centrifugal force.

[Hayes]

Hello, I am somewhat new to this forum but have a basic question that I have had for a long time.
please excuse it if it is a dumb question.

I understand how centrifugal force works - you spin something around and it wants to fly away from the center.
And I understand that there are mathematics equations that explain this fairly well.
My question has to do with multiple centrifugal forces at once, and the only way I know how to describe what i am talking about is to describe an experiment - but I don't know what the outcome would be. Here goes.

If you took a motor with a long arm on it - say 4-5 feet, and spun it around at 30-40 RPM. the end of the rod would have centrifugal force pulling it away from the motor- let's call this the primary rod. What if you then put a smaller motor at the end of that primary rod with another rod on it also spinning (of course weighted properly so it wasn't unbalanced) it would also have its own centrifugal force acting on it call this the secondary rod. The problem is that now the 2 "fields" (for lack of a better term) of centrifugal force would be adding to each other and subtracting from each other with each revolution. And this is because the secondary rod would be swinging away from the primary motor half the time. (lets call this the first problem)

Now what if you added a third motor and rod and at the end of the third rod you had a sealed box with a ping pong ball inside. the question is, could you properly balance out the revolutions of the second rod so that the first problem was negated? could you ever have the centrifugal force from the first and third rod cancel each other out? Could you ever get the ping pong ball to not be squished up against the wall of the box, but rather sitting somewhere in the middle simply due to centrifugal force?

If you are still having trouble seeing what i am talking about, imagine a 6 foot pole with a motor in the middle. the pole had a motor at one end as well attached to it, and the other end has a weight. the motor on the end of the pole has a smaller pole on it with another small motor at the end of the smaller pole (and a weight at the other end.)

 

[jambaugh]

The centrifugal force is often described as a "fictional force" or a "pseudo-force" because when you watch from an inertial frame outside the rotating mechanisms you will not see a centrifugal force at all, you will see the object being accelerated around by centripetal forces.

So in your spinning rod on a spinning rod example If you set it up and watch you will see the rods pulling the object along the looping path with the real tensile force of each rod toward its axis point. But from the perspective of someone sitting on the first rod's axis point he see's this tensile force of the first rod keeping the second rod's center motionless against the centrifugal force that he perceives. But he sees the object being accelerated about that point without any secondary force.

Now consider an observer sitting on the second rods axis point. He feels the effect of the first stage centrifugal force, and the must hold on (feel the centripetal force) to keep from "flying off (along a straight line)". He also, by rotating with the 2nd rod perceives a 2nd order centrifugal force, added to the original one and pushing objects away from him. These will add as vectors and you can solve the problem of what net "force" by adding components. But notice that this second observer is no longer just in a rotating frame, he is in a frame rotating about a point rotating about another point and thus we shouldn't call the composite pseudo-force "centrifugal" any more.
It is just the net apparent force experienced by the observer in a complexly accelerating frame.

Likewise when you incorporate a third axis of rotation, and so on.

Now to answer your specific question, as far as balancing and canceling out, you'd only be able to do so for an instant in time. To have them cancel complete for more than an instant the observed object must (from an outside inertial observer's perspective) be moving in a uniform straight line with constant speed. You can create such an arrangement with multiple bars and axles but I do not think the rotations will be uniform around each axis, the rates of rotation must change as the objects move.

 

[jbriggs444]

Trivial solution: Let the first rod be four feet long. Let the second rod be three feet long.i Let the third rod be one foot long. Set the first rod to spinning. Lock up the second motor with the second rod at a 180 degree angle -- point back up along the first rod. Lock up the third motor with the third rod at a 0 degree angle, continuing the rest of the way to the original axis. The box remains at rest at the axis and the ping pong ball stays at rest in the center.

Less trivial solution. Imagine that you are pedaling a bicycle. The crank shaft rotates at a constant rate with pedals rotating around the shaft (pedal to shaft is rod one). Your lower leg goes from pedal to knee joint (lower leg is rod two). Your upper leg goes from knee joint to hip (upper leg is rod three). You stick a box in your pocket with a ping pong ball inside. The ping pong ball is subject to no centrifugal force.

I do not believe that there is a solution with three constant distinct rotation rates (and three non-zero rod-lengths).

Edit: Note that in the field of astronomy there is a well known term for this sort of arrangement.

 

[Bandersnatch]

Trivial solution: Let the first rod be four feet long. Let the second rod be three feet long.i Let the third rod be one foot long. Set the first rod to spinning. Lock up the second motor with the second rod at a 180 degree angle -- point back up along the first rod. Lock up the third motor with the third rod at a 0 degree angle, continuing the rest of the way to the original axis. The box remains at rest at the axis and the ping pong ball stays at rest in the center.

That's the same solution I was thinking about, but am not sure about your choice to 'lock up' the second and third stage. These motors need to be rotating, at the same angular velocity as the first one, don't they?
That is, you could have a setup with locked motors, which is essentially the same as a single rigid rotating rod with the ping-pong ball sitting at the centre of rotation and thus experiencing no centrifugal forces at its centre of mass. If you, however, allow the 2nd and 3rd stages to freely rotate, without imparting angular velocity (with motors acting against air drag, or without motors in vacuum) they would not stay aligned.
The end effect is the same, though.

 

[Hayes]

It is just the net apparent force experienced by the observer in a complexly accelerating frame.

Likewise when you incorporate a third axis of rotation, and so on.

Now to answer your specific question, as far as balancing and canceling out, you'd only be able to do so for an instant in time. To have them cancel complete for more than an instant the observed object must (from an outside inertial observer's perspective) be moving in a uniform straight line with constant speed. You can create such an arrangement with multiple bars and axles but I do not think the rotations will be uniform around each axis, the rates of rotation must change as the objects move.

Is there a mathematical equation for this that I can look at further? I know that my model is a vast oversimplification of the question I am trying to ask, but I am wondering if it would be possible to achieve the end result of what I am talking about (keep an object in the middle of a closed system). I am also wondering if you could expound on stating that it would only be possible for an instant in time in a straight line?

 

[jbriggs444]

That is, you could have a setup with locked motors, which is essentially the same as a single rigid rotating rod with the ping-pong ball sitting at the centre of rotation

Yes. I was assuming that each motor was attached to one rod and was powering/controlling the relative motion of the next.

 

[Hayes]

I do not believe that there is a solution with three constant distinct rotation rates (and three non-zero rod-lengths).

I can see what you mean, but what if there were more than 3? and also, if you had more than 3, say, 10, 15, 20, although they all have a fixed constant rotation rate, each ones individual rotation rate would effect the "constantness" of the next link in the system would it not? Because the addition/subtraction of forces (yes I know not real forces) would change the relative acceleration/deceleration of the next part of the system. does this mean that there ultimately could be a solution?

 

[jbriggs444]

I can see what you mean, but what if there were more than 3? and also, if you had more than 3, say, 10, 15, 20, although they all have a fixed constant rotation rate, each ones individual rotation rate would effect the "constantness" of the next link in the system would it not? Because the addition/subtraction of forces (yes I know not real forces) would change the relative acceleration/deceleration of the next part of the system. does this mean that there ultimately could be a solution?

I alluded to an approximate solution with the comment about astronomy. Think "epicycles".

Fourier analysis is another way of thinking about it -- sum up a bunch of sine functions to approximate a straight line.

 

[Hayes]

Also, would not the final end point "box with object in the center" be vibrating at a very high speed? as with each subsequent arm that was added, the lateral motion of the final product would be multiplied ... by an exponential amount? or X2? (again I am very ignorant of the math here, but still very curious, so please excuse any dumb questions)