# Uniform Circular motion

What i am going to write is an extremely basic question. It's not a paradox. But why I thought of writing it here, is because meny people go wrong in answering it. Let's see how many of you go wrong here.

OK, the question is as simple as this :-

You are in space surrounded by an isotropic medium of mass such that there is no gravity acting on you and on the small piece of stone you are holding. Somehow, you had, when you were young, borrowed a massless string from the Alpha Centaurians and you have that string in your pocket right now. So you take the string out, tie the mass to one of its ends, hold the other end and rotate it. The scene is this :- the stone is rotating about your hand with a constant speed of 'u'. The entire string is lying in the plane traced by the circular trajectory of the stone. Basically, it's a normal uniform circular motion.

List down the forces acting on the stone.

If you got an answer at this stage, of course, you are wrong because the question is still ambiguous.

The forces will depend on the reference frame from which you are looking at it. So I need to specify the reference frame.

OK, it's the frame from which you look stationary.

Now list the forces.

Aren't the forces independent of any inertial frame of reference?
The gravitational forces cancel each other, so the only force on the stone is the tension of the string that keeps it in circular motion. Am I wrong?

If you got an answer at this stage, of course, you are wrong because the question is still ambiguous.

This statement is cocky and wrong, two things that should never go together in physics. If you are in an accelerated reference frame, it is meaningless to talk about "forces" because forces are a concept from Newtonian Mechanics, and Newtonian Mechanics was only designed to hold true in inertial reference frames.

The forces will depend on the reference frame from which you are looking at it.
This statement is incoherent, and statements like this are apt to mislead people. The forces are invariant in any inertial frame of reference, which are the only frames for which they make sense. If you would like to treat the problem from an accelerated frame of reference, don't call them forces. Study General Relativity, which is a theory of accelerated frames of reference more so then a theory of gravitation.

That being said, the only force acting on the stone is the tension of the string.

To me, it seems like you are a cocky guy who wants to bait a beginner in to saying "the objects speed isnt changing, so the sum of the forces is zero. Therefore the tension in the string balances the centrifugal force." so that you can stride in and tell him he is wrong.

Or, if you have some other point to make, please make it.

dextercioby
Homework Helper
Crosson said:
Study General Relativity, which is a theory of accelerated frames of reference more so then a theory of gravitation.
Incorrect,according to the principle of equivalence (what does this word mean to you ?),the theories are on equal footing...

Crosson said:
That being said, the only force acting on the stone is the tension of the string.

Daniel.

pervect
Staff Emeritus
It's certainly possible to do mechanics in non-inertial frames and arbitrary coordinate systems, and it's not even necessary to use general relativity. Lagrangian mechanics, with its generalized coordinates and generalized forces, is an example.

However, it's generally fairly standard to write down the forces on an object in an inertial frame. When this is done, it's not necessary to specify a coordiante system - any inertial frame will agree about the forces.

Thus the several posters who point out that the only force is the force due to the string are therefore correct, using the usual conventions.

I think we agree on this point dextercioby: General Relativity is a theory that expresses the laws of physics in such a way that is invariant relative to the motion of the observer. Because of the equivalence principle, the "force" of
gravitation is an accelerated reference frame and that is how GR came to be a theory of gravitation. I made the comment I did because Einstein began the theory to express the laws of physics in a way that was invariant with respect to the motion of the observer.

Why is it puzzling that the only force on the stone is the tension in the string? Would you like me to call it the electromagnetic force, because that is of course what it is on a fundamental level. If instead, you expect that there is a "centrifugal force", then tell me what is the cause of such a force?

dextercioby
Homework Helper
The tension IN THE STRING CANNOT ACT ON ANY OTHER OBJECT,BUT THE STRING...Ever heard of REACTION FORCES (u may wanna check the 3-rd principle of Newtonian dynamics).

Daniel.

P.S.It doesn't matter what Einstein thought on the period of 9 years (i haven't read his GR-related articles) ,it matters what resulted and it resulted:gravity on equal footing with general covariance...

The tension IN THE STRING CANNOT ACT ON ANY OTHER OBJECT,BUT THE STRING
Rigid objects don't exert net forces on themselves, by the 3rd law you cite. For any force they exert on themselves, themselves exert an equal and opposite force on they; therefore the sum of the forces is zero.

,it matters what resulted and it resulted:gravity on equal footing with general covariance...
Blah, blah, blah....the point of the theory is that gravity is an accelerated frame of reference. You say it is a theory of Gravity, I say it is a theory of accelerated reference frames. We are both correct, because these are the same thing (actually, when I say it is a theory of accelerated frames I am more correct because this includes gravity as a small subset but also includes centripedal frames etc).

Of course, the forces acting on a body do not depend on the reference frame. But as someone of you said, the Newton's Laws are not valid in accelerated frames. So what some smart people have done, is that they have made the concept of 'pseudo forces'. These are not real forces. But if you are in an accelerated reference frame, you can consider the pseude forces and the newton's laws will become valid.

You may like to argue that if someone asks you to list down the forces acting on a body, you will just tell him the real forces acting on it, not the pseudo forces. But the thing is that the concept of pseodo forces is so common in some places that it becomes misleading if you don't mention them. One of those cases is uniform circular motion (remember centrifugal force?) Another case is rotation (coriollis force) etc. etc.

With circular motion, it is a misconception amongst several people that as soon as a body starts going round with a constant speed, a mesterious force called the centrifugal force suddenly starts acting on it outward. For them, my question would have been good.

But here there are smarter people than I had thought. The answer IS that the only force acting is the tension of the string.

Doc Al
Mentor
dextercioby said:
The tension IN THE STRING CANNOT ACT ON ANY OTHER OBJECT,BUT THE STRING...Ever heard of REACTION FORCES (u may wanna check the 3-rd principle of Newtonian dynamics).
Give the attitude a rest, Daniel. Yes, the tension in the string is a property of the string, but it is common practice to call the force exerted by the string "the tension in the string". The only force acting on the twirling stone is that applied by the string, due to its tension. To object to calling that force "the tension" is overly pedantic.

krab
vinter said:
But here there are smarter people than I had thought. The answer IS that the only force acting is the tension of the string.
It sure is a relief to know that we passed the test.

Now I've a question for you.
vinter said:
You are in space surrounded by an isotropic medium of mass such that there is no gravity acting on you and on the small piece of stone you are holding. ... So you take the string out, tie the mass to one of its ends, hold the other end and rotate it.
With nothing to stand on, how do you propose to start the stone rotating? And once it is rotating, how do you know whether the stone is going around you or you are going around the stone?

I apologise if I sounded arrogant here. That was just for fun..
Anyways, we can still have some good discussions on the topic.
Here are some nice questions posted by krab. Let me try to answer them...

The idea behind you being in space with nothing to stand on was to have zero gravity. If that affected something else, that means my model was inefficient. So to remove this inefficiency, I may say that magically enough, you start rotating the stone. Or I may say that you are somehow fixed to that medium of mass such that you remain stationary with respect to the medium. Or I may give any other explanation. And you may agree, that what exact explanation I give, is not going to affect the actual question.

Once having fixed your position with respect to the frame of reference from which you are observing the stone, there is no question about what is rotating about what. From this particular reference frame, it appears that the stone is 'going around' some fixed point. The definition of 'going around' being that the distance between the stone and the point remains constant.

I think this specifies a unique system quite rigorously.

Indeed, the hypothetical observer would be able to spin the mass, with a small catch: conservation of angular momentum dictates that the observer-string-mass system rotate slowly in the opposite direction that the string-mass is rotating.

Krab, the observer knows he is not going around the stone because he is at rest with respect to the fixed stars. This is mach's principle, that it is the fixed stars that determine acceleration (acceleration is absolute wrt the distant stars).

krab
Crosson said:
Krab, the observer knows he is not going around the stone because he is at rest with respect to the fixed stars. This is mach's principle, that it is the fixed stars that determine acceleration (acceleration is absolute wrt the distant stars).
Exactly. (I was hoping to hear vinter say that...)

Which stars are you talking about? I have not mentioned any stars here.

Even if I say now that there are stars, how do you know whether they are fixed? As we know, the 'red shift' suggests that the universe is expanding and that the stars are going away from us.

Which stars are you talking about? I have not mentioned any stars here.
How funny that you should say this. Ernest Mach, whos principle it I invoked in the first place, reminded us that there are no toy universes; only the real one in which the fixed stars are invariably present. Make no assumptions, in a toy universe devoid of matter (other than the test subject) it is arguable that inertia does not exists.

Even if I say now that there are stars, how do you know whether they are fixed? As we know, the 'red shift' suggests that the universe is expanding and that the stars are going away from us.
The red shift is caused by the expansion of the universe, not by galaxies moving away from us with pseudo velocity. In any case, the expansion is isotropic and I see no reason why this is relevant to mach's principle.

The whole point is that there is only this universe, and we have inertia because of the way the universe is: full of stars.

Now are we talking about determining linear or rotational frames of reference? In both contexts we can verify them experimentally, without relying on the 'distant stars'.

How about the center of mass of all stars in the universe? That should make more sense. As vinter said, they can also be moving.

vinter said:
Which stars are you talking about? I have not mentioned any stars here.

Even if I say now that there are stars, how do you know whether they are fixed? As we know, the 'red shift' suggests that the universe is expanding and that the stars are going away from us.
Vinter,

It does not matter if the stars are fixed. All matters is they are far far away and therefore APPEAR stationery.

Other than that, as you said the only force on the stone was the string-tension, which is right..makes me wonder what ambiguity you expected in the first place when you had asked the question. I mean , you said there was no gravity, so plainly it had to be the tension of the string.

snbose said:
Vinter,

It does not matter if the stars are fixed. All matters is they are far far away and therefore APPEAR stationery.
Then we are doing an approximation. What I presented in my question was a general situation in which I had specified things like 'space filled with an isotropic medium of mass'. And when a question is asked about this space, one should be able to explain it from the definition of 'this space'. So we know that the stone is rotating about the man and not the man about the stone because in the reference frame mentioned there, the man looks stationary and the stone appears to be going around the man. There is no need to include stars here.

snbose said:
Other than that, as you said the only force on the stone was the string-tension, which is right..makes me wonder what ambiguity you expected in the first place when you had asked the question. I mean , you said there was no gravity, so plainly it had to be the tension of the string.
I posted this question just because I have heard few people saying :-
1. Tension in the string
2. Centrifugal force

Anyways, no one here seems to be thinking on such lines, so it's beter that you all completely erase this question from your memory.

krab
vinter said:
Then we are doing an approximation. What I presented in my question was a general situation in which I had specified things like 'space filled with an isotropic medium of mass'. And when a question is asked about this space, one should be able to explain it from the definition of 'this space'. So we know that the stone is rotating about the man and not the man about the stone because in the reference frame mentioned there, the man looks stationary and the stone appears to be going around the man. There is no need to include stars here.
What you are saying here only makes sense if you adopt as true Newton's concept of Absolute Space. This concept is at odds with general relativity. You say "the man looks stationary and the stone appears to be going around the man". Who is doing the looking and to whom does the stone appear to be rotating? If the man cannot tell because of the isotropy of space what direction he is pointed in, how can he even define direction? Conversely, how can the universe tell the difference between the man's two directions? As Crosson says, with isotropic space and no fixed stars, there is no reason to believe that there is any inertia.

I will re-state a few points :-

A man, say Joe, is standing in an isotropic mass medium and there is a stone connected to his hand through a string. This stone is doing some sort of motion which is not defined at this moment. I am standing at a distance r from the man in the same isotropic medium. For me, though, the system is not isotropic, because there is only one direction in which I can find Joe. When I look at him, I find that the distance between me and Joe is not changing. Also, I can keep looking at him without rotating my neck for a long time. I see that there is my companion, the stone, which is doing a similarly absurd thing. It is going around Joe in a circle. I hope you will allow me to use the term 'going around' since I have defined it earlier. From my frame, is there any doubt who is going around what?

Of course, in these conditions, there is an ambiguity remaining. The distance between me and Joe can remain constant and I can keep looking at Joe without rotating my neck even when I am myself orbitting Joe in a 'geostationary orbit' sort of thing. But that will affect questions other than the 'who is going around what' question.