Non-inertial Frames - Newton's Laws of Motion

[TheShrike]

I've been lurking on PF for awhile now, but I reckon I'd benefit by actually trying to participate in the discussion and by asking my own questions once in awhile so, Hi all!

In my second year I took a module on Classical Mechanics, and one of the things we covered was the Coriolis Theorem. Now I know this isn't the only non-inertial frame, and maybe it has even more unusual properties due to being a rotating frame, but I want to know about Newton's Laws in this context and generally in the context of non-inertial frames.

I know that in a rotating frame Newton's First and Second Laws do not hold; this is trivial from the theorem. My understanding is that Newton's Third Law also does not hold. But Newton's #3 encapsulates conservation of momentum, and we certainly don't want to lose that. This must mean that momentum is transferred elsewhere. Is it in the rotation of the frame of reference?

(I have more to ask, but I'll wait for some replies before I convince you I'm totally crazy! )

 

[kmarinas86]

I've been lurking on PF for awhile now, but I reckon I'd benefit by actually trying to participate in the discussion and by asking my own questions once in awhile so, Hi all!

In my second year I took a module on Classical Mechanics, and one of the things we covered was the Coriolis Theorem. Now I know this isn't the only non-inertial frame, and maybe it has even more unusual properties due to being a rotating frame, but I want to know about Newton's Laws in this context and generally in the context of non-inertial frames.

I know that in a rotating frame Newton's First and Second Laws do not hold; this is trivial from the theorem. My understanding is that Newton's Third Law also does not hold. But Newton's #3 encapsulates conservation of momentum, and we certainly don't want to lose that. This must mean that momentum is transferred elsewhere. Is it in the rotation of the frame of reference?

(I have more to ask, but I'll wait for some replies before I convince you I'm totally crazy! )

Ask yourself, "What causes this rotation in the first place?" The answer is something which you also expect to be rotating.

So you can imagine a frame where all the rotations net to zero, or more precisely, where the total angular momentum is zero (i.e. no magnitude, and thus no direction too). For every reaction, there is an equal and opposite reaction. That implies that the number of reactions is even. The total net force on the all that exists is zero (because there is nothing else except that which exists). For every there is a (If both exist...)

 

[Dale]

I know that in a rotating frame Newton's First and Second Laws do not hold; this is trivial from the theorem. My understanding is that Newton's Third Law also does not hold. But Newton's #3 encapsulates conservation of momentum, and we certainly don't want to lose that. This must mean that momentum is transferred elsewhere. Is it in the rotation of the frame of reference?

You are correct. In a rotating frame Newton's first and second do not hold, but Newton's 3rd does. Objects accelerate without experiencing any net force, but every action still has an equal and opposite reaction.

However, it is possible to introduce the concept of "fictitious forces" such as the centrifugal and Coriolis forces. This will patch up Newton's first and second laws, but then break the third law. So now objects accelerate even if they don't experience a real force because they are acted on by the fictitious forces, but those fictitious forces do not have an equal and opposite reaction.

In a non-inertial frame you cannot have all 3 laws satisfied.

 

[TheShrike]

You are correct. In a rotating frame Newton's first and second do not hold, but Newton's 3rd does. Objects accelerate without experiencing any net force, but every action still has an equal and opposite reaction.

However, it is possible to introduce the concept of "fictitious forces" such as the centrifugal and Coriolis forces. This will patch up Newton's first and second laws, but then break the third law. So now objects accelerate even if they don't experience a real force because they are acted on by the fictitious forces, but those fictitious forces do not have an equal and opposite reaction.

In a non-inertial frame you cannot have all 3 laws satisfied.

Thanks. The way I learned this was the second way you have described, (although I think there is also another term, the Euler force (?), which we ignore if the angular velocity is constant). If we introduce the fictitious forces, we lose the third law. So, if we lose the third law, are we unable to use conservation of linear momentum in a rotating reference frame?

Here are my thoughts on this so far:

The reference frame is rotating, which means it is undergoing acceleration, which usually means a force (transfer of momentum) is acting. The momentum from the force causing the frame to rotate could be the source (so to speak) of the momentum of the fictitious forces, thus conserving momentum. However I spot two problems with this reasoning,

1) The reference frame is rotating, so the momentum it has is angular momentum, not linear momentum.
2)More fundamentally, how can a reference frame even be said to have momentum?

 

[A.T.]

The reference frame is rotating, which means it is undergoing acceleration, which usually means a force (transfer of momentum) is acting.

No, you don't need forces to accelerate a reference frame. A reference frame is just an abstract construct, not a massive object.

More fundamentally, how can a reference frame even be said to have momentum?

A reference frame doesn't have momentum. It merely assigns velocities (and thus) momentum to physical objects.

 

[TheShrike]

No, you don't need forces to accelerate a reference frame. A reference frame is just an abstract construct, not a massive object.

A reference frame doesn't have momentum. It merely assigns velocities (and thus) momentum to physical objects.

I suspected as much. So am I unable to apply conservation of momentum from within a rotating frame of reference?

 

[A.T.]

So am I unable to apply conservation of momentum from within a rotating frame of reference?

That is correct.

 

[Dale]

I agree with A.T.

The reference frame is rotating, which means it is undergoing acceleration, which usually means a force (transfer of momentum) is acting. The momentum from the force causing the frame to rotate could be the source (so to speak) of the momentum of the fictitious forces, thus conserving momentum.

The reference frame is a mathematical object. It has no mass, no momentum, and no forces acting on it.

2)More fundamentally, how can a reference frame even be said to have momentum?

Correct, it cannot.

 

[TheShrike]

Alright, that helps untangle the knot I've tied myself into.

So, the reference frame is a mathematical object. Is my understanding correct that a reference frame is a basis, as we understand the term from linear algebra?

How does a reference frame have a velocity?

 

[Dale]

So, the reference frame is a mathematical object. Is my understanding correct that a reference frame is a basis, as we understand the term from linear algebra?

The term "reference frame" is usually used as a synonym for "coordinate system" although technically it is sloppy usage to do that. In actuality, a reference frame is what is called a frame field:
http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity
https://www.physicsforums.com/showthread.php?t=168631

You will probably be fine just thinking of it as a coordinate system until you get well into general relativity.

How does a reference frame have a velocity?

Take two coordinate systems, and express the coordinates of one in terms of the other. Then take the derivative of the spatial coordinates wrt time to get a velocity.

 

[TheShrike]

Take two coordinate systems, and express the coordinates of one in terms of the other. Then take the derivative of the spatial coordinates wrt time to get a velocity.

That makes a lot of sense, thanks.

I'm having a look at your links now. (Haven't studied any GR yet; that's next semester. )