Non-inertial Frames - Newton's Laws of Motion

In summary, in a rotating frame Newton's first and second laws do not hold, but the third law does. This can be explained by the introduction of fictitious forces such as centrifugal and Coriolis forces. However, this breaks the conservation of linear momentum in a rotating reference frame. This is because a reference frame is just an abstract construct and does not have momentum, it merely assigns velocities and momentum to physical objects.
  • #1
TheShrike
44
1
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! :bugeye:)
 
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  • #2
TheShrike said:
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! :bugeye:)

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 o:) there is a :devil: (If both exist...)
 
  • #3
TheShrike said:
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.
 
  • #4
DaleSpam said:
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?
 
  • #5
TheShrike said:
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.
TheShrike said:
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.
 
  • #6
A.T. said:
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?
 
  • #7
TheShrike said:
So am I unable to apply conservation of momentum from within a rotating frame of reference?
That is correct.
 
  • #8
I agree with A.T.

TheShrike said:
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.

TheShrike said:
2)More fundamentally, how can a reference frame even be said to have momentum?
Correct, it cannot.
 
  • #9
Alright, that helps untangle the knot I've tied myself into. :smile:

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?
 
  • #10
TheShrike said:
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.

TheShrike said:
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.
 
  • #11
DaleSpam said:
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. :wink:)
 

1. What is a non-inertial frame?

A non-inertial frame is a reference frame that is accelerating or rotating, meaning Newton's laws of motion do not hold true in this frame. This is in contrast to an inertial frame, where Newton's laws of motion do apply.

2. What are Newton's laws of motion?

Newton's laws of motion are three fundamental principles that describe the behavior of objects in motion. The first law states that an object at rest will remain at rest and an object in motion will remain in motion at a constant velocity unless acted upon by an external force. The second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The third law states that for every action, there is an equal and opposite reaction.

3. How do Newton's laws of motion apply to non-inertial frames?

In non-inertial frames, Newton's laws of motion do not hold true because these frames are accelerating or rotating. This means that the first law does not apply since objects in these frames are not at rest or moving at a constant velocity. Additionally, the second law may not hold true because the acceleration of an object in these frames may not be solely determined by the net force acting on it. The third law may also be affected because the equal and opposite reaction may not occur in these frames due to the acceleration or rotation.

4. What are some examples of non-inertial frames?

Some examples of non-inertial frames include a car making a sharp turn, a roller coaster, and a merry-go-round. In all of these examples, the frames are accelerating or rotating, making them non-inertial.

5. How do we account for non-inertial frames in physics?

In physics, we can account for non-inertial frames by using additional equations and concepts, such as fictitious forces and non-inertial forces. These forces are used to explain the apparent violations of Newton's laws of motion in non-inertial frames. By incorporating these forces, we can still accurately describe and predict the motion of objects in non-inertial frames.

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