# Pseudo force

let me first donate a(A,B) is the acceleration of A relative to B
Observer 2 and a subject A with mass m are falling down from a building
Observer 1 is standing on the ground to observe the motion of these two objects
Ignore the air resistance
In observer 1,he sees the force acting on A
=ma(A,1)
=m[a(A,2)+a(2,1)]
In observer 2,he sees the force acting on A
=ma(A,2)
Obviously a(A,2)=0,but this doesn't make sense
So, we introduce pseudo force f=-ma where a is the acceleration of the frame
The problem is here,which equation should I add the pseudo force into?
I don't know the above assumption is correct or not,please check!

Doc Al
Mentor
The pseudo force is added when viewing things from the accelerated frame.

Andrew Mason
Homework Helper
let me first donate a(A,B) is the acceleration of A relative to B
Observer 2 and a subject A with mass m are falling down from a building
Observer 1 is standing on the ground to observe the motion of these two objects
Ignore the air resistance
In observer 1,he sees the force acting on A
=ma(A,1)
=m[a(A,2)+a(2,1)]
In observer 2,he sees the force acting on A
=ma(A,2)
Obviously a(A,2)=0,but this doesn't make sense
So, we introduce pseudo force f=-ma where a is the acceleration of the frame
The problem is here,which equation should I add the pseudo force into?
I don't know the above assumption is correct or not,please check!
Why does a(A,2) = 0 or ma(A,2) not make sense? They are both falling at the same rate so their relative acceleration is 0. Both are in a state of free-fall and experiencing no inertial effects. As far as I can see, there is nothing that would require introduction of a pseudo force to explain.

AM

Doc Al
Mentor
Leaving general relativity aside, there is a gravitational force acting on the mass yet the acceleration is zero in the falling frame. Thus a pseudo force is introduced to rationalize Newton's laws.

Andrew Mason
Homework Helper
Leaving general relativity aside, there is a gravitational force acting on the mass yet the acceleration is zero in the falling frame. Thus a pseudo force is introduced to rationalize Newton's laws.
But, unlike acceleration due to an electrical force, say, the accelerating observer observes no inertial effects that would require introduction of an inertial force.

The pseudo force that is required in the case of an electrical central force is a centrifugal force. The charged body that is subject to the electrical force experiences an inertial effect that is opposite to the direction of the electrical force. So a centrifugal pseudo force is introduced to explain that effect in the accelerating body's frame. But no such effect appears with a body that is subject only to a gravitational force. The reference frame of a body in gravitational freefall/orbit is equivalent to an inertial frame. If a pseudo force is introduced you would no longer have a frame of reference that is equivalent to an inertial frame.

AM

A.T.
The reference frame of a body in gravitational freefall/orbit is equivalent to an inertial frame.
That's why gravity can modeled as a inertial force, and in General Relativity it is. But this is the classical physics forum, where gravity is an interaction force.
If a pseudo force is introduced you would no longer have a frame of reference that is equivalent to an inertial frame.
Sure you would. In the classical non-inertial free falling frame, the inertial force exactly cancels the force of gravity. So that frame is equivalent to an inertial frame with no gravity acting.

So, we introduce pseudo force f=-ma where a is the acceleration of the frame.

I wouldn't say that we introduce the pseudo force. It simply results from Newton's second law.

Andrew Mason
Homework Helper
That's why gravity can modeled as a inertial force, and in General Relativity it is. But this is the classical physics forum, where gravity is an interaction force.

Sure you would. In the classical non-inertial free falling frame, the inertial force exactly cancels the force of gravity. So that frame is equivalent to an inertial frame with no gravity acting.
If I carry a net charge (one my shoes, say) and I am being whirled around a body by a central electrical force, I experience something that is trying to rip me out of my shoes. So, in my reference frame I postulate a centrifugal force acting on me trying to send me away from my shoes. The pseudo force allows me to analyse motion in my (non-inertial) reference frame.

But I don't have this effect with gravity as the only real force. Since I cannot sense gravity as a force there is no reason to introduce a force to counteract it. Unless one is experiencing tidal effects in free-fall, I have difficulty understanding why it would be necessary to introduce a pseudo force.

In this respect, there is a fundamental difference between gravity and other forces even in classical physics. I don't think you need GR to explain that difference.

AM

A.T.
Since I cannot sense gravity as a force there is no reason to introduce a force to counteract it.
What you can sense is completely irrelevant. In classical mechanics the free falling frame (in a g-field) is an accelerated frame. In an accelerated frame there is an inertial force, per definition. Period.

In this respect, there is a fundamental difference between gravity and other forces even in classical physics.
Yes, it behaves like inertial forces. But in classical physics it is still modeled as a real force.

I don't think you need GR to explain that difference.
It is about how gravity is modeled:
Newton : real force
Einstein : pseudo force

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Andrew Mason
Homework Helper
In an accelerated there is an inertial force, per definition. Period.
There is no inertial "force". Are you suggesting there is an inertial effect? How does it appear?

AM

A.T.
In classical mechanics the free falling frame (in a g-field) is an accelerated frame. In an accelerated frame there is an inertial force, per definition.
There is no inertial "force".
http://en.wikipedia.org/wiki/Fictitious_force

Andrew Mason
Homework Helper
Which is exactly my point. It is not a real "force". It is an inertial effect. But that is just terminology. My question is what inertial effect does a pseudo force explain if the only real force is gravity?

AM

A.T.
There is no inertial "force".
Which is exactly my point. It is not a real "force".
No, that wasn't your point. You said that there is no inertial force, which is wrong.
My question is what inertial effect does a pseudo force explain if the only real force is gravity?
The lack of coordinate acceleration from that single real force acting.

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Andrew Mason
Homework Helper
No, that wasn't your point. You said that there is no inertial force, which is wrong.
No. I said there is no inertial "force". My point in saying that, and my only point, was an inertial effect not a "force". It isn't.
The lack of coordinate acceleration from that single real force acting.
What do you mean? Can you give me an example of a phenomenon observed in the free-falling frame that requires introduction of a pseudo force to explain?

AM

Can you give me an example of a phenomenon observed in the free-falling frame that requires introduction of a pseudo force to explain?

The rest frame of ISS is a free-falling frame. As ISS is at rest there is no net force. But as there is a gravitational force acting between ISS and Earth there must be a pseudo force keeping ISS at its position.

Andrew Mason
Homework Helper
The rest frame of ISS is a free-falling frame. As ISS is at rest there is no net force. But as there is a gravitational force acting between ISS and Earth there must be a pseudo force keeping ISS at its position.
?? Why? The ISS is accelerating. No pseudo force is needed to keep the ISS in its orbit.

In the rest frame of the earth, the motion of the ISS is completely explained by gravitational force.

In the rest frame of the ISS, there are no phenomena appearing that require a pseudo force to explain.

AM

Doc Al
Mentor
?? Why? The ISS is accelerating. No pseudo force is needed to keep the ISS in its orbit.
In the rest frame of the ISS there is no acceleration.

In the rest frame of the earth, the motion of the ISS is completely explained by gravitational force.
Right. And that gravitational force exists in all frames. In the rest frame of the earth, there is a gravitational force and the resulting acceleration; no modifications to Newton's laws are required.

In the rest frame of the ISS, there are no phenomena appearing that require a pseudo force to explain.
You have a gravitational force yet no acceleration.

A.T.
I said there is no inertial "force".
There is an inertial force, because it is a accelerated frame.
Can you give me an example of a phenomenon observed in the free-falling frame that requires introduction of a pseudo force to explain?
Inertial forces are introduced to make Newton's 2nd law applicable to non-inertial frames, not to "explain phenomenons".

Andrew Mason
Homework Helper
You have a gravitational force yet no acceleration.
So what pseudo force do you introduce?

[Comment: Pseudo forces are introduced in order to apply Newton's laws of motion when the accelerating reference frame is treated as an inertial reference frame.

One can apply Newton's laws in the accelerating reference frame, treating it as an inertial frame, without introducing any pseudo forces. You cannot do that if the frame is accelerating due to central mechanical or electric force. You can only do that with a non-inertial reference frame that is in gravitational free-fall.

So my question is: why do we have to introduce a pseudo force in this case?]

AM

Andrew Mason
Homework Helper
Inertial forces are introduced to make Newton's 2nd law applicable to non-inertial frames, not to "explain phenomenons".
But Newton's laws of motion work perfectly in a non-inertial reference frame that is experiencing only gravitational force. You do not need a pseudo force to make them work. If you disagree, tell us what pseudo force is needed in order to make f=ma work in a reference frame that is in gravitational free-fall.

Am

A.T.
what pseudo force is needed in order to make f=ma work in a reference frame that is in gravitational free-fall.
http://en.wikipedia.org/wiki/Fictitious_force
In a coordinate system K which moves by translation relative to an inertial system k, the motion of a mechanical system takes place as if the coordinate system were inertial, but on every point of mass m an additional "inertial force" acted: F = −m a, where a is the acceleration of the system K

Doc Al
Mentor
So what pseudo force do you introduce?
-ma, of course.

[Comment: Pseudo forces are introduced in order to apply Newton's laws of motion when the accelerating reference frame is treated as an inertial reference frame.

One can apply Newton's laws in the accelerating reference frame, treating it as an inertial frame, without introducing any pseudo forces. You cannot do that if the frame is accelerating due to central mechanical or electric force. You can only do that with a non-inertial reference frame that is in gravitational free-fall.

So my question is: why do we have to introduce a pseudo force in this case?]
Once again: If you do not introduce an inertial force, you will have an unbalanced force (gravity just doesn't disappear because you shifted frames) but no acceleration.

Apparently you chose to disregard the gravitational force when you changed frames. Again, this is not general relativity.

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Andrew Mason
Homework Helper
-ma, of course.

Once again: If you do not introduce an inertial force, you will have an unbalanced force (gravity just doesn't disappear because you shifted frames) but no acceleration.

Apparently you chose to disregard the gravitational force when you changed frames. Again, this is not general relativity.
I didn't disregard the gravitational force any more than I disregard the central Coulomb force when I imagine a centrifugal force pulling me away from my charged shoes. In the electrical case I needed a pseudo force to explain why I fly off when my shoelaces break. In the gravitational case I don't. That's the difference.

So what is the inertial "force" that I postulate in the gravitational case. It cannot be the same force as in the electrical case.

AM

A.T.
In the electrical case I needed a pseudo force to explain why I fly off when my shoelaces break.
In the free falling frame you need a pseudo force to explain why free falling objects do not fly down, due to gravity.
In the gravitational case I don't. That's the difference.
There is no difference. The concept of inertial forces is general and applies to all accelerated frames. It is completely irrelevant what types of real forces act.
So what is the inertial "force" that I postulate in the gravitational case.
http://en.wikipedia.org/wiki/Fictitious_force
In a coordinate system K which moves by translation relative to an inertial system k, the motion of a mechanical system takes place as if the coordinate system were inertial, but on every point of mass m an additional "inertial force" acted: F = −m a, where a is the acceleration of the system K

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Andrew Mason
Homework Helper
In the free falling frame you need a pseudo force to explain why free falling objects do not fly down, due to gravity.
But they do fall down due to gravity.
There is no difference. The concept of inertial forces is general and applies to all accelerated frames. It is completely irrelevant what types of real forces act.
That is where we disagree. I say that pseudo forces do not arise if the acceleration of the frame of reference is due to gravity.
http://en.wikipedia.org/wiki/Fictitious_force
"In a coordinate system K which moves by translation relative to an inertial system k, the motion of a mechanical system takes place as if the coordinate system were inertial, but on every point of mass m an additional "inertial force" acted: F = −m a, where a is the acceleration of the system K"
And that is fine if the acceleration of K is due to any force except gravity.

The problem is that "the motion of a mechanical system" in gravitational free-fall "takes place as if the co-ordinate system were inertial" WITHOUT introducing an additional "inertial force" on every point mass m of F = -ma. In fact, it is locally equivalent to an inertial frame of reference (ignoring tidal forces).

If I introduce a force F=-ma on all objects in the frame of reference that is in gravitational free fall, there will be relative motion or tensions within that frame of reference. Where the acceleration of K is due to mechanical or electrical force, this is certainly the case. But not if the acceleration is due to gravity.

AM

Doc Al
Mentor
But they do fall down due to gravity.
Not when viewed from the co-moving, accelerating frame. They are at rest!
If I introduce a force F=-ma on all objects in the frame of reference that is in gravitational free fall, there will be relative motion or tensions within that frame of reference.
No there won't. The inertial force introduces no tensions nor does it produce any relative motion.

A.T.
But they do fall down due to gravity.
Not in the free falling frame
And that is fine if the acceleration of K is due to any force except gravity.
1) Do you have any mainstream references for that exception in classical mechanics?
2) You don't even make sense here. A reference frame is not accelerated by forces.
If I introduce a force F=-ma on all objects in the frame of reference that is in gravitational free fall, there will be relative motion or tensions within that frame of reference.
No, there won't.

Andrew Mason
Homework Helper
This is not complicated but we seem to be talking about different things or using different terminology. Here is what I am using:

1. A frame of reference is defined by a body that has mass. The reference frame is simply the space co-ordinates of any point relative to an origin that is fixed to the body.
2. If the body is not subject to a net external force the reference frame of the body is an Inertial Reference Frame (IRF).
3. If the body is subject to a net external force, the reference frame of the body is a Non-Inertial Reference Frame (NIRF).

Suppose the body used to define the NIRF is subject to an external mechanical force giving the body a constant acceleration a. In order to explain events relative to the NIRF using Newton's laws of motion, I have to invent a pseudo force PF = -ma that applies to all local matter (other than the body that defines the NIRF). With that pseudo force, Newton's laws will work. For example, to keep a mass m at a fixed position in the NIRF (or in uniform motion) I have to apply a force F = ma such that F + PF = 0.

Now suppose that the body used to define the NIRF is subject to an external gravitational force only, which gives the body a constant acceleration a. In order to explain events relative to the NIRF using Newton's laws of motion, I do not have to invent a pseudo force PF = -ma that applies to all local matter. For example, to keep a mass m at a fixed position (or in uniform motion) relative to the NIRF I do not have to apply a force F = ma. If I apply such a force to a local body it will not remain at a fixed position or uniform motion in the NIRF.

AM

A.T.
1. A frame of reference is defined by a body that has mass. The reference frame is simply the space co-ordinates of any point relative to an origin that is fixed to the body.
To define a reference frame you don't need a mass that is at rest in that frame.
Now suppose that the body used to define the NIRF is subject to an external gravitational force only, which gives the body a constant acceleration a. In order to explain events relative to the NIRF using Newton's laws of motion, I do not have to invent a pseudo force PF = -ma that applies to all local matter.
Yes, you have to.
For example, to keep a mass m at a fixed position (or in uniform motion) relative to the NIRF I do not have to apply a force F = ma.
To keep a mass at a fixed position (or in uniform motion) the net force must be zero.
If I apply such a force to a local body it will not remain at a fixed position or uniform motion in the NIRF.
It will remain at a fixed position or uniform motion because the real force of gravity and the pseudo force will cancel exactly.

1. A frame of reference is defined by a body that has mass. The reference frame is simply the space co-ordinates of any point relative to an origin that is fixed to the body.

As A.T. already wrote you do not need a body with mass at rest in that frame. Additionally it is not sufficient to fix the origin to the body because you couldn't distinguish between rotating and non rotating systems.

Now suppose that the body used to define the NIRF is subject to an external gravitational force only, which gives the body a constant acceleration a.

Just to be sure what we are talking about: a is the acceleration of the body in an inertial frame of reference. The acceleration a' of the body in its own rest frame is (of course) zero.

In order to explain events relative to the NIRF using Newton's laws of motion, I do not have to invent a pseudo force PF = -ma that applies to all local matter.

Let's check this for the body the system has been defined with: The acceleration a' is zero (due to the definition of the reference frame) but the gravitational force is not zero. How do you align this with Newton's first law? Everybody else introduces a pseudo force by use of the second law:

$F' = m \cdot a' = 0 = m \cdot g + F_{pseudo}$

$F_{pseudo} = - m \cdot g$

How do you do that without a pseudo force?

Andrew Mason
Homework Helper
To define a reference frame you don't need a mass that is at rest in that frame.
There doesn't have to be an actual mass at rest in that frame, let's call it frame RF. But it is difficult if not impossible to define it without reference to a body with mass. The coordinates of RF are identical to those of a system whose origin is fixed to a body with mass at rest at the origin of RF. If you disagree, tell us how you would define a reference frame.

To keep a mass at a fixed position (or in uniform motion) the net force must be zero.
Only in an IRF. Not in a NIRF. In a NIRF things accelerate relative to the NIRF if the net (real) force on a body is zero.

AM

Doc Al
Mentor
In a NIRF things accelerate relative to the NIRF if the net (real) force on a body is zero.
Thus the need for a pseudo force. Are we finally done?

Andrew Mason
Homework Helper
Thus the need for a pseudo force. Are we finally done?
??
You seem to be agreeing with me AT's statement "To keep a mass at a fixed position (or in uniform motion) the net force must be zero." applies only in an IRF. So that's progress.

I agree that demonstrates the need for a pseudo force to explain the motion of objects relative to a non-gravitationally free-falling NIRF (eg. one that is mechanically accelerated). One must apply a force to keep them from accelerating relative to the NIRF. So you can only treat the NIRF as an IRF if you posit the existence of a mysterious (pseudo) force that is making them accelerate. But in a gravitationally free-falling NIRF one does not have to apply a force to keep them from accelerating relative to the NIRF. It is always there (gravity). So where is the need for a pseudo force?

AM

Dale
Mentor
2021 Award
But in a gravitationally free-falling NIRF one does not have to apply a force to keep them from accelerating relative to the NIRF. It is always there (gravity). So where is the need for a pseudo force?
In a free-falling NIRF (in a uniform gravitational field) a free-falling object has a=0. Therefore, by Newton's 2nd law the net force on the object must be 0. In Newtonian mechanics it is acted on only by a real gravitational force of -mg. Since -mg≠0, there must be another force +mg acting on the mass. This is the pseudo force. With the pseudo force the net force is mg-mg=0.

A.T.