# Force and acceleration

## Main Question or Discussion Point

1.) The tests of falling bodies in a frame under constant acceleration prove all masses accelerate to the floor at the same rate, the rate at which the frame is accelerating.

2.) The same tests in a frame under the constant "force" of acceleration prove the mass of the body dropped determines the rate it accelerates to the floor.

Can a frame be accelerated to show the first results without applying force to the frame?

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Yes. The first frame is being "accelerated" by gravity.

Dale
Mentor
Huh? I don't understand. How can you apply force to a coordinate system? "Reference frame" is just another word for "coordinate system".

If you cannot apply force to a frame, how is it accelerated?

1.) The tests of falling bodies in a frame under constant acceleration prove all masses accelerate to the floor at the same rate, the rate at which the frame is accelerating.

2.) The same tests in a frame under the constant "force" of acceleration prove the mass of the body dropped determines the rate it accelerates to the floor.

Can a frame be accelerated to show the first results without applying force to the frame?
I am guessing that you are referring to that fact that objects of significantly different masses will fall at different rates when dropped one at a time in a "real" gravitational field and will this effect be observed in a lab that is being artificially accelerated by a rocket?

Here is a suggestion that the effect will be observed in the "artificial" case. Say we have a large test mass that is half the total mass of the rocket. When the large test mass is released the rocket motor is temporarily relieved of half its burden and the rocket accelerates faster while the large test mass is in free fall. The time it takes for the large mass to fall inside the rocket will be less than that of a small test mass that is an insignificant fraction of the rockets total mass (if the objects are dropped one at a time). If both masses are dropped at the same time, then they fall at the same rate in a real gravitational field and inside the artifically accelerated lab. Pretty good equivalence eh?

[EDIT] I am assuming statement 1 refers to an artificially accelerated lab and statement 2 refers to a stationary lab in a "real" gravitational field.

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kev, it is pretty good equivalence, but clearly not that good if dropping different masses at different times gives different rates of acceleration in a frame under constant force.
Which is why I'm asking the question. If a frame under force is not as "equivalent" as a frame in constant acceleration, I am assuming the latter is the more "equivalent" to gravitation. If so, how is a frame considered to be in constant acceleration without being under force?

I am refering to the two different explanations of an accelerating frame being equivalent to a gravitational field.

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kev, it is pretty good equivalence, but clearly not that good if dropping different masses at different times gives different rates of acceleration in a frame under constant force.
Which is why I'm asking the question. If a frame under force is not as "equivalent" as a frame in constant acceleration, I am assuming the latter is the more "equivalent" to gravitation. If so, how is a frame considered to be in constant acceleration without being under force?

I gave physical examples, so there is no doubt what I am talking about, but when you talk about frames we have to wonder if you are talking about coordinate systems and it hard to imagine applying force to a mathematical construct.

Usually when we talk about a reference frame under constant acceleration we are talking about an ideal mathematical construct where the test masses are considered to be point masses of insignificant mass so that the acceleration of the frame can be considered constant and it also allows us to ignore the real gravitational mass of the falling particle. The constant acceleration is just a nice ideal that makes the maths easier. Even in Newtonian physics gravitational force of GMm/R^2 and constant acceleration of GM/R^2 is just an aproximation that assumes that M is much greater than m and that R remains constant because dR is much smaller than R.

I understand the purpose of "ideal" mathematical constructs, but they must represent something physically real if they are to prove a physically real phenomena.
I have heard your explanation before but I'm having trouble reconciling it with the definition of an inertial frame. How can Newton's laws be tested if the masses involved are ignored? It seems to me the more physically real the test, the less physically real the equivalence.

I believe this is actually a clasical mechanecs problem.
1) I assume you are considering no real gravatational effects.
2) This expiroment is ment to simulate gravity.
Assuming these 2 things. What happens makes good clasical since, because you are essentually changing the value of G to a G' in the gravation force equation. Lets take a look at Kev's expiroment. First there are 2 parts of a rocket of simular mass.
M1=M2
$$\frac{F = G' M1^{2}}{R^{2}}$$​

Now, when the small mass is droped..

M1>>M2
M1' * M2' ~ M1' + M2' ~ M1'
Inorder to keep the mass of the rockets the same..
M1' = M1 + M2 = 2 * M1
$$\frac{F = 2G' M1}{R^{2}}$$​

From this we see that the attraction between 2 simular masses should be higher then the attraction of a very large and a very small mass. Which is what this thought expiroment proved.

Dale
Mentor
If you cannot apply force to a frame, how is it accelerated?
A frame is just a coordinate system. It has no mass. What force would you apply to something massless to obtain a 1g acceleration?

A reference frame being accelerated simply means that the transformation equations from an inertial frame have a certain form.

I understand the purpose of "ideal" mathematical constructs, but they must represent something physically real if they are to prove a physically real phenomena.
I have heard your explanation before but I'm having trouble reconciling it with the definition of an inertial frame. How can Newton's laws be tested if the masses involved are ignored? It seems to me the more physically real the test, the less physically real the equivalence.
THis wikipedia article http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation quotes the combined gravitational acceleration of two masses to be:

$$a = G {(M + m) \over R^2}$$

For M much greater than m the combined acceleration aproximates to the usual GM/R^2, but when m has the same mass as M the combined acceleration is 2GM/R^2.

This pretty much corresponds with what I described would happen in the physical examples of a real gravitational experiment and an experiment in an artificially accelerated lab when the objects are dropped one at a time.

The full relatavistic calculation is a bit hairy because while the large mass is falling in the artificially accelerated lab the faster acceleration of the lab during the free fall results in greater time dilation of the clock at the floor of the lab and the greater length contraction of the height of the lab, but there should still be a small difference in the fall times of the large and small masses. This difference would for example be very difficult to detect by dropping and timing different masses one at a time in Earths gravitational field.

HallsofIvy
Homework Helper
1.) The tests of falling bodies in a frame under constant acceleration prove all masses accelerate to the floor at the same rate, the rate at which the frame is accelerating.

2.) The same tests in a frame under the constant "force" of acceleration prove the mass of the body dropped determines the rate it accelerates to the floor.
Where did you see that? It doesn't seem right to me!

Can a frame be accelerated to show the first results without applying force to the frame?

1.) The tests of falling bodies in a frame under constant acceleration prove all masses accelerate to the floor at the same rate, the rate at which the frame is accelerating.

2.) The same tests in a frame under the constant "force" of acceleration prove the mass of the body dropped determines the rate it accelerates to the floor.

Can a frame be accelerated to show the first results without applying force to the frame?
Rather than use a frame lets use a physical object such as lab. For the spacetime within the lab to be considered an accelerated reference frame rather than inertial reference frame we would have to place the lab in a gravitational field (not free falling) or accelerate the lab by artificial means such as using a rocket. You seem to be proposing a lab that has accelerated spacetime without artificial or gravitational force being applied to it. How do you propose to physically do that?

kev, you are now paraphrasing my original question.
I am not proposing a "physical" frame of reference (lab) can be accelerated without a "force", inertial or gravitational. Nor am I proposing a non-physical, mathematical construct can be accelerated at all. I am questioning how the non-physical proves the physical when one contradicts the other.

As I understand it, a frame of reference will, with respect to the laws of mechanics tested in it, prove to an internal observer to be under force or not, with the understanding that a frame in free fall is not under force.
If a frame is said to be accelerating it must by the qualifications above, be under force.
If it is under force, the mechanics of equivalence (i.e.: acceleration change during free fall of test mass) prove different masses fall at different rates.
This may seem at first somewhat trivial until you attempt to define a frame of reference from the observations of an internal observer.
If I am in a (lab) under constant acceleration due to constant force (rocket exhaust) I will measure all three of Newton's laws in the mechanics of any test bodies in the lab, but they will not "all" agree with the mechanics of test bodies in a gravitational field.
I will find in this frame, that the rate of acceleration of bodies in free fall depends on their mass.
It is only when I "imagine" the mathematical construct of an accelerating frame exists without a force attributed to its acceleration that I find the equivalence in the mechanics of that frame and a frame in a gravitational field.
So, what am I missing?

Dale
Mentor
Again, I don't know why you keep refering to a frame being under force. A reference frame is simply a set of coordinates, you cannot apply a force to a coordinate system. I can't understand what you are trying to say.

Why don't you use some equations and a concrete example to describe what you are saying?

Hello all.

Is it perhaps that the frame associated with a particular accelerated object or ''tied'' to that object that and is moving with that object, and so a co-moving frame.

Matheinste.

DaleSpam, a frame of reference is a set of coordinates including time, referred to as a coordinate system serving as the "reference" from which an observer measures the mechanics of physical systems. The mathematical construct - coordinate system, cannot be under force, but the physical frame against which the coordinates are measured out can be, such as a rocket ship. To an observer in a frame the coordinates represent the property rest from which they may quantifiably test the motions of bodies against the equations of mechanics.
Einstein's first postulate (below) takes the frame of reference as the "physical" frame from which an observer cannot make any claim to the property "absolute" rest.

"...the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the Principle of Relativity'') to the status of a postulate,..."

When the equations of mechanics are found to "hold good" in a frame of reference, this postulate says the laws of electrodynamics will be valid in such frames.
When the equations of mechanics found to "hold good" are Newton's second law it is because the bodies in or with respect to the frame of reference are accelerating. Having no claim to the property of "absolute" rest, the observer in the frame must concede the principle of relativity makes their measurements "relative" and thus the acceleration of the bodies are equally and accurately defined as resulting from the acceleration of the frame of reference (rocket).
The acceleration of the bodies may then be understood to arise from the acceleration of the frame (a force on it) or from gravitation (a force on the bodies i.e.: free fall). It is understood that the equivalence of all tests done in either situation cannot distinguish between these two possible forces, leaving an observer in an accelerating frame unable to claim they are in inertial or gravitation acceleration.
It is because, as you have pointed out twice now, "you cannot apply force to a coordinate system", that I am asking the question. In all the references I can find, including Einstein's own words, the equivalence of inertial and gravitational acceleration is expressed mathematically as "a frame under [constant] acceleration". But no such "inertial" motion can every be measured by virtue of the fact that "testing" Newton's laws (dropping bodies in an inertially accelerated frame) imparts a change in the acceleration of the very frame in which his laws are upheld.
It is only when we don't test the laws and "imagine" a constantly accelerating frame that the mathematical construct performs according to the equations that are claimed to make the frame equivalent to a frame in gravitational acceleration.
So, I am asking, what am I missing? What have I misunderstood, and where?

Einstein's On the Electrodynamics of Moving Bodies is based on the English translation of his original 1905 German-language paper (published as Zur Elektrodynamik bewegter Körper, in Annalen der Physik. 17:891, 1905)

.....

But no such "inertial" motion can every be measured by virtue of the fact that "testing" Newton's laws (dropping bodies in an inertially accelerated frame) imparts a change in the acceleration of the very frame in which his laws are upheld.
It is only when we don't test the laws and "imagine" a constantly accelerating frame that the mathematical construct performs according to the equations that are claimed to make the frame equivalent to a frame in gravitational acceleration.
So, I am asking, what am I missing? What have I misunderstood, and where?

I think you are missing the fact that "testing" Newton's laws by dropping objects in a real gravitational field such as the Earth's, imparts a change to the acceleration of the Earth frame.

A very massive object dropped from a given height on Earth will fall to the ground in less time than a less massive body dropped from the same height if the test is carried out in a vacuum tube so that air resistance does not complicate the experiment.

While the massive body is falling to the Earth, the Earth is accelerating towards the falling object and the Earth will accelerate faster towards the heavier object reducing the fall time for the heavier object. If the haevy and light object are dropped at the same time, then the Earth will acclerate towards the combined gravitational mass of both objects and the haevy and light object will fall at the same rate (but faster than than either object by itself. The same behavior will be observed in a lab that is being accelerated by a rocket. The ideal accelerating reference frame with no force applied does not exist in reality.

Thank you kev, that is what I was missing. Incredibly obvious now that you mention it.