# Relationship between inertial and gravitational mass?

1. Jun 6, 2015

### Sturk200

I should preface this question by saying that I am not familiar with Einstein's general relativity, so I am trying to understand the relationship between gravitational and inertial mass from a purely classical standpoint.

Newton writes that the gravitational force exerted by an object is proportional to that object's inertial mass. As far as I know the only reliable way to define or measure mass is by measuring the acceleration caused by the application of a known force and quantifying this with respect to a reference mass. Thus inertial mass is defined as an object's ability to resist being accelerated or decelerated by some force.

Applying this definition of mass to the law of universal gravitation, we find that the gravitational force associated with some object is proportional to that object's ability to resist being moved by a force. This seems like too straightforward a connection to be merely coincidental. My question then is this, how are we to understand the relationship between inertial mass and gravitational mass? Particularly, is it possible that an object gains its ability to resist being moved by a force (i.e. gains its inertial mass) by virtue of the fact that it is pulled by a contrary gravitational force towards surrounding objects?

For example, suppose we place a ping pong ball and a brass ball in space and subject them each to an identical force -- we project each with a spring loaded plate. The ping pong ball is projected faster and farther than the brass ball. We want to say this is because the brass ball is more massive. But is it possible that all we are expressing in this statement is that, due to gravity, the brass ball is attracted to the spring loaded plate with a greater force than the ping pong ball, and is therefore endowed with a greater ability to resist the force of the spring?

2. Jun 6, 2015

### Staff: Mentor

Indeed it is, and it intrigued people for better than two centuries after Newton discovered it. It was finally explained by Einstein when he discovered general relativity.
No, that is not the explanation. The way to understand the relationship between inertial and gravitational mass is to learn general relativity; you can't explain it from a classical standpoint for about the same reasons that you can't explain how smoke is formed without considering fire.

3. Jun 6, 2015

### Sturk200

I don't know if the relationship between the curvature of spacetime and universal gravitation is quite as simple as that between fire and smoke, but I take your point and look forward to learning general relativity.

For now, are there concrete reasons that the explanation is incorrect? I wrote the question hoping that somebody with more expert vision might be able to demonstrate why it was wrong -- perhaps by seeing through to an inconsistent consequence -- and thereby correct the idea while enlightening the subject.

4. Jun 6, 2015

### Staff: Mentor

It's tempting to answer that it's incorrect because we already know the correct explanation, and it's something different. Of course, there is always the possibility that our already-known and correct explanation is in fact wrong and there's a better alternative - but such a hypothetical alternative has to make quantitative mathematical predictions at least as good as the current theory, or it's not even on the playing field. Clearly you proposed explanation does not reach that level, whereas general relativity has survived a century of relentless and skeptical scrutiny.

You've already received one warning today for posting personal and speculative theories. That's not what Physics forums is for.

5. Jun 6, 2015

### Sturk200

Thank you. I suppose physics is better served by a forum that discourages independent thought in its students.

6. Jun 6, 2015

### Staff: Mentor

This policy and the reasons for it are discussed at some length in the FAQ and in the various threads in the Forum feedback section. The very quick summary would be something along the lines of "You can't expand the frontiers of science unless you're already standing at the frontier; PF is here to help people get there".

7. Jun 6, 2015

### Sturk200

I apologize if the impressions given was of somebody trying to expand the frontiers, as that is not the intention underlying it. The question posed is simple and avowedly naive. Why can't the resistance to a spring force be explained by the gravitational force set up between the mass to which the force is being applied and the spring mechanism. I assume there is some quite simple demonstration of why this cannot be the case, or else Einstein never would have written a more complicated theory.

8. Jun 6, 2015

### Sturk200

Maybe I can try to clarify the question because I feel like there must be some obvious error in my thinking.

Here is the phenomenon: So imagine we have a spring mounted to a pretty large bowling ball and we take this rig to space. We attach a test mass to the end of this spring rig, compress the spring, and let go. We observe that the test mass is propelled forward with some acceleration. (The bowling ball is included to emphasize the gravitational force of interest, though we could just as well remove the bowling ball and consider the force between the test mass and the spring itself).

How do we analyze this situation? According to the mechanics that I've learned -- assuming I understand it correctly -- the acceleration with which our test mass is propelled will be a function of three things: (1) the force due to the spring (f=-kx), (2) the inertial mass (-kx=ma), and (3) the gravitational force due to the interaction between the bowling ball and the test mass (f=Gmm/r^2, where the two m's refer to the masses of the bowling ball and the test object). The total description of this test object will then be -kx - Gm1m2/r2 = m1a. That is to say, the spring force (which will push in the positive x direction) minus the force due to the gravitational attraction between the two masses (which will pull back in the negative x direction) will be proportional to the conjoined mass and acceleration of the object.

Now here is where my imagination begins to falter. Suppose we try the same experiment again, keeping everything constant, except that we use a more massive test object. The obvious result will be that the more massive object will be propelled with a lower acceleration. I know that the equations tell us that the lower acceleration will be a consequence of two things: the increased gravitational force between the test object and the bowling ball, and the increased inertial mass itself. But I can't see how these two things can be experimentally distinguished from one another, since the two are linked by definition. All we see in the phenomenon is that acceleration decreases in proportion to changes in the mass of the test object. What prevents us from saying that this relationship is entirely due to the gravitational attraction between the mass and the bowling ball, since that measurement already accounts for the inertial mass?

9. Jun 7, 2015

### Staff: Mentor

The thread is reopened, with the reminder that the goal of PF is to learn about existing theories, not invent new ones.

10. Jun 7, 2015

### Staff: Mentor

Look at the equation you posted:
-kx - Gm1m2/r2 = m1a
Or solving for a
a=-kx/m1 - Gm2/r2

Clearly, the acceleration does not "decrease in proportion to" m1, the relationship is more complicated than proportionality. Also, the gravitational term in the equation does not change at all as a function of m1. So gravitation has the wrong form to account for an acceleration which varies as a function of m1.

The algebra simply doesn't support your idea here.

11. Jun 7, 2015

### Sturk200

This is right as long as we consider the gravitational and inertial effects to be two distinct things. What I was trying to ask is why we don't simply say that the inertial resistance to acceleration is entirely a consequence of the gravitational attraction between the test object and the "bowling ball," since that gravitational attraction increases in direct proportion to the inertial mass. Mathematically, this idea might be expressed by replacing the inertial m1 with the gravitational term, and allowing the latter to speak for both, giving something like:
-kx = Gm1m2/r2 ⋅ a.

Of course this equation is an absolute mess, since the maneuver throws off all the units and constants. But what struck and mystified me was that -- and please correct me if I'm wrong here -- it seems to capture the proportionality between the mass of the test object and the inertial resistance to acceleration, and if the proportionality is captured, then the rest of the work is just in fixing the units and constants. Then the thought is, if the inertial effect can be captured entirely by talking about gravitational attraction, then why speak of inertial mass at all?

I think I see now why this wouldn't work, or at least one reason why it wouldn't. The new equation lends a symmetry to the situation that is not present in the phenomenon itself. For if we make the test of increasing the mass of the bowling ball, the equation predicts that this will have an effect on the acceleration of the test mass in equal proportion as would an increased mass of the test mass itself, which is contrary to fact. In fact increasing the mass of the bowling ball would have barely any effect, while increasing that of the test object would have a larger, more noticeable effect. The equation, then, does not allow us to scale the relative importance of the two masses for determining acceleration, and this relative importance is essential.

I hope this counts as understanding existing concepts.

12. Jun 7, 2015

### Staff: Mentor

Yes, this is wrong. Even in your mangled form it is still not a proportionality. It seems very apparent to me, do you not see that directly by looking at the equation (either the incorrect one you wrote or the correct one I wrote)? Neither one is a proportionality.

As you say, the effect of changing the other mass would not be captured correctly either.

Last edited: Jun 7, 2015
13. Jun 7, 2015

### Sturk200

Maybe I'm not getting the meaning of proportionality clear. In my mangled equation, if you increase m1, and hold all else constant, acceleration must decrease proportionally. That means acceleration is proportional to the inverse of the mass, no?

14. Jun 7, 2015

### Bandersnatch

Put the bowling ball in a sling behind the mechanism so that the acceleration due to gravity between the ball and the mechanism is positive. Compare with acceleration when the ball was on top of it.

Or use two identical setups with same propelling force, but different masses of the mechanism.

15. Jun 7, 2015

### Bandersnatch

Proportionality means the following: given X and Y are proportional, when you multiply X by A, you end up with Y multiplied by A.
In the equation:
$a=kx/m_1 - Gm_2/r_2$ (the original had a wrong sign by kx - the 0 point is the neutral position for a top-loaded spring, so for negative gravity you have positive spring force, or vice versa)
when you increase m1 by e.g. a factor of 2, you don't get twice as large nor an inverse proportionality (i.e. 1/2), nor any other such simple relationship.

Your equation (which needs to be said again: is bad algebra) has the same problem.

In concrete terms, the equation above is such, that there exists a mass m1 for which the acceleration changes sign from positive to negative (ball is too heavy to be launched against gravity). In a proportional relationship where everything else is constant, you'd need a negative mass to get that result.

16. Jun 7, 2015

### Staff: Mentor

Ah, my apologies I misread your formula (the multiplication dot looked like a minus sign on my mobile device). Your formula is indeed a proportionality.

As such it is experimentally wrong for any scenario where both the spring and gravitational forces are important. In other words, experiments show that the relationship is not proportional, so your mangled equation cannot be fit to the data.

I don't think it is appropriate to continue discussing a known wrong equation. Let's stick to correct ones. You can ask questions about how we can experimentally determine the correct equation without requiring discussion of the incorrect one.

Last edited: Jun 7, 2015
17. Jun 7, 2015

### Sturk200

Summary: Experiments show that the relationship between inertial mass and acceleration is proportional. Experiments also show that the relationship between inertial mass and the gravitational force is proportional. Bad idea: Woah, maybe inertial mass is just due to the gravitational force! Definitive objection to bad idea: But experiments show that the relationship between the gravitational force and acceleration is not proportional when gravitation is not the only relevant force.

18. Jun 7, 2015

### Staff: Mentor

Good summary.

19. Jun 7, 2015

### Sturk200

Thanks for all the help.