I don't know if this will make any sense, or if there's some elementary answer I'm overlooking but here goes:

I know that acceleration due to gravity is constant, independent of an objects mass. I.e., the whole tennis ball v. bowling ball dropping thing (not accounting for wind resistance/aerodynamics).

My question is this; if acceleration is the same, then doesn't the force acting upon the body have to scale based on the object's mass? Doesn't there need to be more force in order for a 10kg object to be accelerated the same speed as is needed to accelerate a 1kg object?

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

Right. Through some strange coincidence, an object's gravitational mass (which determines how strongly gravity acts on it) happens to be the same as its inertial mass (which determines its resistance to acceleration).

"Coincidence" is not generally a very friendly term in science; was wondering if there's a reason why? Or if not, what current theories suggest.

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

Yes, and that force is what we usually term the weight. And it is proportional to mass, the equivalence principle basically.

Yeah I think I just diagramed my answer out for me. If A (earth) pulls B and C, and B and C also pull on A, then the net force on B would be A + the pull of B, and since B is also (proportionate to) the mass of B, then it cancels out. Same goes for C. therefore you're left with the same acceleration for B and C.

Sorry if I didn't word that clearly but yeah I think I see what I was missing.

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

Force of gravity F(g)

F(g)=GmM/r^2

Generic force accelerating an object

F=ma

We know in a field that F=F(g)

ma=GmM/r^2
a=GM/r^2

So, we know that F(g) is a larger magnitude with a larger m, but that the mass cancels, giving equal a for any m

This does assume that the m in F and F(g) are the same. So far no experiment has shown any difference between the two. The guy who can prove or disprove the equvalence principle will have his name remembered with Newton and Einstein

Okay brain, help me put this in words!

Okay, with gravity it's agreed that objects of mass have an attracting force towards one another. That means, a satellite orbiting the earth pulls on the earth as well as the earth pulls on it.

Now if the pull from the satellite isn't enough to overcome the earth's inertia, then it makes sense that it would translate into a movement towards the earth.

Example would be a man pulling on a rope anchored to a brick wall. If you can't pull strong enough to move the wall towards you, it doesn't just cancel out. You will actually pull yourself towards the wall.

So if gravitational pull is dirrectly related to an object's mass, as is an objects inertia, it makes sense that this pull (satellite->main body) would cancel out, and you'd be left with the tenisball bouncing off your left foot at the same time the bowling ball flattens your right foot.

At least, that makes sense to me. Let me look at it from a formula standpoint.

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

General relativity is founded on the assumption that this is not a coincidence at all. Instead, Einstein treated gravity not as a force, but as a curvature of spacetime, with all objects following the nearest thing to a straight line in curved space. That ends up with all objects accelerating equally, regardless of mass.

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Conserve energy. Commute with the Hamiltonian.

I probably misunderstood your question, because I don't really understand how the rest of the posts in this thread answer it very directly.

So for what it's worth, here's how I've always "understood" it (I know I know very little):


Gravity, being a field (or a curve in space, or whatever) is affecting all mass in that (area of the) field by some acceleration.

Every gram of that 10kg object is in that field, and feels the acceleration due to that field. Every gram of that 1kg object is in that field, and feels the acceleration due to that field.

Gravity does not provide a force per object.

So both objects accelerate the same, as every unit of mass in each object feels the same acceleration. The 10kg object could be 10 x 1kg objects that happen to be next to each other.

Basically, the bigger object does feel more force (by f=ma). It's the sum of force exerted on each unit of mass, by the acceleration of gravity.


That's just my amateur/layperson impression.


(I think I've read that near a black hole the gradient of the change in gravitational acceleration means one side of the 10kg object might experience so much more acceleration than the other, that it would pull apart...)

Cheers,

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It also works if you use a hammer and a feather.

Your answer is a good one, but it is really only part of the story. You have explained why a brick falls the same if broken in two pieces, but that's not anything unique to gravity. What's so special is that a brick falls the same as a bucket of water, or an iron bar.

Do you mean I should word it more like "every gram in a 10kg brick feels the same acceleration as every gram in a 1kg iron bar (at a given location in a given gravitational field)"?


Or is there something "deeper" I'm missing? (Sorry, I don't understand what's so special about the other cases you mention).

Thanks,

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My take on Ken G's post (well, one take; there are a few): you could count the exact number of electrons, protons, and neutrons in each (assume each weighs exactly 10 kg). You could consult a table giving the rest mass of each particle, to 6 or even 10 significant digits. With some multiplication and addition, you would get three numbers, one each for the brick, bucket of water, and iron bar. The numbers will not be the same, even though they weigh the same - do you know why?

Yet - and this is truly marvellous - all three 'fall the same'!

Okay I think I see my problem. I was thinking of gravity as a dirrect (strong?) force. Should really think of it as a field. I can't wait for my PolSci class to be done so I have time to get back to reading this stuff.

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

No. If you have 3 objects massing 10kg, all weight 98N at the surface of the Earth. (If you want it 10.00000000 kg weighs 98.something to 8 digits, I'm not going to look it up)

The thing Fazor seems to be looking at is more:
(impact is energy from 1 m fall)
mass weight acceleration impact
1kg 9.8N 9.8m/s^2 9.8 J
10kg 98N 9.8m/s^2 98J
100kg 980N 9.8m/s^2 980J

So yes, a heavier object will hit harder, even tho it has the same acceleration and same speed if the start conditions are the same.

This is actually where the ATMer really fail. Generally they dont know the math well enough to know what it is actually saying.

The issue is, if you word it like that, you are assuming what you are trying to show. If you have a brick, and you break it in two pieces, there's no way to "tell" gravity that the brick is now in two pieces, that much is true-- so it argues that different amounts of the same thing should fall the same. But if you further assume that the gravitational force depends only on how many grams you have, and not what it is made of (or the chemical processes going on inside, as alluded to by Nereid ), then you are assuming what you are trying to show. If grams are defined using the force of gravity, then you have a circular argument, and if grams are defined in terms of the measured inertia of the object, then you still have no way to know if different substances, rather than different amounts of the same substance, will fall the same. For example, why does a proton fall the same as a neutron? In other words, even though we expect both gravity and inertia to be additive properties of aggregate objects, you still can't break protons up into neutrons without changing them in a more fundamental way.

Having said that, I should point out that nevertheless your suggestion is a very good one for explaining why big bricks fall the same as small bricks, and that's really the main misconception people have, not the issue of why different substances fall the same.

Actually (korjik, Ken G), what I was alluding to (and one of the things I thought Ken G was too) is the various Eötvös experiments (example), and tests of the strong equivalence principle: "The strong equivalence principle applies to all laws of nature, and implies that even gravitational self-energy must obey the equivalence principle."

To me, that is marvellous.

The thing Fazor seems to be looking at is more:
(impact is energy from 1 m fall)
mass weight acceleration impact

Actually, disregarding impact. Obviously two objects moving at the same velocity with two seperate masses, the object of larger mass has more KE, and more momentum.

What I was looking at is this: If you have a 10-ton safe, and a barstool, both at rest on the gymnasium floor (why a gym, safe, and stool? I dunno I'm random like that): it takes more force to accelerate the 10-ton safe at the same rate as it would to accelerate the bar stool (unless it's one of those rare 10-ton bar stools). But with gravity, the force is the same, yet the resulting acceleration is also the same. That's what I was asking about, but examining the answers, I think I'm just considering gravity in the wrong manner.


The question would be, in the absense of any other gravitational fields, would a 1kg object be attracted to a 10kg object at the same rate that a 10kg object is attracted to a 100kg object? would the 1kg object be attracted to the 100kg object at the same rate as the 10/100? (I know the answer is yes, but the question is why).

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

Technically, my impact energy is the KE after a 1m fall.

There are two parts to the why. First is the assumption that you are in the frame of the larger mass, and therefore its movement is zero. This isnt actually necessary, but it makes the math alot easier.

When you use the first assumption, The equations of motion become very simple

ma=GmM/r^2

The cancellation of the m, in this case the smaller mass, makes the acceleration independent of the smaller mass. That really is all there is to it. the rest of the quantities, like p, KE, v are all dependent to some degree or another on a, so they all scale in one way or another.

Nothing to do with the stong equivalence principle then? After all, safes and bar-stools have different compositions, different mixes of the EM, strong (and weak?) forces ...

Oh dear ... but do you think it's marvellous?

Yes. For simplicity of math, lets go back to the 10 kg and 1 kg object (say a Jack Russel terrier and a rat). From the gravitational force equation, we know that the earth pulls on the terrier 10 times as hard as it pulls on the rat.

F_g = mMG/r² (1)

Where:

F_g = force of gravity on a given object
m = gravitational mass of the small object
M = mass of the big object (Earth in this case)
G = Gravitational constant
r = the distance between the centers of gravity of the Earth and the object in question.

Assuming both objects are at the same distance from the center of the Earth, the only number that changes is the mass of the object. If object 1 is 10 times more massive than object 2, then F_g will be 10 times greater for object 1.

No. We just showed that the forces are different for different masses.

Yes, the acceleration is the same. This is where we look at relationship between Force and acceleration:

F = ma (2)

Where:

F = any force
m = inertial mass of the object
a = acceleration

or a = F/m (3)

Now, for a given mass, increased force means increased acceleration, but if you increase the mass as you increase the force (keeping them proportional), you will see that acceleration remains the same. So, if both the force and mass are increased by a factor of 10, the acceleration is constant. Note in our problem that this is exactly what happens. The 10 kg object has 10 times as much mass as the 1 kg object, and also has 10 times as much force acting on it from the Earth!

This is why the acceleration is the same for both objects. All this hinges on inertial mass being equal to gravitational mass (the equivilancy principle), which so far appears to be true based on experimental evidence.

No. You are starting to mix apples and oranges. The OP deals with objects attracted to the Earth. This is a special case where M from Equation (1) remains the same. When you start looking at mulitple sets of objects being attracted to each other, you need to work out the math. In this case, the force between the 1 and 10 kg objects is 100 times less than the force between the 10 and 1000 kg objects. Since the 10 kg object has 10 times the mass of the 1 kg object, it will be accelerated 10 times as fast towared the 100 kg object.

Yes. Because M remains the same in this case. It is exactly analogous to the case of each object being attracted to the earth.

No, the force on the heavier object will be greater.

Newton's law of gravitation is F= G ( M1 M2 ) r^2, so the force of gravity acting between two objects is proportional to the product of their masses.

Nothing to do with the stong equivalence principle then?
It would help if I better understood what the strong equivalence principle was...

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

Yes, that's what I'm getting at. So the force of gravity is really the result of both masses together, not independantly. Which if I would have paid more attention is clearly stated in the formula. Silly me

So, the force of gravity (G(m1*m2)r^2) related to m2 always produces a constant acceleration if you substitute m2 for any other mass.

So could you start to combine acceleration/force formulas? Like:
F = m*a, but above we state F is (G(m1*m2)r^2, so:
(G(m1*m2)r^2=m*a

Now the m we're dealing with is m2, so pop that in and get:
(G(m1*m2)r^2=m2*a

m1 in this case is earth, so we know that a = 9.8m/s^2 (roughly at the surface). Which would leave us at:
(G(m1*m2)r^2=m2*9.8m/s^2.

Leaving the only part of the above equasion left to be solved G (i know it's already solved, technically).

Did I set this up right? If I pick a mass for m2 and do the math (which I'm about to try anyway) will I come up with the correct grav-constant for G? or did I go way off track somewhere?

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

you have it backwards. in the equations you are using m2 cancells out leaving

Gm1/r^2=a

technically, you should measure a, m1 and r, and then use them to figure G

This wiki page seems to provide a pretty good overview.

The 'composition difference' (Eötvös) tests are actually of the weak equivalence principle; the Eöt-Wash Group has also done tests of the strong one - another case of 'they fall the same' (inverse square law, in this case).

Is it marvellous yet?

isn't a 9.8m/s^2, m1 (estimated) 6.0e24, and r (estimated) 6356.75km?
That gives me about 6.6e-11, which is roughly G. Good.

Anyway that means I can work through the formulli to show everything. Anyway slightly derailed from my OP question, although I think what I was trying to get at in the OP doesn't really have an answer, at least not yet. (what exactly is/causes gravity. I guess that's one thing super-string theory attempts an answer to? way over my head, and from what I understand purely theoretical and untested).

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

Yeah I skimmed through that, but most of it was beyond my quick-reading comprehension level I'll haveta give it a more thorough look when I have the time.

As for "is it marvellous?" yes, all physics is. That's what origonally brought me to this forum (i'm not all that interested in astronomy ((don't stone me!!)) but i love physics).

My problem is, I've never really formally studied it, outside of my highschool classes (which has been nearly a decade ago! hard to believe). So I'm working with an incomplete (and oft mis-remembered) knowlege of basic physics. Which when I get into more intermediate stuff, I tend to badly mangle it. And the more advanced stuff, I tend to glaze over while the drool slowly makes its way out the corner of my mouth. I'll be done with college in two weeks though, which should allow more time to go back and re-introduce myself to the basics.

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I'm like one of those idiot savants...well, except for the savant part.
"In order to increase awareness of the homeless, security have been given binoculars."

Me too. That was what I was referring to in terms of chemical processes-- the chemical self energy of everyday objects far outweighs (no pun intended) the gravitational self energy.

As usual, it seems we're close ... I had in mind the strong nuclear self-energy (and, maybe, the weak too), as well as the 'chemical' (which is due to electromagnetism).

How marvellous that gravity seems quite blind to the particular composition of mass-energy: the (rest) mass of particles, the (binding) energy of globs of particles (whether nuclei, atoms, molecules, ...), and even the gravitational self-energy!

Also marvellous is the ingenuity of the experimenters ...

Oh, and an implication: photons 'weigh' something too (photonic self-energy?).