Tom Sawyer

I was having a conversation with a friend and he said something about Newtonian physics that I want someone to verify, because it sounds reasonable to me but I don't know if it's true or not. Understand that I haven't taken a physics class since high school, so please don't laugh at me if this a a really simple question that everyone should know the answer to.

Newton found that when two objects of different masses were dropped from the same height, they hit the ground at the same time. My friend says that this was due to an observational error and that when two large objects come together in space, their gravities will pull them towards each other faster than two small objects. The reason it seemed like the two weights Newton dropped fell at the same speed was because the difference in masses between a one pound rock and a ten pound rock is irrelevant when compared to however many billions of tons the Earth weighs, so it just looked like they fell at the same speed.

This seems to make logical sense to me, but like I said, I'm not a physics guy. Can anyone out there tell me if there's any truth to this?

Abacus

That sounds correct.

If I pick up a 10 lb bowling ball and drop it towards the earth, it will be accelerated towards the earth by a force of 10 lbs. Also the earth will be accelerated towards the bowling ball by a force of 10 lbs. The point is, a force of 10 lbs acting on the earth will have no observable effect, even though it is there.

Now imagine that the moon is the bowling ball. The tug of the moon on the earth is observable. The ocean tides are one effect of this.

[ August 02, 2002: Message edited by: Random Number Generator ]</p>

gabalski

Nope, that's not correct.

Remember that acceleration depends on 2 factors, force and mass (a=f/m). While the force acting on the 10 lb weight is 10x as great as the force acting on the 1 lb weight, the additional mass offsets that force resulting in the acceleration remaining constant.

This is not due to the small difference between the 2 masses relative the mass of the earth. If we head out to Jupiter with 2 weights, a 1 lb and a 500,000,000 lb, they would also accelerate torward Jupiter at exactly the same rate.

Mike

Abacus

But what about the acceleration of Jupiter towards a 1 lb weight as opposed to a 500,000,000 lb weight? Wouldn't those accelerations be different by a factor of 500,000,000?

The difference isn't in the acceleration of the small weights, but in the acceleration of the large planet. The 500,000,000 lb weight and Jupiter will come together faster than the 1 lb weight and Jupiter will acting under mutual gravitation.

beausoleil

Actually it is correct, if I'm reading it right. I 1kg mass would accelerate towards the asteroid Eros more slowly than it would towards the Earth or Jupiter. However a 1 kg mass and a 10 kg mass would both accelerate at the same rate towards the asteroid Eros.

Describing it as an observational error is a bit harsh - when the small body is much less massive than the large body the acceleration of the large body is too small to detect. For 2 masses M and m, the mass M accelerates towards m with g1=GM/r^2 while mass m acclerates towards M with g2=Gm/r^2. If M is large enough compared to m, g2 can be neglected and everything looks to be falling with the same acceleration. If not, not.

What was said about Jupiter above is correct, but not really what the guy was getting at.

gabalski

No, the force would be 500,000,000 times greater. The additional mass resists any additional acceleration.

The force due to gravity between any 2 objects is F=GMm/(r)**2

where
G is gravitational constant
M and m are the masses of the 2 objects
r is the distance between the 2 objects

since a = F/m, we would have [GMm/(r)**2]/m, and the m cancels out. So for any m you put in there, the acceleration is constant.

Hope this helps clarify.

Mike

EDITED TO ADD:

I just noticed beausoleils post after I posted, and he is correct. I misunderstood the original question.

This is absolutely correct.

But when you compare the acceleration of 2 different masses against 1 mass, the accelerations will be equal.

[ August 02, 2002: Message edited by: gabalski ]</p>

Abacus

Yes, the force on Jupiter will be 500,000,000 times greater in the case of a 500,000,000 lb pound weight as opposed to a 1 lb weight. The mass of Jupiter is still the same. Thus, the acceleration of Jupiter towards the weight, according to a=F/m will be 500,000,000 times greater with the 500,000,000 lb weight as opposed to the 1 lb weight.

However, the acceleration of the weight towards Jupiter will be the same, regardless of whether it is the 500,000,000 lb weight or the 1 lb weight.

See the difference?

Feather

This is poorly woorded, I think. Here's how I interpret this situation, based on the original post and some replies:

m1 = small mass 1
m2 = small mass 2

M1 = large mass 1 (M1 &gt; m1 and M1 &gt; m2)
M2 = large mass 2 (M2 &gt; m1 and M2 &gt; m2)

Now, in Newtonian Physics, the acceleration of m1 toward m2 will be less than that M1 toward M2. But the acceleration between M2 and any other object is a constant at a given separation distance.

So the acceleration of M1 toward M2 is the same as the acceleration of m1 (and m2) toward M2.

Likewise, the acceleration of M2 toward M1 will be the same as that of m1 and m2.

And so on.

beausoleil

In one experiment you let a 1 kg mass fall towards Jupiter.

In another experiment you let a point mass the same as Jupiter's mass fall towards Jupiter from the same place.

In the second experiment the collision will occur sooner than in the first, because the acceleration of Jupiter towards the point mass will be significant. Simplifying acceleration due to gravity as a constant called 'g' only works as long as the 'falling' bodies are vastly smaller masses than the attracting masses.

Feather

The value of 'g' is only constant at a given distance.

A more accurate statement is that every mass in the universe is attracted toward a given mass with the same acceleration function.

Find g(r) for a given mass and that function never changes--it is independent of every other mass in the universe. It's the gradient of the potential function for a given mass.

What I think you really mean, beausoleil, is that the motion of both bodies must be considered.

So in the given problem, the two large masses will move toward one another faster than one of the large masses and the smaller mass. Hence the two large masses will collide sooner than the one large mass and the small mass, given the same drop heights.