Freethought & Rationalism Archive

beausoleil · 08-04-2002, 12:11 PM

Quote:

Originally posted by Feather:
<strong>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.</strong>

Exactly.

DNAunion · 08-05-2002, 03:07 PM

DNAunion: Didn't follow the whole thread, but let me explain what I was taught. (I believe the second or third post stated this, but then was argued against).

Newton's Universal Law of Gravitation is expressed as F = Gm1m2/d^2, where F is the force of attraction, G is the universal Gravitation constant, m1 and m2 are the masses, and d is distance between the two object's centers of mass. Since we are interested in how two different masses (a "light" object and a "heavy" object") would accelerate towards a common mass (the Earth or Jupiter, for example), the we can set m1 equal in both situations. And since we are interested in both bodies being the same distance from the Earth or Jupiter, we can set d (and therefore d^2) equal in both situations. What we are left with is that F is proportional to m2. Thus, the Earth's (or Jupiter's) gravity/mass pulls on a more massive object more strongly than it pulls on a less massive object. That will tend to make a more massive object accelerate more quickly.

However, Newton's second law of motion is a = F/m, where a is the acceleration, F is the applied force, and m is the mass. This shows that a more massive object is harder to accelerate.

So although the gravitational force exerted on m2 is greater for a greater mass of m2, the greater mass of m2 also resists acceleration to a greater degree: in fact, to the same exact degree. Mass, in a sense, cancels out of the equations.

This has been verified on the Moon (where air resistance is absent), where a feather and a hammer were dropped and both accelerated at the same rate.

08-05-2002, 10:13 PM

I think the original question is a bit hard on Newton. There is no evidence I am aware of that he ever performed an experiment of dropping two weights as described. It is to Newton that we owe the formula F = Gm1m2/r**2, which is about motion in vacuo. He derived the formula in order to find something that would be compatible with Kepler's (observational) laws of planetary motion, and he invented the differential calculus to do so. He was a great mathematician.

If you apply the formula to any two masses, it is obvious that both are affected by the presence of the other. This means that the earth does not revolve round the centre of the sun, but round their mutual centre of mass, to a first approximation that ignores the gravitational contribution of other nearby masses. But the sun is so much more massive than the earth that the centre of mass is still near enough to the centre of the sun not to make much of a difference.

When you drop two weights near the surface of the earth, their acceleration due to gravity is the same, but because of the existence of the earth's thick atmosphere, gravity is not the only significant force,and hence acceleration, acting on the weights.

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08-05-2002, 03:07 PM	#12
DNAunion Veteran Member Join Date: Jan 2001 Location: USA Posts: 1,072	DNAunion: Didn't follow the whole thread, but let me explain what I was taught. (I believe the second or third post stated this, but then was argued against). Newton's Universal Law of Gravitation is expressed as F = Gm1m2/d^2, where F is the force of attraction, G is the universal Gravitation constant, m1 and m2 are the masses, and d is distance between the two object's centers of mass. Since we are interested in how two different masses (a "light" object and a "heavy" object") would accelerate towards a common mass (the Earth or Jupiter, for example), the we can set m1 equal in both situations. And since we are interested in both bodies being the same distance from the Earth or Jupiter, we can set d (and therefore d^2) equal in both situations. What we are left with is that F is proportional to m2. Thus, the Earth's (or Jupiter's) gravity/mass pulls on a more massive object more strongly than it pulls on a less massive object. That will tend to make a more massive object accelerate more quickly. However, Newton's second law of motion is a = F/m, where a is the acceleration, F is the applied force, and m is the mass. This shows that a more massive object is harder to accelerate. So although the gravitational force exerted on m2 is greater for a greater mass of m2, the greater mass of m2 also resists acceleration to a greater degree: in fact, to the same exact degree. Mass, in a sense, cancels out of the equations. This has been verified on the Moon (where air resistance is absent), where a feather and a hammer were dropped and both accelerated at the same rate.

08-05-2002, 10:13 PM	#13
DMB Guest Posts: n/a	I think the original question is a bit hard on Newton. There is no evidence I am aware of that he ever performed an experiment of dropping two weights as described. It is to Newton that we owe the formula F = Gm1m2/r2, which is about motion in vacuo. He derived the formula in order to find something that would be compatible with Kepler's (observational) laws of planetary motion, and he invented the differential calculus to do so**. He was a great mathematician. If you apply the formula to any two masses, it is obvious that both are affected by the presence of the other. This means that the earth does not revolve round the centre of the sun, but round their mutual centre of mass, to a first approximation that ignores the gravitational contribution of other nearby masses. But the sun is so much more massive than the earth that the centre of mass is still near enough to the centre of the sun not to make much of a difference. When you drop two weights near the surface of the earth, their acceleration due to gravity is the same, but because of the existence of the earth's thick atmosphere, gravity is not the only significant force,and hence acceleration, acting on the weights.

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