Newton's law of universal gravitation

[MrOnak]

I stumbled over something that I can't really wrap my head around... The wikipedia page for Newton's law of universal gravitation lists the following formula to calculate the gravitational pull:

F = G * ((m1 * m2) / r^2)

where

G = gravitational constant

m1 = mass of object 1

m2 = mass of object 2

r = distance between the centers of m1 and m2

That's fine at first glance. I understand what that means and it fit my understanding at first.

But one thing is odd about that formula: It implies that the gravitational pull for m1 towards m2 is the same as the one for m2 towards m1, regardless of their mass ratio. That would mean that a very heavy m1 would move the same distance towards m2 as a very light m2 would move towards m1 in the same timeframe due to the gravitational pull? I'm almost 100% sure that's a misinterpretation on my part but the formula certainly hints that way. By the way the image with F1 = F2 = ... is more explicit if my example is hard to understand.

Can anyone point out why my interpretation of that formula is wrong and keep my view of the world of physics intact? Thanks

[Xenotheos]

I haven't done physics in years and I wasn't all that great at it when I was in practice, but the link said F1 = F2 which means both their forces are the same. If you look at the equation for force, F = ma, where m is mass this means that while the force is equal the larger body has a lower acceleration than the smaller body.

[MrOnak]

Physics is 20 years away so yeah.. I know where you're coming from.

I think I understand what you're saying. Inertia comes to mind? The pull F is the same on both bodies but the smaller one is affected more due to lesss inertia? *scratches head*. It's late... I'll try that again tomorrow

[TheShadow1138]

Xenotheos is correct. Though the forces acting on both objects are the same, the accelerations they experience are very different. As an example, let's calculate the accelerations experienced by the Earth and a 100 kg mass.

F = (-6.67e-11*5.97e24*100)/6371000^2 = 0981.04 N

So we have the Earth and the 100kg mass exerting a 981.04 N force on one another. Now let's calculate the accelerations. (NOTE: I omit the negative sign from the force in the calculations as it serves only to indicate direction of motion.)

F/m_100kg = 981.03N/100kg = 9.81 m/s^2 (which is what we expect,the acceleration due to gravity at Earth's surface.)

F/m_Earth = 981.04N/5.97e24 kg = 1.64e-22 m/s^2

This second number is why they would not move the same distance in the same time. The 100kg mass's velocity will increase by 9.81 m/s every second it falls while the Earth's velocity increases by a very, very small amount, 1.64e-22 m/s every second. I hope that helps clear things up for you, if only a little.

[MrOnak]

That'll help me a lot TheShadow... tomorrow

Thanks a bunch

[YNM]

The total acceleration of any of the mass involved is exactly the sum of the pulling force/mass (ie. it's pull on the other mass) and pulled force/mass (other mass pull on this mass), so it doesn't matter who's pulling who. The difference one will see is on acceleration of the respective mass, as F = m.a, so the larger mass accelerates slower. And if it's going to collide, the larger mass will move shorther than the smaller mass (due to the smaller acceleration it has). This explains why Moon is bound to Earth, and why Moon makes tidal wave on Earth.

[Yoha]

To add to the discussion, this is the difference between the inertial and the gravitational masses. What we call the gravitational mass is a property of objects than makes them attract one another according to gravitational law (GMm/r²). On the other hand, the inertial mass is the tendency to an object to rest as it is, meaning keeping a linear movement with constant speed. We have tested and measured the values of these two different masses and, as far as we know, there are just the same. Thus, the acceleration of an object of mass m subject to the gravity of an object of mass M is GM/r^2, which does not depends on the mass of the former (ignoring air drag, all objects fall at the same speed; the Earth is barely moved by the gravity pull of a falling object).

[MrOnak]

Turns out a couple of hours of sleep is all I needed. Stupid me didn't recognize that F is measured in Newtons which, of course, means that its effect is relative to mass. TheShadow brought the key to that lock Rep duely added.

Thanks a bunch to everyone. Now back to my N-Body simulation code *sigh*

[Tex]

Ah, but you're forgetting the 1st law: An object at rest will stay at rest, an object in motion will stay in motion, yadda yadda yadda. The more massive object is harder to move, and so needs a stronger force to move the same distance. So while it is true that the force acting on both of the objects is the same, you would need more than that to move the more massive object the same distance. So the smaller one covers the distance.

In addition, if both objects had the exact same mass, then both objects would cover the same amount of distance.