Imagine there are two bodies of mass 1 Kg each separated by some distance.Then by Newton's formula of calculating gravitational force we can see that the gravitational force between them is very small.Now as we know that $a=\frac Fm$ we can find the acceleration of any one of the two bodies which is too very small but $not=0$.Does that mean that the two bodies will eventually come an meet (maybe after 1000 years) because of this acceleration?Please help.
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$\begingroup$ They won't necessarily meet if the are separating with escape velocity or more. $\endgroup$– John AlexiouCommented Aug 14, 2015 at 19:05
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3 Answers
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If they're at rest to start with, and if there is nothing else in the Universe, then Yes. The gravitational force increases their velocity to a finite value (which still increases) after a finite time which is enough for them to collide at some moment. The time may be very very long if the objects are very light and/or very distant, however.
(You may avoid it. If the two objects are a ball and a ring, with the hole ready to make the ball go through, they will not collide but they will get through, back and forth.)
If the initial velocity of the objects is nonzero, and has a component perpendicular to the separation of the objects, the relative separation of the two objects will draw a hyperbola, parabola, or an ellipse. They are very likely to avoid each other. The simplest example of this scenario is that the two objects may end up orbiting each other. The Earth orbits the Sun and they will never hit each other. (The Sun will die before it would happen due to the perturbations of the orbits by other bodies.)
You may also consider some corrections from general relativity. Orbiting bodies in general relativity emit gravitational waves which makes them lose energy and their distance decreases (because the potential energy does). So they eventually hit each other, anyway. After a time scale that is even longer than the previous ones, this effect wins and the two objects collapse into one.
In practice, one usually neglects some or most or all of these tiny effects. In the real world, there are usually much stronger or faster effects than some very weak gravitational force between two very light bodies. This makes the idealization considered above irrelevant and inapplicable.
answered Aug 14, 2015 at 17:39
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It just depends upon the masses of the two bodies: not on the ground because static friction would also be acting on it.
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$\begingroup$ It would also depend on the initial velocities of the two bodies. $\endgroup$– Bill NCommented Aug 14, 2015 at 20:26
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If there is nothing that stops both masses from accelerating towards each other (like a barrier or other larger gravitational influences, for instance), then yes: they will very slowly accelerate towards each other, albeit depending on actual values of mass and distance at a very, very slow rate.
You might want to look the Cavendish' Experiment, sometimes called 'Weighing the Earth' which used surprisingly small masses close together which then really moved closer together. Basically from that small motion the size of gravitation could then be deduced.
