Gravitation and a Falling Stone
As said by this law of universal gravitation every object in this universe is attracting every other object towards it. This force of gravitational attraction between any two objects in the universe is inversely proportional to the square of the distance between the objects. This force is again directly proportional to the product of the masses of these two objects involved. Here we will derive the formula of the Gravitational Force first and then analyze the system made of a falling stone and the earth to find out their individual acceleration.
Gravitation: Deriving the Formula of Force of Attraction
Revisiting what we discussed in the last post once more for a quick memory loading:
Say FG is the force of gravitational attraction (force of attraction due to Gravitation) between any two objects.
And say, m1 is the mass of one object and m2 be the mass of the second one. d is the distance between the centers of these two objects.
(Objects are assumed to be spherical.)
Then FG ∞ m1.m2
FG ∞ 1/d2
So FG ∞ m1.m2/ d2
So force of attraction due to Gravitation between 2 objects= FG = (G.m1.m2)/ d2
where G is a constant, called Gravitational constant.
Falling stone: A Case Study
That means this force due to gravitation between any two objects will decrease as the distance between them increases.
If the distance is decreased i.e. the objects are brought closer to each other, then the gravitational attraction between them will increase.
Similarly objects with more masses will experience more of this attractive force and vice versa.
As we know if there is a force applied on any object (net force), an acceleration of the object is produced.
Now let’s consider a system consisting of a stone in air (say when released from the roof of a house) with mass m and the earth with mass M with a distance between them being d at some instance. Undoubtedly M is many times larger than m.
Acceleration of the stone and the earth
Each of these (the stone and the earth) will experience a force of attraction, which is say F. F = (G.m.M)/ d2
Now we know a force will create an acceleration. Acceleration of a body = Force on the body/mass of the body.
Here the acceleration of the stone with mass m =a1 = force/mass= GM/d2 ___________ (1)
And the acceleration of the earth with mass M =a2 = force/mass= Gm/d2 ____________(2)
Now from the above 2 equations we can easily find out the ratio of the acceleration of the earth and that of the stone, which is found to be m/M or mass of the stone/mass of the Earth.
This value is very very small as M is much greater than m.
From the above 2 expressions, we can see very clearly that the acceleration of the earth is very very negligible compared to that of the stone. Credit for this goes to huge difference between the mass of earth and that of the stone.
Because of this we see the stone to fall towards the earth with an acceleration of nearly 10 meter/second2 and the movement of the earth towards the stone is not observable.