To understand the universal law of gravitation, let us first read an interesting story. Newton was sitting under an apple tree, and an apple fell on him; this made him think that the apple had fallen due to the downward pull of the Earth on the apple. When a body is thrown up, it reaches a certain height and then falls down.
The downward pull of the Earth on the body decreases its velocity until it becomes zero and the same downward pull causes the apple to fall down from the tree. Newton thought that if Earth can attract an apple, it can also attract the moon. Is the force the same in both cases? Yes, it is the same type of force, responsible in both cases.
Newton argued that at each point of its orbit, the moon falls towards the Earth, instead of going off in a straight regular path. So, he thought that the Moon must be attracted to the Earth. But the question arises, why do we not really see the moon falling towards the Earth?
To understand this, we will do an activity– Take a piece of a stone tied to one end of a string. Hold the other end of the string in your hand and whirl it round like shown.
You will notice that the stone moves in a circular path with a certain speed but its direction of motion changes continuously. The change in direction of motion involves change in the velocity of the stone. Thus it is an accelerated motion.
The external force ‘F’ that causes this acceleration and that keeps the stone moving uniformly along the circular path is acting towards the centre of the circular path. This is called a centripetal force (center seeking force).
At any instant, if we release the string, the stoneflies along the tangent to the circular path at that instant. This is because, the moment the string is released, centripetal force is no longer provided and the stone is free to fly off along the tangent. Based on this activity, we can conclude that the motion of the moon around the earth is due to the centripetal force provided by the force of attraction of the earth on the moon.
So Newton concluded that-
There is a force of attraction acting between any two bodies in the universe which may be terrestrial or celestial. This phenomenon is called Gravitation.
On the basis of the factors on which this force depends, Newton formulated the Universal Law of Gravitation. In this blog, we will study Newton’s Law of Gravitation.
Newton’s Universal Law of Gravitation
Define Universal Law of Gravitation: Every object in the Universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The direction of the force is along the line joining the centers of two objects.
The formula reveals that when m1 and m2 are large, F is large, i.e. the gravitational force of attraction between heavier bodies is larger than that between the lighter bodies.
If we double the distance between two bodies, the gravitational force of attraction between them becomes one fourth.
If the distance between the two bodies is halved, the gravitational force of attraction between them becomes four times.
Mathematical Expression of Newton’s Universal Law of Gravitation:–
Let A and B are two objects of masses ‘m1’ and ‘m2’ respectively and r is the distance between their centers.
If F is the gravitational force of attraction between these objects, then according to Newton’s Law of Gravitation, the force between the two objects is directly proportional to the product of their masses.
And the force is inversely proportional to the square of the distance between their centers.
Combining both the equations:-
F (m1 * m2)/ r2
Replacing the sign of proportionality
F = G * ( (m1 * m2)/ r2 )
Where G is a constant of proportionality and is called a universal gravitational constant.
From the above we have,
G = F * (r2/ (m1 *m2) )
So SI unit of G is
G = N m2/kg2
If m1 = m2 = 1 kg and r= 1 meters
Then G = F * ( (1)2 / (1 * 1)) = F
The value of G was found out by Henry Cavendish (1731- 1810)
Numerically G is equal to 6.67 * 10-11 N m2 / kg2
The value of G is always constant. It neither depends on the masses of the objects nor on the distance between their centers nor on their shape, size, nature or medium separating the bodies.
As the value of G is very small, therefore gravitational force between any two ordinary objects will be extremely weak.
From above we can say that, Universal Gravitational constant G is numerically equal to the gravitational force of attraction between two bodies, each of 1 Kg mass and kept at a distance of 1 meter from each other.
This law is applicable to all the bodies having mass, so it is called Newton’s Law of gravitation.
Note: ∞ = Proportional
What is the Importance of Universal Law of Gravitation?
Universal Law of Gravitation explains several phenomenon such as:-
1. The binding of the terrestrial objects around the Earth.
2. The holding of the atmosphere around the Earth.
3. Rainfall and snowfall on the Earth.
4. The flow of water in the rivers.
5. Revolution of artificial Satellites around the Earth.
6. Revolution of the Planets around the Sun.
7. The formation of tides, by the rising and falling of water level in the oceans due to the gravitational force of attraction, which the sun and the moon exert on the seawater.
8. The predictions about solar and lunar eclipse are made on the basis of this law.
Why an apple falls towards the Earth and why not the earth moves towards the apple?
The gravitational force of attraction between any two objects is always mutual, irrespective of the mass, size, shape, or distance between the objects. It means that if the earth attracts an apple, the apple also attracts the Earth with an equal and opposite force. Now the question arises, why an apple falls towards the Earth and why not the earth moves towards the apple?
To find the answer, let us understand this fact,
We know that according to the second law of motion, for a given force, acceleration is inversely proportional to the mass. This means that if the mass is less, acceleration is more and if the mass is more acceleration is less. Because, the mass of an apple is negligibly small as compared to that of the Earth, so we do not see the Earth moving towards the apple.