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Many modern technologies, such as computers and smartphones, are built on the principles of Physics Topics such as quantum mechanics and information theory.

## Newton’s Universal Law of Gravitation with Examples

In the previous chapter we have studied that a force is necessary to make a body move. That is, a force is necessary to produce motion in a body. Now, if we drop a piece of stone from some height, the stone falls down towards the earth. Since the stone starts moving downwards (it is in motion), therefore, a force must be acting on it. This force is due to the attraction between the earth and the stone and it is called the gravitational force of earth (or gravity of earth). Thus, a stone dropped from a height falls towards the earth because the earth exerts a force of attraction (called gravity) on the stone and pulls it down.

It is said that once the great English scientist Isaac Newton was sitting in his garden under an apple tree when an apple from the tree fell on him. Newton said that an apple falls down from the tree towards

the earth because the earth exerts a ‘force of attraction’ on the apple in the downward direction. This force of attraction exerted by the earth is called its gravity. Similarly, a leaf falls down from a tree due to gravity of earth. In fact, the earth attracts (or pulls) all the objects towards its centre. The force with which the earth pulls the objects towards it is called the gravitational force of earth or gravity (of earth). It is due to the gravitational force of earth that all the objects fall towards the earth when released from a height.

The gravitational force of earth (or gravity of earth) is responsible for holding the atmosphere above the earth; for the rain falling to the earth; and for the flow of water in the rivers. It is also the gravitational force of earth (or gravity of earth) which keeps us firmly on the ground (and we do not float here and there). Similarly, a ball thrown upwards also falls back to the earth due to the gravitational force of the earth. Since the gravitational force of earth (or gravity of earth) pulls the objects in the downward direction, therefore, a force has to be applied by us to raise an object to a height above the surface of earth (to overcome the gravitational force of earth).

### Every Object in the Universe Attracts Every Other Object

When we drop an object, it falls towards the earth. This means that the earth attracts the various objects towards its centre. Newton said that it is not only the earth which attracts the other objects, in fact, every object attracts every other object. So, according to Newton, every object in this universe attracts every other object with a certain force. The force with which two objects attract each other is called gravitational force (or gravity). The gravitational force between two objects acts even if the two objects are not connected by any means.

If the masses of the objects (or bodies) are small, then the gravitational force between them is very small (which cannot be detected easily). For example, ‘two stones’ lying on the ground attract each other but since their masses are small, the gravitational force of attraction between them is small and hence we do not see one stone moving towards the other stone. If, however, one of the objects (or bodies) is very big (having a very large mass), then the gravitational force becomes very large (and its effect can be seen easily). For example, a stone (lying at a height) and the earth attract each other, and since the earth has a very large mass, the gravitational force of attraction between them is very large due to which when the stone is dropped, it moves down towards the earth. Please note that the ‘gravitational force’ or ‘gravity’ is always a force of attraction between two objects (or two bodies). We will now study the universal law of gravitation.

The universal law of gravitation was given by Newton. So, it is also known as Newton’s law of gravitation. According to universal law of gravitation : Every body in the universe attracts every other

body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The direction of force is along the line joining the centres of the two bodies.

Suppose two bodies A and B of masses m_{1} and m_{2} are lying at a distance r from each other (see Figure). Let the force of attraction between these two bodies be F. Now, according to the universal law of gravitation:

(i) the force between two bodies is directly proportional to the product of their masses. That is,

F ∝ \(\frac{m_1 \times m_2}{r^2}\) …….. (1)

(ii) the force between two bodies is inversely proportional to the square of the distance between them. That is,

Gravitational force, F = G × \(\frac{m_1 \times m_2}{r^2}\)

where G is a constant known as “universal gravitational constant”. The value of gravitational constant G does not depend on the medium between the two bodies. It also does not depend on the masses of the bodies or the distance between them.

The above formula gives the gravitational force of attraction F between two bodies of masses m_{1} and m_{2} which are at a distance r from one another. This formula is applicable anywhere in this universe, and it is a mathematical expression of universal law of gravitation. Since the gravitational force between two bodies is inversely proportional to the square of the distance between them, therefore, if we double the distance between two bodies, the gravitational force becomes one-fourth and if we halve the distance between two bodies, then the gravitational force becomes four times.

Newton’s law of gravitation is called universal law of gravitation because it is applicable to all the bodies having mass : whether the bodies are big or small; whether the bodies are terrestrial (which are on earth) or celestial (which are in outer space) such as the stars, planets and satellites.

### Units of Gravitational Constant, G

According to universal law of gravitation, the gravitational force F between two bodies of masses m_{1} and m_{2} placed at a distance r apart is given by :

F = G × \(\frac{m_1 \times m_2}{r^2}\)

This can be rearranged to get an expression for the gravitational constant G as follows :

G = F × \(\frac{r^2}{m_1 \times m_2}\)

Now, the unit of force F is newton (N), the unit of distance r is metre (m), and the unit of masses m_{1} and m_{2} is kilogram (kg). So, the SI unit of gravitational constant G becomes :

\(\frac{\text { newton(metre) }^2}{(\text { kilogram })^2}\) or \(\frac{\mathrm{Nm}^2}{\mathrm{~kg}^2}\) or Nm^{2}/kg^{2} or Nm^{2}Kg^{-2}

### Value of Gravitational Constant, G

If the masses m_{1} and m_{2} of the two bodies are 1 kg each and the distance r between them is 1 m, then putting m_{1} = 1 kg; m_{2} = 1 kg and r = 1 m in the above formula, we get:

F = G (when m_{1} = m_{2} – 1 kg and r = 1 m)

Thus, the gravitational constant G is numerically equal to the force of gravitation which exists between two bodies of unit masses kept at a unit distance from each other. The value of universal gravitational constant G has been found to be 6.67 × 10^{-11}Nm^{2}/kg^{2}. The extremely small value of gravitational constant (G) tells us that the force of gravitation between any two ordinary objects will be very, very weak.

The value of gravitational constant G has been determined by performing careful experiments with two gold balls. Two heavy gold balls were suspended near each other by strong but delicate threads. The force between the two gold balls was then measured. Knowing the force, the masses of the gold balls, and the distance between them, the value of G was calculated by using Newton’s gravitation formula. The experiment with gold balls gave the value of gravitational constant G as 6.67 × 10^{-11} Nm^{2}/kg^{2}. Please note that though the exact value of gravitational constant G is 6.67 × 10^{-11} Nm^{2}/kg^{2} but many times it is taken as 6.7 × 10^{-11} Nm^{2}/kg^{2} in the numerical problems just for the sake of convenience in calculations.

The formula : F = G × \(\frac{m_1 \times m_2}{r^2}\) can be used to calculate the gravitational force anywhere in this universe.

In all the cases, the value of G is 6.67 × 10^{-11} Nm^{2}/kg^{2} and m_{1} and m_{2} are the masses of the two bodies and r is the distance between the centres of the two bodies. The force of gravitation is a vector quantity and it acts along the line joining the centres of mass of the two bodies.

### Gravitational Force Between Objects of Small Size and Big Size

Suppose two balls of 1 kg each are placed with their centres 1 m apart, then they attract each other with a force given by,

F = G × \(\frac{m_1 \times m_2}{r^2}\)

Putting G = 6.67 × 10^{-11} Nm^{2}/kg^{2}; m_{1} = 1 kg; m_{2} = 1 kg and r = 1 m

We get : F = \(\frac{6.67 \times 10^{-11} \times 1 \times 1}{(1)^2}\) newtons

or F = 6.67 × 10^{-11} newtons

It is obvious that the gravitational force of attraction between twö balls of 1 kg each and 1 m apart is

6.67 × 10^{-11} newtons and this is a very small force.

Though the various objects on this earth attract one another constantly, they do not cause any motion because the gravitational force of attraction between them is very small. If, however, at least one of the bodies is large (like the sun or the earth), then the gravitational force becomes very large. For example, the earth has a very large mass and this force is quite large between the earth and other objects. Hence, objects when thrown up, fall back to the earth. This point will become more clear from the following example in which we are calculating the gravitational force between a ball having 1 kilogram mass and the earth.

**Example Problem 1.**

Calculate the force of gravitation due to earth on a ball of 1 kg mass lying on the ground. (Mass of earth = 6 × 10^{24} kg; Radius of earth 6.4 × 10^{3} km; and G = 6.7 × 10^{-11} Nm^{2}/kg^{2})

**Solution.**

The force of gravitation is calculated by using the formula:

F = G × \(\frac{m_1 \times m_2}{r^2}\)

Here, Gravitational constant, G = 6.7 × 10^{-11} Nm^{2}/kg^{2}

Mass of earth, m_{1} = 6 × 10^{24} kg

Mass of ball, m_{2} = 1 kg

And, Distance between centre, of earth and ball r = Radius of earth

= 6.4 × 10^{3} km = 6.4 × 10^{3} × 1000 m

= 6.4 × 10^{6} m

Now, putting these values in the above formula, we get:

F = \(\frac{6.7 \times 10^{-11} \times 6 \times 10^{24} \times 1}{\left(6.4 \times 10^6\right)^2}\)

or F = 9.8 newtons

Thus, the earth exerts a gravitational force of 9.8 newtons on a ball of mass 1 kilogram. This is a comparatively large force. It is due to this large gravitational force exerted by earth that when the 1 kg ball is dropped from a height, it falls to the earth.

It is clear from tire above discussion that we can ignore the gravitational force between two balls of 1 kilogram each because it is very small. But we cannot ignore the gravitational force between a 1 kilogram ball and the earth because it is quite large (due to the large mass of earth). And when both the objects (or bodies) are very big, having very large masses, then the gravitational force of attraction between them becomes extremely large. For example, the sun, the earth and the moon have

extremely large masses, therefore, the gravitational force of attraction between the sun and the earth (or other planets), or between the earth and the moon, is extremely large. This point will become more clear from the following example in which we are calculating the gravitational force between the earth and the moon.

**Example Problem 2.**

The mass of the earth is 6 × 10^{24} kg and that of the moon is 7.4 × 10^{22} kg. If the distance between the earth and the moon be 3.84 × 10^{5} km, calculate the force exerted by the earth on the moon. (G = 6.7 × 10^{-11} Nm^{2} kg^{-2})

**Solution.**

The force exerted by one body on another body is given by the Newton’s formula :

F = G × \(\frac{m_1 \times m_2}{r^2}\)

Gravitational constant, G = 6.7 × 10^{-11} Nm^{2} kg^{-2}

Mass of the earth, m_{1} = 6 × 10^{24} kg

Mass of the moon, m_{2} = 7.4 × 10^{22} kg

And, Distance between the earth and moon, r = 3.84 × 10^{5} km

= 3.84 × 10^{5} × 1000 m

= 3.84 × 10^{8} m

Putting these values in the above formula, we get:

F = \(\frac{6.7 \times 10^{-11} \times 6 \times 10^{24} \times 7.4 \times 10^{22}}{\left(3.84 \times 10^8\right)^2}\)

F = 2.01 × 10^{20} newtons

Thus, the gravitational force exerted by the earth on the moon is 2.01 × 10^{20} newtons. And this is an extremely large force. It is this extremely large gravitational force exerted by the earth on the moon which makes the moon revolve around the earth.