## Key Concepts:

- The universal law of gravitation
- Nature of gravitational force
- Importance of gravitational force

## Introduction:

Sir Isaac Newton said all things fall towards the Earth. He was puzzled by the problem that everything falls when released. It is believed that the sight of a falling apple made Newton wonder if the force that caused the apple to fall might extend to the Moon or even beyond.

When a small object is tied and whirled around the object, it moves in a circular path and changes its direction of motion at every point. We see that the force which is causing it to move in a circular path is always directed towards the center of the path. The force which is pulling the object towards the center of the circular path is known as the centripetal force.

The tension in the string will provide the necessary centripetal force. When the string breaks, the object moves along the straight line tangentially to the circular path.

Similarly, Newton reached a conclusion that the Moon revolves around the Earth in a circular orbit, and at each point of its orbit, the Moon is attracted by the Earth. Instead of moving in a straight line.

The necessary centripetal force is provided by the gravitational force exerted by the Earth.

### Explanation:

Newton discovered that not only the objects that are falling and the moon, but all objects in this universe attract each other, and the same force is responsible for this attraction. He called it a Gravitational force.

In the year 1666, Newton studied Kepler’s work on planetary motion. He found that the magnitude of the force F on a planet, due to the sun, varies inversely with the square of the distance **r** between the center of the planet and the sun.

F α 1/r^{2}

This force acts in the direction of the line connecting the centers of the two objects.

### Proof:

The gravitational force follows the inverse square law; for example, if the weight of an apple on the surface of the Earth (radius = d) is 1 N, then as we increase the distance to 2d, 3d, 4d, and 5d, the weight of the apple becomes (1/4) N, (1/9) N and (1/16) N.

### Example:

Similarly, suppose the force between two bodies is 80 N when they are at a distance of 0.1 m. If we keep increasing the distance between the two masses as shown in the table, we get a graph between force and distance, as shown in the figure.

On the basis of his own third law of motion and experimental findings, Newton reached the conclusion that the gravitational force of attraction between two objects must be proportional to the product of their masses.

If A and B are two bodies with masses m_{1} and m_{2}, separated by a distance r. The force F between two objects is given by,

F α m_{1} x m_{2}

Thus, the Gravitation force acting between two bodies depends on the product of the mass of the bodies and the distance between the two bodies. If we increase the masses, the force increases, and if we increase the distance between the bodies, the gravitational force decreases.

**The universal law of gravitation in mathematical form:**

The figure below describes how the universal law of gravitation is derived mathematically.

F α m_{1}————-(i)

F α m_{2} ————(ii)

F_{ }α 1 / r^{2} ————-(iii)

From the above equation, we can rewrite them as the following:

F_{ }α m1 m2 / r^{2} ————-(iv)

If we remove the proportionality sign, we get proportionality constant G as the following:

F = G m1 m2 / r^{2}

The above equation is the mathematical representation of Newton’s universal law of gravitation.

Hence, G = Fr^{2} / m1 m2

SI Unit of G is: Nm^{2} kg^{-2}

Value of G = 6.673 × 10^{–}^{11} Nm^{2} kg^{-2} (was found out by Henry Cavendish (1731–1810)

The proportionality constant G is also known as the Universal Gravitational Constant.

**The universal law of gravitation:**

Every object in the universe attracts every other object with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them.

The force is along the line joining the centers of two objects. The law is applicable to all bodies, whether the bodies are big or small, whether they are celestial or terrestrial. Hence, it is universal.

**Nature of Gravitational force:**

- Gravitational force is a non-contact force.
- Gravitational force is always attractive in nature.
- Gravitational force acts because of mass.
- Gravitational force is a long-range force.
- It is always directed towards the center of mass.
- Gravitational force doesn’t depend upon the medium between the objects.
- Gravitational force is the weakest fundamental force in nature

**Importance of gravitational force:**

- Gravitational force binds us to the Earth.
- Gravitation causes tides due to the Moon and Sun on Earth.
- The Moon revolves around the Earth because of gravity.
- Planets move around the sun because of gravity.

**Numerical:**

Calculate the gravitational force of attraction between the Moon and the Earth. Given the mass of the Earth is 5.972 × 10^{24 }kg, the mass of the Moon is 7.35 × 10^{22} kg, and the distance between them is 384,400 km.

**Solution:**

Given that,

m_{1} = 5.972 × 10^{24} kg

m_{2} = 7.35 × 10^{22} kg,

r = 384,400 km

G = 6.673 × 10^{-11 }Nm^{2}/kg^{2}

r = 3.84 × 10^{5} km = 3.84 × 10^{8} m

**F = G (m**_{1}**m**_{2}**/r**^{2}**)**

F = (6.673 × 10^{-11}) (5.972 × 10^{24} × 7.35 × 10^{22})/ (3.84 × 10^{8})^{2}

= 19.86 × 10^{19} N

**F = 1.986 × 10 ^{20} N **

The gravitational force of attraction between the Moon and the Earth is approximately **1.986 × 10 ^{20} N**, which is huge. That is why it is apparent.

### Summary:

**Centripetal force**: The force which is pulling the object towards the center of the circular path is

known as the centripetal force.

The Moon revolves around the earth in a circular orbit and at each point of its orbit, the moon

is attracted by the earth. Instead of moving in a straight line. The necessary centripetal force is

provided by the gravitational force exerted by the earth.

**The universal law of gravitation**: Every object in the universe attracts every other object with a

force that is proportional to the product of their masses and inversely proportional to the square of the distance between them.

F=Gm_{1}m_{2} / r^{2}

Nature of gravitational force:

- Gravitational force is a non-contact force.
- Gravitational force is always attractive in nature.
- Gravitational force acts because of mass.
- Gravitational force is a long-range force.
- It is always directed towards the center of mass.
- Gravitational force doesn’t depend upon the medium between the objects.
- Gravitational force is the weakest fundamental force in nature.

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