**What is Gravitation?**

Gravitation is a fundamental force of nature that describes the force of attraction between all objects present in the universe.

**Concept of Gravitation**

Nature comprises many mysteries. One such mystery that remained unsolved for quite a long time is gravitation. No one knew how we are staying on the surface of the earth, or why any object when thrown up, always returns to the ground.

In the 17th Century, legendary scientist Sir Isaac Newton first came up with the concept of gravity. His law of gravitation changed science and helped it shape as we know it now. Newton’s law of gravitation describes that two isolated bodies in the universe always attract each other. Later, another legendary scientist Albert Einstein established the theory of general relativity to describe gravity. Einstein stated that the gravitational force among two objects follows relativity, or their force of attraction is relative to each other.

Our planet Earth’s gravity is so strong, that it keeps us on the surface. Not only that, but the water present in the water bodies, and the atmosphere which supports life on Earth are all present due to the gravitational force of our planet. Nature has four fundamental forces, and gravity is one of them.

## What is Newton’s Law of Gravitation?

Newton’s law of gravitation states that every matter particle that exists in this universe has an attraction for the other particles. Einstein later established the same theories with his theory of general relativity. This force of attraction between them is called gravitation, and this force varies-

- Directly to the product of the masses between the said two particles.
- Inversely to the square of the distance between the said two particles.

This force of attraction between these two particles is a fundamental force, and it has been termed gravitation. The action of this force is always to the line segment that joins the centers of the two different particles. Also, both the particles exert gravity, which means it is a mutual force.

### Formula

In Physics, the force of gravitation is expressed with formulas. For instance, let us consider the two objects between which we are measuring gravity are m_{1} and m_{2} respectively. Let their centers are joined by a straight line, and the length of this line is r. Also, let the force of attraction between these two objects has a magnitude of F. Therefore, we can write the following statements as per the law of gravitation-

$\begin{array}{l}\text{F}\propto {m}_{1}{m}_{2},\\ \text{and}\\ \text{F}\propto \frac{1}{{r}^{2}}\end{array}$

If we combine both of these factors, we get-

$\text{F}\propto \frac{{m}_{1}{m}_{2}}{{r}^{2}}$

The above expression can also be written as-

$\text{F}=\text{G}\frac{\left({m}_{1}{m}_{2}\right)}{\left({r}^{2}\right)}$

Here, we define the term G as the gravitational constant. It is also known as the constant of universal gravitation and is used as a constant of proportionality. This is an important value that helps us solve many complex equations related to general relativity and gravitation.

## What is the Gravitational Constant G?

In the above equation, if we consider the value of both m_{1} and m_{2} as 1, and we consider the radius of the distance between them to be 1 as well, then the above equation becomes F = G.

Hence, we can define the constant of universal gravitation G as the magnitude of the attractive force that exists between two objects that both have a unit mass, and whose distance between the centres is unity as well.

Usually, the standard value of G is taken as 6.67 x 10^{-11} N m^{2}/kg^{2}.

## What is Gravitational Potential Energy?

The work that needs to be done in arranging two or multiple bodies in their places when they interact with each other is known as the gravitational potential energy.

This potential energy is nothing but the work done on the bodies to bring them into their current positions from infinity.

## What is Gravitational Potential Energy of a System?

When multiple bodies are assembled in their respective positions, then the work done to do so can be said to be the system’s gravitational potential energy. This energy is a change in gravitational potential energy.

By definition, it has a magnitude equal to the negative value of the work done by the force of gravitation. This is because the system changes configuration in doing so.

We can express it as follows-

${\text{U}}_{\text{f}}-{\text{U}}_{\text{i}}={\text{W}}_{\text{gr}}$

Where U_{f} ,U_{i} are final and initial gravitational potential energy and W_{gr} is the work done by force of gravity. If we are talking about two-point masses m_{1} and m_{2}, and if their radius of the distance of separation is changed from r_{1} to r_{2}, then the change occurring in the gravitational potential energy is given by-

$\text{U}\left({r}_{2}\right)-\text{U}\left({r}_{1}\right)=\text{G}{m}_{1}{m}_{2}\left[\frac{1}{{r}_{1}}-\frac{1}{{r}_{2}}\right]$

If such a case arrives when the two objects are at infinite separation, and the value of the gravitational potential energy is considered to be zero, then we can write the gravitational potential energy for two mass objects at a radius of the distance of r between them is-

$\text{U}\left(r\right)=-\text{G}\frac{{m}_{1}{m}_{2}}{r}$

## What is Gravitational Potential?

When we consider any material object, then the gravitational field around it can be defined by the gravitational intensity vector, or ‘$\overrightarrow{\text{E}}$’. However, a scalar function can also be used to describe the gravitational field around the object. This is known as the gravitational potential, and it is denoted by V.

By definition, the gravitational potential of a test mass object at any particular point is the potential energy per unit mass, when the object is placed at that point.

We denote ii as-

$\text{V=}\frac{\text{U}}{m}$

Here, m is the test mass and U denotes the gravitational potential energy.

When we take the point of reference far enough, or when the distance of the reference point is infinity, then the potential of the particular point which lies in the gravitational field is given by the total work completed by an external agent per unit mass. This work done is measured while the test mass is brought to the point from infinity.

When we solve the equations, we see-

$\text{V=}\frac{\text{G}m}{r}$

We know that the potential quantity is scalar in nature. Hence, when many object particles are present in a point that lies in the field of gravitation, we find the arithmetic sum of all the potentials that are generated by the object particles at that point. This is the resultant potential.

The potential and the gravitational field are closely related, and they can be expressed as-

$\text{E=-}\frac{dV}{dr}$

## What are Kepler’s Laws of Planetary Motion?

Scientist Johannes Kepler in the 17^{th} Century proposed that our planets are part of the solar system, and each planet revolves around the sun in a planetary motion. He came up with three laws. These laws governed the motion of the planets around the sun. The laws are as follows-

- The planets revolve around the sun in an elliptical shape. One focus of the ellipse contains the sun’s center.
- If we join an imaginary line between the centers of the sun and planets, then with equal intervals of time, we will get equal areas.
- If we consider the ratio of the square of periods between two planets in the solar system, and the ratio between the cubes of their distance with respect to the sun, then the value of these ratios is equal.

Gravitation helps in keeping the planets in orbit. It helps in holding the Earth’s atmosphere, keeps the artificial and natural satellites into orbit, and helps in sustaining life on our planet. With the help of the theories presented by Newton and Einstein, many complex gravitational problems are solved that aid interplanetary missions.

## Practice Problem

If two objects weighing 5000 kg and 4000 kg are separated at a distance of 1 m, then calculate the force of attraction between them. Take the value of G = 6.67 x 10^{-11} N m^{2}/kg^{2}.

Here, we have m_{1} = 5000 kg, m_{2} = 4000 kg, and r = 1 m.

By using the formula

$\text{F=G}\frac{\left({m}_{1}{m}_{2}\right)}{\left({r}^{2}\right)}$, we get-

$\begin{array}{c}\text{F}=\text{G}\frac{{m}_{1}{m}_{2}}{{r}^{2}}\\ =\frac{6.67\times {10}^{-11}\times 5000\times 4000}{{1}^{2}}\text{N}\\ =\text{0}\text{.001334N}\end{array}$

This is the required force of attraction between the two objects.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelors in Science (Physics)
- Masters in Science (Physics)

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