# Gravitational Force Equation and Definition

Gravitation is one of the four fundamental forces of nature. Of all four basic forces, gravitational force is the one that we are particularly familiar with. The other forces are the weak nuclear force, electromagnetic force and strong nuclear force. The weak and strong forces play a special role at the level of subatomic particles. They are important to understand the cohesion of the matter around us.

## Gravity is the weakest of the four fundamental forces

If you compare the strengths of the four fundamental forces of physics, it becomes clear that gravitational force is the weakest of all forces. For example, a roof tile falling from a house does not reach the center of the earth, but splinters on the ground. This is because the electromagnetic forces between the atoms of the brick and the floor have caused a sudden repulsion. It is a force that gravitation couldn’t overcome. The strong force is even stronger than electromagnetism, and it even manages to hold protons of the same electrical charge together in an atomic nucleus. We owe the variety of chemical elements to the strong force, but also the weak force (radioactivity).

But the attribute weak is not synonymous with unimportant. Gravity is very important to us since it is the dominant force in the universe. Because gravitation and electromagnetic force have an infinite range. However, in contrast to electromagnetism, gravity has the property that it can not be shielded. The consequence is: gravitational forces dominate the universe. It is gravity that forms the large-scale structures. It makes the planets orbit the sun, it compresses massive stars into black holes at the end of their existence, and even brings galaxies and galaxy clusters together.

## Who discovered gravity?

Gravity is familiar to us in everyday life, it is also puzzling. It is by no means easy to understand what the nature of gravity is. Even in the 21st century, physicists and astronomers know a lot about the gravitational force, but even today we are still far from having understood everything. Is gravity a force at all?

### Ancient pioneer: Aristotle

One of the most well-documented pioneers in gravity research is the Greek scholar Aristotle (384 – 322 BC). Aristotle is actually better known as an important humanist, as a student of Plato and as an educator of Alexander the Great. Aristotle, however, also tried to explain the movement of the sun, the moon and other planets of the solar system known at that time by using rather simple models. He had geocentric view of the world: the resting earth was at the center of this model, and the sun, moon and planets orbit around it.

Aristotle also tried to explain falling objects. For him, the straight fall path was proof that the earth was at rest. These first considerations with the principle of ‘nature observation – explanatory model’ are in the spirit of the Enlightenment epoch almost two thousand years later (experiment – theory). Aristotle thus presents the first (if not entirely convincing) phenomenological model for gravity.

### Galileo – gravitational researcher and pioneer of astronomy

The Italian physicist, mathematician and philosopher Galileo Galilei (1564 – 1642) was the first to systematically and mathematically research gravitation. Galileo is said to have performed fall experiments on the Leaning Tower of Pisa to test his hypothesis whether the weight or density of a body determines how quickly the body falls. Galileo also carried out numerous mechanical experiments with pendulums and rolling objects on an inclined plane. He explained the trajectory of projectiles in the gravitational field by superimposing two movements, namely an evenly accelerated falling movement and a uniform rectilinear projectile movement (superposition principle) and proved the parabolic trajectory.

### Newton’s law of universal gravitation

The study of gravitational force has achieved a tremendous breakthrough through the English universal scholar Sir Isaac Newton (1643- 727). Newton has become famous for many achievements: the foundation of differential and integral calculus, discoveries in optics (color theory, corpuscular theory of light) and the theory of gravity named after him today.

Newtonian gravitational physics is the first gravitational theory worthy of the name theory because it is a consistent, comprehensive approach and not merely phenomenology or hypothesis. Newton presented this theory in his work Mathematical Principles of Natural Philosophy. This is the first standard work of theoretical physics.

Newton devoted himself to optics and gravitation in 1665. The story of the apple falling from the tree, which is said to have hit Newton on the head and inspired him to study gravitational theory, is probably a myth. Newton only wrote that a falling apple made him think about gravitation. At the age of 27 (on the recommendation of the predecessor of this office) Newton became a Lucasian Professor of Mathematics – a chair at the University of Cambridge, which, incidentally, was also held by Stephen Hawking in recent past.

### Gravitational force equation

F = G*((m sub 1*m sub 2)/r^2)

• F is the force of attraction between two bodies
• G is the universal gravitational constant
• m sub 1 is the mass of the first object
• m sub 2 is the mass of the second object
• r is the distance between the centers of each object

### Planetary motion and gravity

Newton was familiar with Johannes Kepler’s discovery that the planets move around the sun in elliptical orbits. In 1666, Newton began to seek a physical explanation of these purely empirical Kepler laws. He was able to explain Kepler’s second law in 1679 by theorizing that a central attractive force originates from the sun.

A meeting of three members of the Royal Society in 1684 was to become a key event. Newton’s adversary Robert Hooke, astronomer Edmond Halley and architect Christopher Wren met here. They discussed a force that is proportional to the inverse distance square and determines the planetary motion. Inspired by this discussion, Halley asked Newton about the orbital shape of a celestial body that resulted from this force law. Newton had calculated this question years ago and knew that it had to be an elliptical orbit. The detailed elaboration of this calculation finally led to a year and a half of Newton’s creative phase and the publication of the Principia in 1687.

Book I of the Principia contains the three laws of motion that are now taught as Newtonian laws:

• The first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. Also known as inertia.
• The second law explains how the velocity of an object changes when it is subjected to an external force.
• The third law states that for every action (force) in nature there is an equal and opposite reaction.

Book II is a textbook on fluid mechanics. Finally, in Book III, Newton presents his law of gravitation and demonstrates the validity of this theory of gravitational forces using the movements of planets and comets.

### Einstein’s general theory of relativity

In 1916 Albert Einstein (1879 – 1955) presented a completely new view of gravity. This year he published the General Theory of Relativity, a theory of gravitation that sees gravity not as a force but as a geometric property of space and time. This theory of gravity was preceded by the special theory of relativity, which is not a theory of gravity. Nevertheless it initiated the revolutionary redefinition of the terms energy, mass, time and space. According to Einstein, space and time are linked together to form a four-dimensional structure: the Spacetime.

In general, spacetime is dynamic. It constantly changes its curvature properties. This happens especially with gravitational waves, which can also be described with Einstein’s theory. They are bumps in space-time that change with the speed of vacuum spread. This is an important difference to Newtonian physics: In Einstein’s theory, gravitation does not spread at random, but exactly with the curvature. Einstein’s theory has proven itself many times in experiments.

## Cosmological meaning of gravitation

The importance of gravitation for the entire universe is based on two properties that were mentioned at the beginning. In principle, gravitation has an arbitrary range and it can not be shielded. Because of the first property, gravitation can also work over extremely large distances. Up to billions of light years, i.e. scales that are as large as the cosmos itself. Because of the second property, non-shield ability, there is hardly any way to stop gravitation (unless through anti-gravity).

The consequence is what astrophysicists call gravitational instability. This can be descriptively described in such a way that matter begins to ‘clump’ due to the influence of gravity. It explains the way in which structures evolve in the universe. When an astronomical body contracts due to the influence of its own gravity, gravitational collapse sets in. Later, this lump fragmented into smaller ones. These scenarios are essential to understand the origin of the large-scale structure in the cosmos.

The first generation of stars was formed by gravitational instabilities from the initial distribution of the ‘primary gas’ that arose in primordial gas clouds. The first galaxies emerged from this under the influence of gravity – at least in the hierarchical growth scenario.

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