# Newton's Law of Gravity

Newton's law of gravity defines the attractive force between all objects that possess mass. Understanding the law of gravity, one of the fundamental forces of physics, offers profound insights into the way our universe functions.

## The Proverbial Apple

The famous story that Isaac Newton came up with the idea for the law of gravity by having an apple fall on his head is not true, although he did begin thinking about the issue on his mother's farm when he saw an apple fall from a tree. He wondered if the same force at work on the apple was also at work on the moon. If so, why did the apple fall to the Earth and not the moon?

Along with his Three Laws of Motion, Newton also outlined his law of gravity in the 1687 book Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), which is generally referred to as the Principia.

Johannes Kepler (German physicist, 1571-1630) had developed three laws governing the motion of the five then-known planets. He did not have a theoretical model for the principles governing this movement, but rather achieved them through trial and error over the course of his studies. Newton's work, nearly a century later, was to take the laws of motion he had developed and applied them to planetary motion to develop a rigorous mathematical framework for this planetary motion.

## Gravitational Forces

Newton eventually came to the conclusion that, in fact, the apple and the moon were influenced by the same force. He named that force gravitation (or gravity) after the Latin word gravitas which literally translates into "heaviness" or "weight."

In the Principia, Newton defined the force of gravity in the following way (translated from the Latin):

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

Mathematically, this translates into the force equation:

FG = Gm1m2/r2

In this equation, the quantities are defined as:

• Fg = The force of gravity (typically in newtons)
• G = The gravitational constant, which adds the proper level of proportionality to the equation. The value of G is 6.67259 x 10-11 N * m2 / kg2, although the value will change if other units are being used.
• m1 & m1 = The masses of the two particles (typically in kilograms)
• r = The straight-line distance between the two particles (typically in meters)

## Interpreting the Equation

This equation gives us the magnitude of the force, which is an attractive force and therefore always directed toward the other particle. As per Newton's Third Law of Motion, this force is always equal and opposite. Newton's Three Laws of Motion give us the tools to interpret the motion caused by the force and we see that the particle with less mass (which may or may not be the smaller particle, depending upon their densities) will accelerate more than the other particle. This is why light objects fall to the Earth considerably faster than the Earth falls toward them. Still, the force acting on the light object and the Earth is of identical magnitude, even though it doesn't look that way.

It is also significant to note that the force is inversely proportional to the square of the distance between the objects. As objects get further apart, the force of gravity drops very quickly. At most distances, only objects with very high masses such as planets, stars, galaxies, and black holes have any significant gravity effects.

## Center of Gravity

In an object composed of many particles, every particle interacts with every particle of the other object. Since we know that forces (including gravity) are vector quantities, we can view these forces as having components in the parallel and perpendicular directions of the two objects. In some objects, such as spheres of uniform density, the perpendicular components of force will cancel each other out, so we can treat the objects as if they were point particles, concerning ourselves with only the net force between them.

The center of gravity of an object (which is generally identical to its center of mass) is useful in these situations. We view gravity and perform calculations as if the entire mass of the object were focused at the center of gravity. In simple shapes — spheres, circular disks, rectangular plates, cubes, etc. — this point is at the geometric center of the object.

This idealized model of gravitational interaction can be applied in most practical applications, although in some more esoteric situations such as a non-uniform gravitational field, further care may be necessary for the sake of precision.

## Gravity Index

• Newton's Law of Gravity
• Gravitational Fields
• Gravitational Potential Energy
• Gravity, Quantum Physics, & General Relativity

## Introduction to Gravitational Fields

Sir Isaac Newton's law of universal gravitation (i.e. the law of gravity) can be restated into the form of a gravitational field, which can prove to be a useful means of looking at the situation. Instead of calculating the forces between two objects every time, we instead say that an object with mass creates a gravitational field around it. The gravitational field is defined as the force of gravity at a given point divided by the mass of an object at that point.

Both g and Fg have arrows above them, denoting their vector nature. The source mass M is now capitalized. The r at the end of the rightmost two formulas has a carat (^) above it, which means that it is a unit vector in the direction from the source point of the mass M. Since the vector points away from the source while the force (and field) are directed toward the source, a negative is introduced to make the vectors point in the correct direction.

This equation depicts a vector field around M which is always directed toward it, with a value equal to an object's gravitational acceleration within the field. The units of the gravitational field are m/s2.

## Gravity Index

• Newton's Law of Gravity
• Gravitational Fields
• Gravitational Potential Energy
• Gravity, Quantum Physics, & General Relativity

When an object moves in a gravitational field, work must be done to get it from one place to another (starting point 1 to endpoint 2). Using calculus, we take the integral of the force from the starting position to the end position. Since the gravitational constants and the masses remain constant, the integral turns out to be just the integral of 1 / r2 multiplied by the constants.

We define the gravitational potential energy, U, such that W = U1 - U2. This yields the equation to the right, for the Earth (with mass mE. In some other gravitational field, mE would be replaced with the appropriate mass, of course.

## Gravitational Potential Energy on Earth

On the Earth, since we know the quantities involved, the gravitational potential energy U can be reduced to an equation in terms of the mass m of an object, the acceleration of gravity (g = 9.8 m/s), and the distance y above the coordinate origin (generally the ground in a gravity problem). This simplified equation yields gravitational potential energy of:

U = mgy

There are some other details of applying gravity on the Earth, but this is the relevant fact with regards to gravitational potential energy.

Notice that if r gets bigger (an object goes higher), the gravitational potential energy increases (or becomes less negative). If the object moves lower, it gets closer to the Earth, so the gravitational potential energy decreases (becomes more negative). At an infinite difference, the gravitational potential energy goes to zero. In general, we really only care about the difference in the potential energy when an object moves in the gravitational field, so this negative value isn't a concern.

This formula is applied in energy calculations within a gravitational field. As a form of energy, gravitational potential energy is subject to the law of conservation of energy.

Gravity Index:

• Newton's Law of Gravity
• Gravitational Fields
• Gravitational Potential Energy
• Gravity, Quantum Physics, & General Relativity

## Gravity & General Relativity

When Newton presented his theory of gravity, he had no mechanism for how the force worked. Objects drew each other across giant gulfs of empty space, which seemed to go against everything that scientists would expect. It would be over two centuries before a theoretical framework would adequately explain why Newton's theory actually worked.

In his Theory of General Relativity, Albert Einstein explained gravitation as the curvature of spacetime around any mass. Objects with greater mass caused greater curvature, and thus exhibited greater gravitational pull. This has been supported by research that has shown light actually curves around massive objects such as the sun, which would be predicted by the theory since space itself curves at that point and light will follow the simplest path through space. There's greater detail to the theory, but that's the major point.

## Quantum Gravity

Current efforts in quantum physics are attempting to unify all of the fundamental forces of physics into one unified force which manifests in different ways. So far, gravity is proving the greatest hurdle to incorporate into the unified theory. Such a theory of quantum gravity would finally unify general relativity with quantum mechanics into a single, seamless and elegant view that all of nature functions under one fundamental type of particle interaction.

In the field of quantum gravity, it is theorized that there exists a virtual particle called a graviton that mediates the gravitational force because that is how the other three fundamental forces operate (or one force, since they have been, essentially, unified together already). The graviton has not, however, been experimentally observed.

## Applications of Gravity

This article has addressed the fundamental principles of gravity. Incorporating gravity into kinematics and mechanics calculations is pretty easy, once you understand how to interpret gravity on the surface of the Earth.

Newton's major goal was to explain planetary motion. As mentioned earlier, Johannes Kepler had devised three laws of planetary motion without the use of Newton's law of gravity. They are, it turns out, fully consistent and one can prove all of Kepler's Laws by applying Newton's theory of universal gravitation.

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