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Newton's Law of Gravity

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Newton's Law of Gravity

The force equation for two particles interacting through gravity

Andrew Zimmerman Jones

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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 apply 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 shown to the right. 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. Click on the picture to see an illustration of two particles interacting through gravitational force.

In this picture, you will see that, despite their different mass and sizes, they pull on each other with equivalent force. 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.

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