Newton's law of universal gravitation

Newton's law of universal gravitation describes gravity as a force. It says that every particle in the universe pulls on every other particle. The strength of this pull depends on two things: how heavy the particles are and how far apart they are. The heavier the particles, the stronger the pull. The farther apart they are, the weaker the pull.

This idea was part of classical mechanics and was written down by Isaac Newton in his book called Philosophiæ Naturalis Principia Mathematica. That book came out on July 5, 1687. Newton's work brought together what people knew about gravity on Earth and what they saw happening in space.

Later, Albert Einstein developed a new idea called general relativity, which gives a more detailed view of gravity. But Newton's law is still very useful for most everyday situations. It works well for calculating things like how planets move around the Sun or how objects fall to the ground.

History

Main article: History of gravitational theory

Before Newton’s law of gravity, many ideas tried to explain why things fall down. Early thinkers like Aristotle believed objects fell because it was their natural purpose.

Later, during the 1600s, scientists used careful experiments and observations. Galileo Galilei measured how objects fall, and Johannes Kepler described how planets move around the sun based on Tycho Brahe’s star watching.

In 1687, Isaac Newton brought everything together. He showed that the same force that makes apples fall from trees also keeps the Moon orbiting Earth. Newton’s work proved that gravity acts as if all of Earth’s mass were at its center. His ideas united Earth’s gravity with the motions of planets — a big step in understanding our universe.

Modern form

In modern language, Newton’s law of universal gravitation explains how objects attract each other. The force depends on how heavy the objects are and how far apart they are.

The constant that helps us measure this force was first found through an experiment done by a British scientist named Henry Cavendish in 1798. This experiment tested Newton’s ideas about gravity using small masses in a lab. It happened many years after Newton’s work was published, so Newton himself could not use this constant in his calculations.

Every point mass attracts every single other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:
F = G m 1 m 2 r 2   {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ } where F is the force between the objects; G is the Newtonian constant of gravitation (6.674×10−11 m3⋅kg−1⋅s−2); m1 is the mass of the first object; m2 is the mass of the second object; r is the distance between the centers of the masses.

Bodies with spatial extent

When objects take up space rather than being like tiny points, we figure out the gravity between them by thinking of them as many tiny points added together. If we make these tiny points smaller and smaller, we can find the exact force by adding up all their effects.

For objects that are perfectly round and evenly spread out, the gravity they create acts as if all their mass is at the center. This special rule does not work for objects that are not perfectly round.

Newton’s shell theorem helps us understand gravity inside such round objects. It tells us two important things:

• The mass that is closer to the center than a point acts as if all that mass is at the center.
• The mass that is farther away from the center than a point does not pull on that point at all — the forces balance out.

Because of this, there is no gravity pulling you if you are inside a hollow, round shell.

Vector form

Newton's law of universal gravitation can also be shown as a vector equation to include the direction of the gravitational force. In this version, bold letters show vectors. The formula shows how the force between two objects depends on their masses and the distance between them, now including direction.

The formula helps us understand both the strength and the direction of the gravitational pull between any two objects. It works the same way as the simpler version, but now we see F as a vector, and the right side includes a unit vector to show direction. We also see that F₁₂ = −F₂₁, meaning the forces are equal in strength but opposite in direction.

Gravity field

Main article: Gravitational field

The gravitational field explains the pull of gravity on an object at any point in space, measured for each unit of the object's mass. It is the same as the gravitational acceleration at that spot.

When more than two objects are involved, like a rocket between Earth and the Moon, this idea helps us understand gravity better. For just two objects, we can describe the gravitational field as depending on the mass of the object creating the field and the distance away from it.

Gravitational fields are special because the work done by gravity does not depend on the path taken. This means there is a gravitational potential field linked to the gravitational field.

Extensions

Scientists have looked for ways to change Newton’s idea about gravity using special tools called neutron interferometers. They are trying to see if gravity works in ways that are different from what we usually think.

Solutions

Predicting how objects move when pulled by gravity is called the n-body problem. We can fully solve it for two objects, but with more objects, the motion can become very unpredictable and we usually need computers to find answers. The most studied case is the three-body problem, where we know some special solutions, like those that create Lagrange points.