# 13: Gravitation


In this section, we study the nature of the gravitational force for objects as small as ourselves and for systems as massive as entire galaxies. We show how the gravitational force affects objects on Earth and the motion of the Universe itself. Gravity is the first force to be postulated as an action-at-a-distance force, that is, objects exert a gravitational force on one another without physical contact and that force falls to zero only at an infinite distance. Earth exerts a gravitational force on you, but so do our Sun, the Milky Way galaxy, and the billions of galaxies, like those shown above, which are so distant that we cannot see them with the naked eye.

• 13.1: Prelude to Gravitation
Our visible Universe contains billions of galaxies, whose very existence is due to the force of gravity. Gravity is ultimately responsible for the energy output of all stars—initiating thermonuclear reactions in stars, allowing the Sun to heat Earth, and making galaxies visible from unfathomable distances.
• 13.2: Newton's Law of Universal Gravitation
All masses attract one another with a gravitational force proportional to their masses and inversely proportional to the square of the distance between them. Spherically symmetrical masses can be treated as if all their mass were located at the center. Nonsymmetrical objects can be treated as if their mass were concentrated at their center of mass, provided their distance from other masses is large compared to their size.
• 13.3: Gravitation Near Earth's Surface
The weight of an object is the gravitational attraction between Earth and the object. The gravitational field is represented as lines that indicate the direction of the gravitational force; the line spacing indicates the strength of the field. Apparent weight differs from actual weight due to the acceleration of the object.
• 13.4: Gravitational Potential Energy and Total Energy
The acceleration due to gravity changes as we move away from Earth, and the expression for gravitational potential energy must reflect this change. The total energy of a system is the sum of kinetic and gravitational potential energy, and this total energy is conserved in orbital motion. Objects with total energy less than zero are bound; those with zero or greater are unbounded.
• 13.5: Satellite Orbits and Energy
Orbital velocities are determined by the mass of the body being orbited and the distance from the center of that body, and not by the mass of a much smaller orbiting object. The period of the orbit is likewise independent of the orbiting object’s mass. Bodies of comparable masses orbit about their common center of mass and their velocities and periods should be determined from Newton’s second law and law of gravitation.
• 13.6: Kepler's Laws of Planetary Motion
Johannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of time. From this analysis, he formulated three laws: Kepler’s first law states that every planet moves along an ellipse. Kepler’s second law states that a planet sweeps out equal areas in equal times. Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit.
• 13.7: Tidal Forces
Earth’s tides are caused by the difference in gravitational forces from the Moon and the Sun on the different sides of Earth. Spring or neap (high) tides occur when Earth, the Moon, and the Sun are aligned, and neap or (low) tides occur when they form a right triangle. Tidal forces can create internal heating, changes in orbital motion, and even destruction of orbiting bodies.
• 13.8: Einstein's Theory of Gravity
According to the theory of general relativity, gravity is the result of distortions in space-time created by mass and energy. The principle of equivalence states that that both mass and acceleration distort space-time and are indistinguishable in comparable circumstances. Black holes, the result of gravitational collapse, are singularities with an event horizon that is proportional to their mass.
• 13.E: Gravitation (Exercises)
• 13.S: Gravitation (Summary)

Thumbnail: Our visible Universe contains billions of galaxies, whose very existence is due to the force of gravity. Gravity is ultimately responsible for the energy output of all stars—initiating thermonuclear reactions in stars, allowing the Sun to heat Earth, and making galaxies visible from unfathomable distances. Most of the dots you see in this image are not stars, but galaxies. (credit: modification of work by NASA).

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