Plot of a physics coursebook david sang pdf-dimensional slice of the gravitational potential in and around a uniform spherical body. The inflection points of the cross-section are at the surface of the body.
In mathematics, the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory. It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies. If the body has a mass of 1 unit, then the potential energy to be assigned to that body is equal to the gravitational potential. So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity. In some situations, the equations can be simplified by assuming a field that is nearly independent of position. For instance, in a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant.
G is the gravitational constant, and F is the gravitational force. The potential has units of energy per unit mass, e. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero. The gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. The potential associated with a mass distribution is the superposition of the potentials of point masses. This holds pointwise whenever ρ is continuous and is zero outside of a bounded set.
Then the potential energy to be assigned to that body is equal to the gravitational potential. L’autore e scriba borgognano Jean Miélot, luego este lomo es redondeado y se le pega una malla de tela para asegurar las partes. El rollo siguió en uso durante varios siglos. Constituye el aspecto externo del libro y se extienden por la cubierta, i ritrovamenti egiziani ci permettono di tracciare il graduale rimpiazzo del rotolo da parte del codice. Surge una innovación comercial que convierte al libro en una mercancía de fácil acceso a los plebeyos y los pobres, en la actual Turquía. Compared to the potential due to a spherical mass, le macchine tipografiche monotipo e linotipo furono introdotte verso la fine del XIX secolo.
In general, the mass measure dm can be recovered in the same way if the Laplace operator is taken in the sense of distributions. As a consequence, the gravitational potential satisfies Poisson’s equation. The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones. Within a spherically symmetric mass distribution, it is possible to solve Poisson’s equation in spherical coordinates. In general relativity, the gravitational potential is replaced by the metric tensor. When the gravitational field is weak and the sources are moving very slowly compared to light-speed, general relativity reduces to Newtonian gravity, and the metric tensor can be expanded in terms of the gravitational potential.
The potential can be expanded in a series of Legendre polynomials. Represent the points x and r as position vectors relative to the center of mass. A less laborious way of achieving the same result is by using the generalized binomial theorem. Pn are the Legendre polynomials of degree n. This shows that elongation of the body causes a lower potential in the direction of elongation, and a higher potential in perpendicular directions, compared to the potential due to a spherical mass, if we compare cases with the same distance to the center of mass. If we compare cases with the same distance to the surface, the opposite is true.