Physics Formulae/Gravitation Formulae

From Wikiversity
Jump to navigation Jump to search

Lead Article: Tables of Physics Formulae


This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Gravitation.

Gravitational Field Definitions[edit]

A common misconseption occurs between centre of mass and centre of gravity. They are defined in simalar ways but are not exactly the same quantity. Centre of mass is the mathematical descrition of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts. They are only equal if and only if the external gravitational field is uniform.

Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues.


Quantity Name (Common) Symbol/s Defining Equation SI Units Dimension
Centre of Gravity

(Symbols can vary

enourmously)

ith moment of mass


Centre of gravity for a descrete masses



Centre of a gravity for a continuum of mass



m [L]
Standard Gravitation

Parameter of a Mass

N m2 kg-1 [L]3 [T]-2
Gravitational Field, Field

Strength, Potential Gradient,

Acceleration

N kg-1 = m s-2 [L][T]-2
Gravitational Flux m3 s-2 [L]3[T]-2
Absolute Gravitational Potential J kg-1 [L]2[T]-2
Gravitational Potential Differance J kg-1 [L]2[T]-2
Gravitational Potential Energy J [M][L]2[T]-2
Gravitational Torsion Field

Hz = s-1 [T]-1
Gravitational Torsion Flux N m s kg-1 = m2 s-1 [M]2 [T]-1
Gravitomagnetic Field Hz = s-1 [T]-1
Gravitomagnetic Flux N m s kg-1 = m2 s-1 [M]2 [T]-1
Gravitomagnetic Vector Potential [1] m s-1 [M] [T]-1


Gravitational Potential Gradient and Field

Laws of Gravitation[edit]

Modern Laws[edit]

Gravitomagnetism (GEM) Equations:

In an relativley flat spacetime due to weak gravitational fields (by General Relativity), the following gravitational analogues of Maxwell's equations can be found, to describe an analogous Gravitomagnetic Field. They are well established by the theory, but have yet to be verified by experiment [2].

Einstein Tensor Field (ETF) Equations

where Gμυ is the Einstien tensor:

GEM Equations

Gravitomagnetic Lorentz Force

Classical Laws[edit]

It can be found that Kepler's Laws, though originally discovered from planetary observations (also due to Tycho Brahe), are true for any central forces.

For Kepler's 1st law, the equation is nothing physically fundamental; simply the polar equation of an ellipse where the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, centred on the central star.


e = elliptic eccentricity

a = elliptic semi-major axes = planet aphelion

b = elliptic semi-minor axes = planet perihelion



Newton's Law of Gravitational Force
Gauss's Law for Gravitation
Kepler's 1st Law

Planets move in an ellipse, with the star at a focus

Kepler's 2nd Law
Kepler's 3rd Law

Gravitational Fields[edit]

The general formula for calculating classical gravitational fields, due to any mass distribution, is found by using Newtons Law, definition of g, and application of calculus:


Uniform Mass Corolaries[edit]

For uniform mass distributions the table below summarizes common cases.

For a massive rotating body (i.e. a planet/star etc), the equation is only true for much less massive bodies (i.e. objects at the surface) in physical contact with the rotating body. Since this is a classical equation, it is only approximatley true at any rate.


Superposition Principle for

the Gravitational Field

Gravitational Acceleration
Gravitational Field for

a Rotating (spinning about axis) body

= azimuth angle relative to rotation axis

= unit vector perpendicular to rotation

axis, radial from it

Uniform Gravitational Field, Parabolic Motion = Initail Position

= Initail Velocity

= Time of Flight

Use Constant Acc. Equations to obtain

Point Mass
At a point in a local

array of Point Masses

Line of Mass = Mass

= Length of mass distribution

Spherical Shell = Mass

= Radius

Outside/at Surface


Inside

Spherical Mass Distribution = Mass

= Radius

Outside/at Surface


Inside

Gravitational Potential Energy of a

Physical Pendulum in a Uniform Field

= seperation between pivot and centre of mass

= length from pivot to centre of gravity


= mass of pendulum

= mass moment of pendulum

Gravitational Torque on a physical

Pendulum in a Uniform Field


For non-uniform fields and mass-moments, applying differentials of the scalar and vector products then integrating gives the general gravitational torque and potential energy as:



Gravitational Potentials[edit]

Potential Energy from gravity
Escape Speed
Orbital Energy


See also[edit]

References[edit]

  1. Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, ISBN 0-691-03323-4
  2. Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, ISBN 0-691-03323-4