Gravitational potential
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In celestial mechanics, the gravitational potential is a scalar field created by any mass, such as the earth or the Sun. The potential at a distance r from a point mass M is given by
where G is the gravitational constant.
The gravitational force is the gradient of this potential, and follows an inverse square law. In this way, the gravitational potential is analogous to the electrostatic potential and the gravitational force is analogous to the electrostatic force. In mathematics, the potential is known as the Newtonian potential, and is fundamental in the study of potential theory.
The magnitude of the observed acceleration is then
The direction of the acceleration is towards the center of mass of the object. On the surface of the Earth, the so-called standard gravity g is approximately 9.8 m/s2, although this varies with position and altitude. The acceleration is a little higher at the poles, and a little lower at the equator because the Earth is an oblate spheroid.
[edit] Potential energy
The gravitational potential (P) should not be confused with the potential energy (U). Whereas the gravitational potential is the scalar field associated with a massive body, the potential energy refers to the energy that a small test mass has as a result of its position within the field. If the large mass is represented as an idealized point mass, then one has
- U = mP
where m is the mass of the small object.
In some applications, the equations can be simplified by assuming a field independent of position. For instance, near the surface of the Earth, the gravitational acceleration can be considered roughly constant, and to a good approximation the change in potential energy of a small mass m as it moves from one height to another is just linearly related to the height difference:
- ΔU = mgΔh
[edit] References
- Peter Dunsby (1996-06-15). "Mass in Newtonian theory". Tensors and Relativity: Chapter 5 Conceptual Basis of General Relativity. Department of Mathematics and Applied Mathematics University of Cape Town. http://www.mth.uct.ac.za/omei/gr/chap5/node4.html. Retrieved on 2009-03-25.
- Lupei Zhu Associate Professor, Ph.D. (California Institute of Technology, 1998). "Gravity and Earth's Density Structure". EAS-437 Earth Dynamics. Saint Louis University (Department of Earth and Atmospheric Sciences). http://www.eas.slu.edu/People/LZhu/teaching/eas437/gravity.ppt. Retrieved on 2009-03-25.
- Charles D. Ghilani (2006-11-28). "The Gravity Field of the Earth". The Physics Fact Book. Penn State Surveying Engineering Program. http://surveying.wb.psu.edu/sur351/geoid/grava.htm. Retrieved on 2009-03-25.
