Gravitational acceleration

In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass.[1]

This equality is true regardless of the masses or compositions of the bodies. At different points on Earth, objects fall with an acceleration between 9.78 and 9.82 m/s2 depending on latitude, with a conventional standard value of exactly 9.80665 m/s2 (approx. 32.174 ft/s2). Objects with low densities do not accelerate as rapidly due to buoyancy and air resistance.

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Classical mechanics

Main article: Classical mechanics

The barycentric gravitational acceleration at a point in space is given by:

\mathbf{\hat{g}}=-{G M \over r^2}\mathbf{\hat{r}}

where:

M is the mass of the attracting object, \scriptstyle \mathbf{\hat{r}} is the unit vector from center of mass of the attracting object to the center of mass of the object being accelerated, r is the distance between the two objects, and G is the gravitational constant.

The relative acceleration of two objects in the reference frame of either object or the center of mass is:

\mathbf{\hat{g}} = -{G( M%2Bm ) \over r^2}\mathbf{\hat{r}}

Thus, for a given total mass, relative gravitational acceleration does not depend on each mass separately. As long as one mass is much smaller than the other, relative gravitational acceleration is almost independent of the smaller mass.

All small masses brought in from far away and dropped one at a time will experience the same acceleration, relative to an inertial frame or the frame of the large mass. Disregarding air resistance, all small masses dropped simultaneously from the same height will hit the ground at the same time; for example, during Apollo 15 an astronaut on the Moon simultaneously dropped a feather and a hammer and they reached the ground at the same time.

General relativity

Main article: General relativity

In Einstein's theory of general relativity, gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. In Einstein's theory, masses distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. The gravitational force is a fictitious force. There is no gravitational acceleration, in that the proper acceleration and hence four-acceleration of objects in free fall are zero. Rather than undergoing an acceleration, objects in free fall travel along straight lines (geodesics) on the curved spacetime.

References

  1. ^ Gerald James Holton and Stephen G. Brush (2001). Physics, the human adventure: from Copernicus to Einstein and beyond (3rd ed.). Rutgers University Press. p. 113. ISBN 9780813529080. http://books.google.com/books?id=czaGZzR0XOUC&pg=PA113&dq=%22gravitational+acceleration%22+%22all+objects%22+neglecting+air&hl=en&ei=KSb8TIDVIofGsAOUsNX2DQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCgQ6AEwAA#v=onepage&q=%22gravitational%20acceleration%22%20%22all%20objects%22%20neglecting%20air&f=false. 

See also