Gravitational acceleration: Wikis

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In physics, gravitational acceleration is the specific force or acceleration on an object caused by gravity. In a vacuum, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass. This is true regardless of the mass or composition of the body. On the surface of the Earth, all 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. In a vacuum all small objects have same acceleration regardless of density. [1][2]

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,
$\mathbf{\hat{r}}$ 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.
G is the gravitational constant of the universe.

The relative acceleration of two the objects in the reference frame of the attracting object is:

$\mathbf{\hat{g}} = -{G( M+m ) \over r^2}\mathbf{\hat{r}}$

The relative acceleration depends on both masses.

Disregarding air resistance and the Earth's movement towards falling objects, all masses (large or small) dropped simultaneously will hit the ground at the same time. All masses lifted one at a time and dropped will hit the ground at the same time.

Notes

1. ^ The international system of units (SI) - United States Department of Commerce, NIST Special Publication 330, 2001, p. 29
2. ^ The International System of Units (SI) - Bureau international des poids et mesures, 8th edition, 2006, pp. 142–143

In physics, gravitational acceleration is the specific force or acceleration on an object caused by gravity. In a vacuum, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass.[1][2] This is true regardless of the mass or composition of the body. On the surface of the Earth, all 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.

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

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

where:

M is the mass of the attracting object,
$\mathbf\left\{\hat\left\{r\right\}\right\}$ 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.
G is the gravitational constant of the universe.

The relative acceleration of two the objects in the reference frame of the attracting object is:

$\mathbf\left\{\hat\left\{g\right\}\right\} = -\left\{G\left( M+m \right) \over r^2\right\}\mathbf\left\{\hat\left\{r\right\}\right\}$

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.

Disregarding air resistance, all small masses dropped simultaneously will hit the ground at the same time.

All small masses lifted one at a time and dropped will take the same amount of time to hit the ground.

All small masses brought in from far away and dropped one at a time will experience the same acceleration relative to an inertial frame. However, larger masses will take less time to hit the ground than smaller ones. Introducing additional mass increases the relative acceleration and decreases the free fall time. During Apollo 15 an astronaut on the Moon simultaneously dropped a feather and a hammer and they reached the ground at the same time.

In 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; the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime.

Notes

1. ^ The international system of units (SI) - United States Department of Commerce, NIST Special Publication 330, 2001, p. 29
2. ^ The International System of Units (SI) - Bureau international des poids et mesures, 8th edition, 2006, pp. 142–143