Gravitational Acceleration - Classical Mechanics

Classical Mechanics

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

where:

M is the mass of the attracting object, 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.

When this calculation is done, one has to account that the earth is rotating, and the centrifugal force has to be reduced from this. For example, the equation above gives the acceleration at 9.820 m/s², when GM = 3.986E14 m³/s², and R=6.371E6 m. The centripetal radius is r = R cos(latitude), and the centripetal time unit is approximately (day / 2pi), reduces this for r = 5E6 metres, to 9.79379 m/s², which is closer to the true value.

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

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.