Not entirely true. Close enough for government work though. Here is the applicable equation:
FG = Magnitude of the Gravitational Force (N)
m1 = Mass of one body (kg)
m2 = Mass of a second body (kg)
G = 6.672x10-11 N.m2/kg2 (Universal Gravitational Constant)
r = Distance between the centers of mass of the two objects (m)
You also have to allow for the
F = ma part. The force that accellerates the smaller mass is the gravitational force you've shown, and when equated to "ma", the "m" term for the smaller mass drops out, so the acceleration due to gravity of the smaller mass becomes:
Fg = m
2a = Gm
1m
2/r^2 ---> a = Gm
1/r^2
so
a, the acceleration due to gravity, is a constant so long as
m1 and
r are constant. For things relatively close to the surface of the Earth,
r is as close to constant as to not matter much, so
a = g = 9/8 m/s^2, for all practical purposes.
ETA: And here's another point, to emphasise to the woos why science works so well: it's all interconnected. You can't just poke one hole in it and expect it all to come crashing down. The fact that the acceleration due to gravity is only dependent on the mass of the large object and the distance between the objects is how we know the mass of the planets. So long as they have a moon, we can tell the mass of the planet, as the orbital speed of an object at a given distance will be the same regardless of the mass of the orbiting object. So if you have a planet with a moon who's period and orbital distance can be determined, we then know the mass of the planet. But, note, not the mass of the moon, unless it also has it's own sattelites.
There's some fiddling you can do if the system is like the Earth-Moon system, where the moon is big enough (relative to the Earth) to produce a measurable wobble in the Earth, so as to figure out what the mass of the moon is, but that's a special case.
