Gravity is usually given in units of acceleration: if you let something fall near the surface, how fast does it accelerate?
Newton gave us two equations (he gave us a lot more, but we only need two for now):
F = m a (Force = mass times acceleration)
F = G M m / d^2
For our purposes,
M is the mass of the planet,
m is the mass of the body at or near the surface (don't worry, it cancels out)
G is a constant to make units come out right
d is distance (in this case, the radius of the planet).
We are looking for a (the acceleration near the surface)
Since the F is the same in both cases, the equations are equal to each other:
m a = G M m / d^2
m cancels out
a = G M/ d^2
G = 6.672x10^-11 N m^2/kg^2
M (Earth) = 6x10^24 kg (I rounded)
d (Earth) = 6,371,000 m (you have to use metres)
d^2 = 40.6 million million square metres= 40.6x10^12 m^2
a = 6.672x10^-11 (N m^2/kg^2) * 6x10^24 kg / 40.6x10^12 m^2
a = (6.672 * 6 / 40.6) * (10^-11 * 10^24 / 10^12) * (N / kg)
(From F = m a, we get that 1 N = 1 kg * 1 m/s^2, so we can change N to kg*m/s^2)
a = 0.98 x 10 * (kg * m / (kg * s^2) = 9.8 m/s^2
If you increase the mass of the planet without changing its radius, a increases.
If you increase the radius without increasing the mass, a decreases.
The Moon's mass is 0.0123 that of Earth.
The Moon's radius is 0.2724 times Earth's radius.
gravity on Moon = gravity on Earth * 0.0123 / (0.2724^2)
gravity on Moon = 9.8 m/s^2 * 0.1657 = 1.62 m/s^3
(1/6 that of Earth).
Take an object from Earth to the Moon and it will only weigh 1/6 as much.
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