# What determines a planet or moons gravity

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

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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.

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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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