Newton’s Law of Gravity

This equation describes the strength of gravitational forces based on the masses of two objects and the distance between them.

Masses attract each other, and the attraction is stronger if the objects are more massive and if they are closer together.

$F_g = \cfrac{G m_1 m_2}{r^2}$

Where:
F_g = Gravitational Force (N)
G = gravitational constant = 6.67 x 10⁻¹¹ Nm²/kg²
m₁ = mass of object 1 (kg)
m₂ = mass of object 2 (kg)
r = distance between the objects’ centers of mass (m)

The constant G (capitalized) is not the same as acceleration due to gravity, g (lowercase). Capital G represents the strength of the gravitational force anywhere in the universe. It’s quite a small number – this means that it takes a lot of mass to create any kind of a noticeable gravitational force.

The equation shows that gravitational force is proportional to either of the two masses. For example, if you replace m₁ with something that has five times as much mass, that results in a gravitational force that’s also five times greater.

The force and distance follow an inverse square relationship. For example, if the distance between two objects is tripled (x3), the force will be 1/3² or 1/9 as strong as before. This means that the force gets stronger very quickly as objects get close together, and the force likewise gets weaker very quickly as the objects move farther apart.

Equal and Opposite Forces

When two objects experience a gravitational force, each object always pulls on the other with the same amount of force, but in opposite directions. For example, the earth’s pull on the moon is equal to the moon’s pull on the earth. Because the earth is much more massive, it accelerates much less than the moon as a result of the force (a = F_net/m).

Gravitational Field Strength

$g = \cfrac{Gm}{r^2}$

Where:
g = gravitational field strength (N/kg)
m = mass of the central object (i.e. the earth in our case)
r = distance to the center of mass of the central object (m)
(distance to the center of the earth in our case)

This equation lets us calculate earth’s gravitational field. Earth’s mass is 5.97 x 10²⁴ kg, and its radius is about 6.378 x10⁶ m. You can plug the numbers in along with G (6.67 x 10⁻¹¹ Nm²/kg²) and calculate the familiar value of g on earth’s surface:

g = 9.8 N/kg

Notice that the units are N/kg instead of m/s². Actually, these units are the same, but it’s convenient to use N/kg when we calculate Force of Gravity (Weight) on earth’s surface:

F_g = mg

You can also calculate your weight using the full gravitational force equation, but it’s more work:

$F_g = \cfrac{G m_1 m_2}{r^2}$

When you use F_g = mg, you bundle most of the terms into “g” as the number 9.8. If you calculate weight somewhere else, such as Mars or the Moon, you would need a different value of g, which you can calculate using the mass and radius for those astronomical objects.

Acceleration Due to Gravity

You can calculate earth’s acceleration due to gravity from Newton’s Law of Gravity.

To change that equation into an acceleration equation instead of a force equation, we use Newton’s 2nd Law of Motion:

F_net = ma

The “a” in that equation will be our acceleration due to gravity, which we will call “g.”

$F_g = \cancel{m} g = \cfrac{G \cancel{m_1} m_2}{r^2}$

One of the masses cancels out. This shows that objects near a planet’s surface accelerate at the same rate due to gravity regardless of the object mass. We are left with this equation:

$g = \cfrac{Gm}{r^2}$

Earth’s mass is about 5.97 x 10²⁴ kg, and its radius is about 6,378,000 m. If you plug those numbers into the equation, you can find that the acceleration due to gravity on earth’s surface is about 9.8 m/s².

Altitude vs Gravity

If you want to calculate the gravitational force or gravitational acceleration a certain distance away from earth’s surface, you will need to add that distance to earth’s radius. This is the distance between the centers of the two objects involved, so that is the number that you plug into the gravity equation.