Newton's Law of Universal Gravitation

All objects with mass experience attraction towards each other as a result of the gravitational force between them. The strength of this force between two objects is given by

         G (m₁ m₂)
    F_g = —————————
            r²

where

• F_g is the force of gravity
• G is the universal constant of gravitation
• m₁ is the mass of one object
• m₂ is the mass of a second object
• r is the distance between the two objects

The gravitational force is then directly proportional to the mass of each interacting object (or to the product of them), and inversely proportional to the distance between them, squared.

Recall from Newton's Laws that

    F = m a

where 'a' is the acceleration; if the force 'F' is due to gravity, the 'a' may be written as 'g' ( F = m g )

If, say, the first mass is you, and the second mass is the planet you are standing on, then the gravitational acceleration you experience is directly proportional to the mass of the planet, and inversely proportional to the square of its radius; i.e.,

         m
    g ∝ ————
         r²

To find the ratio of the gravitational accelerations experienced by a person or object on two different planets, a helpful formula would be:

    g_A     (m_A/m_B)
   ———— = —————————
    g_B     (r_A/r_B)²

You may also discover that if you were to take the ratio of the gravitational accelerations experienced by two objects of different masses being influenced by a third body, their values of g would be the same. The reason is that although the strength of the gravitational force between higher-mass objects is greater than for lower masses, the higher-mass object requires a greater force be applied to have the same overall acceleration. This is how, on an airless body like Earth's Moon, an astronaut can drop a feather and a hammer and watch them both hit the ground at the same time!

See also:

• Newton's Laws of Motion

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