Newton and Gravitation: The Universal Law of Gravitation

Newton's Law

Qualitatively Newton's Law of gravitation states that:

Every massive particle attracts every other massive particle with a force directly proportional to the product of their masses and inversely proportional to the square of distance between them

In vector notation, if

is the position vector of mass m₁ and

is the position vector of mass m₂, then the force on m₁ due to m₂ is given by:

The difference of the two vectors in the numerator gives the direction of the force. The appearance of a cube, instead of a square, in the denominator is in order to cancel this direction-giving factor of |

| in the numerator.

Figure %: Direction of force is the difference of the position vectors.

This force has some remarkable properties. First, we note that it acts at a distance , meaning that irrespective of any intervening matter, every particle in the universe exerts a gravitational force on every other particle. Furthermore, gravity obeys a principle of superposition. This means that to find the gravitational force on any particle it is necessary only to find the vector sum of all the forces from all the particles in the system. For example, the force of the earth on the moon is found by vector summing all the forces between all the particles in the moon and earth. This sounds like an immense task, but actually simplifies calculation.

Gravity as a central force

Newton's Universal Law of Gravitation produces a central force. The force is in the radial direction and depends only on the distance between objects. If one of the masses is at the origin, then () = F(r). That is, the force is a function of the distance between the particles and completely in the direction of . Obviously, the force is also dependent on G and the masses, but these are just constant--the only coordinate on which the force depends is the radial one.

It is easy to show that when a particle is in a central force, angular momentum is conserved, and motion takes place in a plane. First, let us consider the angular momentum:

(

) =

×(m

) +

= 0

The last equality follows because the cross product of

with itself is zero, and since

is entirely in the direction of

, the cross product of these two vectors is zero also. Since angular momentum does not change over time it is conserved. This is essentially a more general expression of Kepler's Second Law, which we saw (here) also asserted the conservation of angular momentum.

At some time t₀, we have the position vector and velocity vector of the motion that define a plane P with a normal given by = ×. In the previous proof we showed that × does not change in time. This means that = × does not change in time either. Therefore, × = for all t. Since must be orthogonal to , it must always lie in the plane P.