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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.
= 
= 
(
) = 
F(r)
= 
(
×
) = 
×
= 
×(m
= 0
= 
= 
