Newton derived from his observations that all
objects attract each other with a force directly proportional to the product
of their masses (m,M)
and inversely proportional to the square of the distance (r)
between them. Namely:
where G is a universal constant =
G = 6.67 x 10-11 m3.kg-1.s-2
Since the earth (or any other planet) is very
massive compared to the objects on its surface, the gravitational force
is directed toward the center of the earth. Every object which is displaced
upwards or away from the surface will fall back to the surface if released.
The acceleration due to gravity which is observed at the surface of the
earth is given by:
There is a gravitational potential energy related
to the gravitational force. At the surface of the earth this is just the
force times distance, or
where g = 9.8 m.s-2.
In general, the gravitational potential energy
of an object due to the force of gravity when far from the surface of the
planet or sun or star is:
where the negative sign means that the potential
energy at a very large almost infinite distance from the surface is zero,
and the force is attractive so that potential energy can be converted to
kinetic energy as the object moves closer to the surface.
If an object of mass m is in circular
orbit about another object (say a planet or star of mass M)
at a distance r
from the center, then the energy relations are:
in which we have used the relation for the centripetal
acceleration shown elsewhere.
The escape velocity is determined by the requirement
that the kinetic energy imparted to the escaping object at the surface
of the planet should just equal the magnitude of its potential energy,
such that the total energy is zero.
The result is:
is the radius of the planet or star from which the object is escaping.
If an object moves in a circle about a center
point at constant speed, its velocity is always changing (despite the constancy
of its speed, which is only the magnitude of its velocity and, thus, it
needs to be accelerated. This acceleration or change in the direction of
the velocity must be provided by a force directed toward the center of
the circle. This force -- called the centripetal force -- is given by:
is the mass of the object, v
is the magnitude of the velocity or speed,and r
is the radius of the circle.