Motion in a Circle, Acceleration and Gravity
Acceleration is defined as “rate of change of velocity”. When a particle is moving in a circle it is constantly accelerating because it's direction, hence it's velocity is changing. If it is accelerating it must be experiencing a force.
to give
is the magnitude of the velocity
around the circle and
is the acceleration a hence we may write
(1)
Now we can use
and
where
may be due to gravity or electricity of any other nature of force to eliminate the acceleration in terms of quantities which may be easier to measure. For example, the force of gravity is given by
so the acceleration of body 1 is
(2)
We can equate (1) and (2)
(3)
We can also derive Kepler's Law of planetary motion, that the square of the period of a plant's orbit around the Sun is proportional to the radius of the orbit cubed.
Sub this into (3).
where
is the mass of the central body that is being orbited