I have two balls connected to each other by a string. The string goes through a tube. One ball is heavier than the other. Holding on to the tube I can swing the light ball causing the heavier ball to be lifted upwards. Qualitatively what is happening here is that the gravitational pull in the heavy ball creates tension in the string which in turn yields a center seeking net force on the ball in orbit, thus permitting it to remain in uniform circular motion.
We draw a vector diagram of the forces involved. Applying Newton's second
law to the hanging mass which is at rest we obtain
To start off with we shall neglect the gravitational pull in the
light ball in comparison to the force it feels due to the tension in the string.
This implies that the net force on the ball equals the tension in the
string. On the other hand if the light ball is performing
uniform circular motion we know that
Where in order to not confuse it with the tension in the string
I have defined the period of the circular motion to be 
Combining the equations we can solve for an expression for the period
for given radius of orbit and mass ratio:
Note that we derived this equation under the assumption that T is
much larger than 
so the equation only holds
for 
. Other cases are not alot harder but rather than
consider those we look at the conical pendulum.