Closely related to the previous problem is the conical pendulum.
A ball which is attached to a central rod with a cord swings around in
a circle. The string forms a conus, hence the name.
The forces acting on the mass are gravitation and tension in the string.
The net force is given by
When the ball is performing uniform circular motion the net force acting
on it is
Equating the two expressions for the net force and resolving in two
scalar equations we get
We are not interested in the tension but rather in an expression that
determines the angle of the conus. We get this by dividing the two equations:
We notice that as usual the mass of the conical pendulum vanishes.
Interestingly we have that the height, h from suspension point to the
plane of the circle formed by the pendulum is given by
Inserting this expression into the former equations yields an expression for
the height:
We see that h only depends on the free fall acceleration and the angular
frequency of the circular motion. We confirm this beautifully simple results by
observing the corresponding experiment.