The Conical Pendulum With Constant Radius

A conical pendulum can be made to rotate in a circle with a constant radius while varying its length.

The diagram shows a conical pendulum of mass m rotating in a circle of radius r. The centripetal force making the pendulum rotate in a circle is equal toand is due to the horizontal component of the tension in the string.

Resolving vertically for the pendulum of massgives


Resolving horizontally for the pendulum of massgives


Dividing (2) by (1) gives

From the diagram above


But(the circumference of the orbit divided by the period) so

Cancelling r from both sides and inverting gives

A graph ofon the y axis againston the x axis will have gradient

is then equal to