Conical Pendulum

A conical pendulum consists of a mass on a string that moves in a horizontal circle. The string traces out a cone as the mass rotates.

The mass experiences two forces: tension $T$ along the string and weight $mg$ downward. The key insight is that tension does two jobs at once through its components.

We resolve tension into a vertical component $T_{y} = T \cos \theta$ and a horizontal component $T_{x} = T \sin \theta$, where $\theta$ is the angle the string makes with the vertical.

The diagram shows the forces on the mass. The dashed vectors are the components of tension. Notice that the angle $\theta$ in this force diagram is the same as the angle the string makes with the vertical.

The geometry shows that the angle between the string and vertical at the ceiling equals the angle between the tension force and vertical at the mass. Both angles are $\theta$.

We will now apply Newton's second law to the following free body diagram: