Conical pendulums

A conical pendulum is a body on the end of a string that moves about a horizontal circle. To solve these systems just consider the tension in the string, or both strings if their are two.

Example

Given the following system, $P$ has mass $0.3\text{ kg}$, $XP$ has length $0.8\text{ m}$, the tension is $XP$ is twice the tension in $YP$. Find the angular speed of $P$

Find the tension

$$\begin{align} 2T\cos(55^{\circ}) &= 0.3g + T\cos(25^{\circ})\\ 2T\cos(55^{\circ}) - 0.3g - T\cos(25^{\circ}) &= 0\\ T(2\cos(55^{\circ})-\cos(25^{\circ})) - 0.3g &= 0\\ T(2\cos(55^{\circ})-\cos(25^{\circ})) &= 0.3g\\ T &= 12.207\\ &= 12.2 \text{ N} \end{align}$$

Find the angular speed

$$\begin{align} F &= ma\\ F &= m r \omega^2\\ 2 \times 12.2 \times \sin(55^{\circ}) + 12.2 \times sin(25^{\circ}) &= 0.3 \times 0.8\sin(55^{\circ}) \times \omega^2 \\ 25.157... &= 0.196... \times \omega^2\\ \omega^2 &= 127.966...\\ \omega &= 11.3\text{ rad s}^{-1} \end{align}$$