Two smooth beads A and B, free to move on a vertical smooth circular wire, are connected by a string. Weights W1, W2, and W are suspended from A, B, and a point C of the string respectively. In equilibrium, A and B are in a horizontal line.

If $ \angle B A C=\alpha \text { and } \angle A B C=\beta \text {, What is the ratio of } \tan \alpha: \tan \beta ? $

Solution:

Resolving forces horizontally and vertically at the points A, B, and C respectively, we get

$ \begin{aligned} &T \cos \alpha=R_{1} \sin \gamma....(i) \\\\ &T_{1} \sin \alpha+W_{1}=R_{1} \cos \gamma...(ii) \\\\ &T_{1} \cos \beta=R_{2} \sin \gamma...(iii) \\\\ &T_{2} \sin \beta+W_{2}=R_{2} \cos \gamma...(iv) \\\\ &T_{1} \cos \alpha=T_{2} \cos \beta...(v) \\\\ &T_{1} \sin …