A bead of weight w can slide on a smooth circular wire in a vertical plane. The bead is attached by a light thread to the highest point of the wire, and in equilibrium the thread is taut and makes an angle θ with the vertical. Then the tension of the thread and the reaction of the wire on the bead are:
Solutions · 1
Official solution
With the geometry giving ∠RQW=2θ (isosceles triangle from the circle's radius), applying Lami's theorem at the bead Q:
\[\frac{T}{\sin2\theta}=\frac{R}{\sin(180^{\circ}-\theta)}=\frac{w}{\sin(180^{\circ}-\theta)} \implies R=w,\qquad T=2w\cos\theta\]