Chapter 5, Problem 5.104CP

A mobile is formed by supporting four metal butterflies of equal mass m from a string of length L. The points of support are evenly spaced a distance ℓ apart as shown in Figure P5.54. The string forms an angle θ₁ with the ceiling at each endpoint. The center section of string is horizontal. (a) Find the tension in each section of string in terms of θ₁, m, and g. (b) In terms of θ₁, find the angle θ₂ that the sections of string between the outside butterflies and the inside butterflies form with the horizontal. (c) Show that the distance D between the endpoints of the string is

$D=\frac{L}{5}\left\{2\cos {\theta }_1+2\cos \left[{\tan }^{-1}\left(\frac{1}{2}\tan {\theta }_1\right)\right]+1\right\}$

Figure P5.54