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\displaystyle \text{EPE} = \frac{\lambda a^2 (2 \sin \frac{\theta}{2} - 1)^2}{2a} = \frac{1}{2} \lambda a(2 \sin \frac{\theta}{2} - 1)^2. \\[br]\text{GPE} = 2mga \cos \theta. \\ \\[br][br]\text{Potential energy of the system:} \\[br]V = 2mga \cos \theta + \frac{1}{2} \lambda a(4 \sin^2 \frac{\theta}{2} - 4 \sin \frac{\theta}{2} + 1) \\[br]= mga(2 - 4 \sin^2 \frac{\theta}{2}) + 2\lambda a \sin^2 \frac{\theta}{2} - 2\lambda a\sin \frac{\theta}{2} + \frac{1}{2}\lambda a \\[br]= 2(\lambda a - 2mga)\sin^2 \frac{\theta}{2} - 2 \lambda a \sin \frac{\theta}{2} + \text{constant} \\[br]\frac{\text{d}V}{\text{d}\theta} = 2(\lambda a - 2mga) \sin \frac{\theta}{2} \cos \frac{\theta}{2} - \lambda a\cos \frac{\theta}{2} \\[br]= a\cos \frac{\theta}{2} (2(\lambda - 2mg)\sin \frac{\theta}{2} - \lambda). \\[br]\frac{\text{d}V}{\text{d}\theta} = 0 \Rightarrow \cos \frac{\theta}{2} = 0 \text{ (n/a)}, \; \sin \frac{\theta}{2} = \frac{\lambda}{2(\lambda - 2mg)}. \\[br]\lambda > 4mg \Rightarrow \lambda - 2mg > 2mg \\[br]\Rightarrow 0 < \frac{mg}{\lambda - 2mg} < \frac{1}{2} \\[br]\Rightarrow \frac{1}{2} < \sin \frac{\theta}{2} < 1 \\[br]\Rightarrow 30 < \frac{\theta}{2} < 90 \\[br]\Rightarrow 60 < \theta < 180.
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