Problems

Two light elastic springs each have natural length \(a\). One end of each spring is attached to a particle \(P\) of weight \(W\). The other ends of the springs are attached to the end-points, \(B\) and \(C\), of a fixed horizontal bar \(BC\) of length \(2a\). The moduli of elasticity of the springs \(PB\) and \(PC\) are \(s_1 W\) and \(s_2 W\) respectively; these values are such that the particle \(P\) hangs in equilibrium with angle \(BPC\) equal to \(90^\circ\).

1. Let angle \(PBC = \theta\). Show that \(s_1 = \dfrac{\sin\theta}{2\cos\theta - 1}\) and find \(s_2\) in terms of \(\theta\).
2. Take the zero level of gravitational potential energy to be the horizontal bar \(BC\) and let the total potential energy of the system be \(-paW\). Show that \(p\) satisfies \[ \frac{1}{2}\sqrt{2} \geqslant p > \frac{1}{4}(1+\sqrt{3}) \] and hence that \(p = 0.7\), correct to one significant figure.