Problems

Two particles, of masses \(m_1\) and \(m_2\) where \(m_1 > m_2\), are attached to the ends of a light, inextensible string. A particle of mass \(M\) is fixed to a point \(P\) on the string. The string passes over two small, smooth pulleys at \(Q\) and \(R\), where \(QR\) is horizontal, so that the particle of mass \(m_1\) hangs vertically below \(Q\) and the particle of mass \(m_2\) hangs vertically below~\(R\). The particle of mass \(M\) hangs between the two pulleys with the section of the string \(PQ\) making an acute angle of \(\theta_1\) with the upward vertical and the section of the string \(PR\) making an acute angle of \(\theta_2\) with the upward vertical. \(S\) is the point on \(QR\) vertically above~\(P\). The system is in equilibrium.

1. Using a triangle of forces, or otherwise, show that:
   1. \(\sqrt{m_1^2 - m_2^2} < M < m_1 + m_2\)\,;
   2. \(S\) divides \(QR\) in the ratio \(r : 1\), where \[ r = \frac{M^2 - m_1^2 + m_2^2}{M^2 - m_2^2 + m_1^2}. \]
2. You are now given that \(M^2 = m_1^2 + m_2^2\). Show that \(\theta_1 + \theta_2 = 90^\circ\) and determine the ratio of \(QR\) to \(SP\) in terms of the masses only.