A particle moves in a plane under a force directed towards an origin \(O\); using polar coordinates with \(O\) as origin to describe the position of the particle at time \(t\), prove that \(r^2\dot{\theta}\) is constant. A particle of mass \(m\) moves on a smooth horizontal table, and is attached to a point \(O\) of the table by a light elastic string of natural length \(a\) and modulus \(\lambda\). Initially the particle rests at a point \(P\) at distance \(a\) from \(O\). It is projected with velocity \(\sqrt{\left(\frac{4\lambda a}{3m}\right)}\) in a direction at right angles to \(OP\). Shew that the greatest elongation of the string in the subsequent motion is \(a\).