A particle of mass \(m\) is attached by a string to a point on the circumference of a fixed circular cylinder of radius \(a\) whose axis is vertical, the string being initially horizontal and tangential to the cylinder. The particle is projected with velocity \(v\) at right angles to the string along a smooth horizontal plane so that the string winds itself round the cylinder. Shew (i) that the velocity of the particle is constant, (ii) that the tension in the string is inversely proportional to the length which remains straight at any moment, (iii) that if the initial length of the string is \(l\) and the greatest tension the string can bear is \(T\), the string will break when it has turned through an angle \[ l/a-mv^2/aT. \]
Shew that if a number of particles connected by inelastic strings move under no forces, their linear momentum and energy are constant. Three equal particles \(A, B, C\) connected by inelastic strings \(AB, BC\) of length \(a\) lie at rest with the strings in a straight line on a smooth horizontal table. \(B\) is projected with velocity \(V\) at right angles to \(AB\). Shew that the particles \(A\) and \(C\) afterwards collide with relative velocity \(\frac{2V}{\sqrt{3}}\). If the coefficient of restitution is \(e\), find the velocities of the three particles when the string is again straight.