State and prove the principle of the conservation of linear momentum for a system of particles. A smooth inclined plane of angle \(\alpha\) and mass \(M'\) can slide on a smooth horizontal plane. Another smooth inclined plane of the same angle \(\alpha\) and of mass \(M\) is placed on the first so that the upper face of \(M\) is horizontal. A particle of mass \(m\) is placed on this horizontal face and the whole system is then released from rest. Find the acceleration of \(m\).
Prove that \(v^2/\rho\) is the acceleration inwards along the normal, when a particle describes a plane curve with velocity \(v\). A smooth parabolic tube has its axis vertical and vertex upwards. A particle inside the tube is projected from the vertex with a velocity \(\sqrt{2g(a+h)}\). When the particle is at a depth \(z\) below the directrix prove that the pressure on the tube is \(mgh\sqrt{az^{-3}}\), where \(m\) is the mass of the particle and \(4a\) is the latus rectum of the parabola.