Show that when a particle describes a curve its acceleration components along and perpendicular to the curve are \(\frac{d v}{d t}\) and \(\frac{v^2}{\rho}\), where \(v\) is the velocity of the particle, and \(\rho\) is the radius of curvature at the instantaneous position of the particle. \par Two equal particles are connected by a light inelastic string of length \(\pi a\), and are placed on a smooth circular cylinder of radius \(2a\) which has its axis horizontal, so that they rest in unstable equilibrium with the string passing over the top of the cylinder. If the equilibrium is slightly disturbed, find the tension in the string and the reactions of the particles on the cylinder in terms of the angular displacement, and show that the lower particle leaves the cylinder when at an angular distance of approximately 77\(^\circ\) 20' from the top.