Two equal, perfectly elastic, smooth spheres are suspended by vertical strings so that they are in contact, with their centres at the same level. A third equal sphere falls vertically and strikes the other two spheres symmetrically with velocity \(u\), in such a way that at the moment of impact the three centres are in a vertical plane. Shew that the falling sphere rebounds with velocity \(\dfrac{5u}{7}\) and that the other two spheres begin to move outward with velocity \(\dfrac{2\sqrt{3}u}{7}\). Find the impulsive tension in the strings, given that \(w\) is the weight of each sphere.
A reel consists of a cylinder of radius \(r\) and two rims of radius \(R (>r)\). The mass of the reel is \(M\) and its radius of gyration about its axis is \(k\). It is placed on a perfectly rough horizontal table and the thread is drawn out horizontally from beneath it and passes over a light pulley at the edge of the table. The free end is then attached to a mass \(m\) which is allowed to descend under gravity. Shew that the mass \(m\) descends with acceleration \[ \frac{mg(R-r)^2}{M(R^2+k^2)+m(R-r)^2}. \]