A light uniform rod of length \(2l\) is freely suspended from one end \(A\) and carries a concentrated mass \(m\) at the other end \(B\). A light spring, producing a force \(\frac{16mg}{l}\) per unit extension or compression, is attached to the rod at its mid-point and lies horizontally so that the rod is vertical when in equilibrium. Find the amplitude and period of small oscillations of the end \(B\) (i) when it is given a small displacement \(a\) in a direction perpendicular to the spring and then set free; and (ii) when it is projected with a small velocity \(v\) in a direction parallel to the spring. Sketch the projection on a horizontal plane of the path of \(B\) when these two oscillations are superposed, assuming them to begin simultaneously.