Two particles of mass m are connected by a light inextensible string of length l. One of the particles rests on a smooth horizontal table in which there is a small hole. The string passes through the hole so that the second particle hangs vertically below the hole. Take the position of the hole as the origin and describe the position of the particle on the table by plane polar coordinates r and ?, as shown in figure.

(i) Write down a formula for the angular momentum of the particle on the table about the origin in terms of r and d?/dt.

(ii) Explain why this angular momentum is a constant.

(iii) Write down a formula for the sum of the energies of both masses in terms of r, dr/dt and d?/dt. Using the result that this energy is constant, eliminate d?/dt and show that

(dr/dt)² = ? ? ?/?² ? gr.

where ? and ? are constants and g is the acceleration due to gravity.

(iv) Initially the particle on the table is at a distance l/2 from the origin and is travelling with speed a in a direction at right angles to the string. Obtain a formula for (dr/dr)² when r = l.

(v) Hence find the condition such that the particle initially below the table does not pass through the hole. (The string always remains taut.)