A particle of mass m sliding on a smooth horizontal table is attached to a light inextensible string which passes through a hole in the table and is connected to another particle of equal mass which hangs immediately below the hole. At time t = 0, the particle on the table is projected with a speed √2gh perpendicular to the line joining it to the hole and at a distance a from the hole. There is no friction in the system. (a) Write down the angular momentum of the particle about the hole, and show that it is conserved. dr ¹(ª)² = dt (b) Write down the total mechanical energy and show that = gh ¹h (1–2²7) + 1 (1) Show that the tension of the string is: ½ mg (1 + 24h). + g(a-r). (c) Differentiate the equation and find an expression for d²r/dt². (d) Show that if 2h = a, the particle on the table moves in a circle of radius a centred on the hole. (e) Show that the particle below the table will be pulled up to the hole if 2h > a and the length of the string is less than h/2 + √ah+h²/4.

A particle of mass m sliding on a smooth horizontal table is attached to a light inextensible string which passes through a hole in the table and is connected to another particle of equal mass which hangs immediately below the hole. At time t = 0, the particle on the table is projected with a speed √2gh perpendicular to the line joining it to the hole and at a distance a from the hole. There is no friction in the system. (a) Write down the angular momentum of the particle about the hole, and show that it is conserved. dr ¹(ª)² = dt (b) Write down the total mechanical energy and show that = gh ¹h (1–2²7) + 1 (1) Show that the tension of the string is: ½ mg (1 + 24h). + g(a-r). (c) Differentiate the equation and find an expression for d²r/dt². (d) Show that if 2h = a, the particle on the table moves in a circle of radius a centred on the hole. (e) Show that the particle below the table will be pulled up to the hole if 2h > a and the length of the string is less than h/2 + √ah+h²/4.