PROBLEM 13-53 (page 146, 14th edition) PROBLEM 13-48 (page 138, 12th edition) In the diagram below, particles A and B are connected by a fixed length cord which passes through a frictionless hole on the horizontal frictionless table. A hangs 0.3 m above the floor. B is 0.5 m from the hole in the table. There is a hole in the floor directly under A. The mass of A is 4 kg. The mass of B is 2 kg. The sizes of A and B may be neglected. Motion begins when B is given velocity 1.1 m/s perpendicular to its radial line from the hole. Determine the speed of B when A passes through the hole in the floor. · Determine the maximum speed of B. B V B A 1
13-53 (page 146, 14th edition) 13-48 (page 138, 12th edition) OVERVIEW OF THE MOTION In the text version of this problem, A doesn't move and B moves in a circular path about the hole in the table. The tension in the cord is constant and equal to the weight of A. In the PHYS 170 version of this problem, the motion of A and B is more interesting. The tension in the cord is not constant and is initially less than the weight of A. A initially accelerates downwards and B is pulled inwards. The sum of the kinetic energies of A and B and the gravitational potential energy of A is remains constant since there is no friction. The tension in the cord transfers energy between A and B but doesn't contribute to the total energy of A and B. The angular momentum of B about the vertical axis remains constant since the tension force on B is in the radial direction. B loops around the hole while A falls then proceeds back out to its original distance from the hole. A moves upwards to its original height above the floor. The motion continues in a periodic fashion. The path of B is an ellipse with one focus at the hole when the tension force is inversely proportional to square of the distance of B from the hole.
13-53 (page 146, 14th edition) 13-48 (page 138, 12th edition) SUMMARY OF RESULTS Data m = 4kg mp = 2kg g = 9.81m/s2 A h is the height of A above the floor A r is the radial distance of B from the hole B Position 1 Start of motion h = 0.3 m `A1 r = 0.5 m B1 V .= 0 A1 B1r V = 0 B1! = 1.1 m/s Position 2 A passes through the hole in the floor h = 0 m A2 `B2 = 0.2 m A2 =1.34m/s V B2 = 3.06m/s Position 3 Maximum speed of B A3 h = ! 5.95 cm r = 0.1405m B3 'A3 V= 0 B3 V =3.91m/s