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Problem 4 A block of mass m is relased from rest on a frictionless triangular block of mass M and angle of inclination (as shown below). The triangular block rests on a frictionless horizontal surface. m M Ꮎ Part a In this problem, we will use the horizontal coordinates of the masses as our generalized coordinates (with x₁ positive to the left and x2 positive to the right). X1 M m X2 Describe an alternative set of generalized coordinates that you can use for this problem. Part b What are the center of mass positions of the blocks as functions of the generalized coor- dinates x1 and x2? Part c velocities. Write an expression for the velocities of the particles as a function of the generalized Part d What is the total kinetic energy of the system in terms of the generalized velocities? 1 Part e What is the total potential energy of the system in terms of the generalized coordinates? Part f Show that the Lagrangian of the system is therefore given by 1 L = 1 = 1 ½ Mಠ+ ½ m (i² + (à₁ + 2)² tan²0) + mg (x1+x2) tan ə

Problem 4 A block of mass m is relased from rest on a frictionless triangular block of mass M and angle of inclination (as shown below). The triangular block rests on a frictionless horizontal surface. m M Ꮎ Part a In this problem, we will use the horizontal coordinates of the masses as our generalized coordinates (with x₁ positive to the left and x2 positive to the right). X1 M m X2 Describe an alternative set of generalized coordinates that you can use for this problem. Part b What are the center of mass positions of the blocks as functions of the generalized coor- dinates x1 and x2? Part c velocities. Write an expression for the velocities of the particles as a function of the generalized Part d What is the total kinetic energy of the system in terms of the generalized velocities? 1 Part e What is the total potential energy of the system in terms of the generalized coordinates? Part f Show that the Lagrangian of the system is therefore given by 1 L = 1 = 1 ½ Mಠ+ ½ m (i² + (à₁ + 2)² tan²0) + mg (x1+x2) tan ə

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Problem 4 A block of mass m is relased from rest on a frictionless triangular block of mass M and angle of inclination (as shown below). The triangular block rests on a frictionless horizontal surface. m M Ꮎ Part a In this problem, we will use the horizontal coordinates of the masses as our generalized coordinates (with x₁ positive to the left and x2 positive to the right). X1 M m X2 Describe an alternative set of generalized coordinates that you can use for this problem. Part b What are the center of mass positions of the blocks as functions of the generalized coor- dinates x1 and x2? Part c velocities. Write an expression for the velocities of the particles as a function of the generalized Part d What is the total kinetic energy of the system in terms of the generalized velocities? 1 Part e What is the total potential energy of the system in terms of the generalized coordinates? Part f Show that the Lagrangian of the system is therefore given by 1 L = 1 = 1 ½ Mಠ+ ½ m (i² + (à₁ + 2)² tan²0) + mg (x1+x2) tan ə
Transcribed Image Text:Problem 4 A block of mass m is relased from rest on a frictionless triangular block of mass M and angle of inclination (as shown below). The triangular block rests on a frictionless horizontal surface. m M Ꮎ Part a In this problem, we will use the horizontal coordinates of the masses as our generalized coordinates (with x₁ positive to the left and x2 positive to the right). X1 M m X2 Describe an alternative set of generalized coordinates that you can use for this problem. Part b What are the center of mass positions of the blocks as functions of the generalized coor- dinates x1 and x2? Part c velocities. Write an expression for the velocities of the particles as a function of the generalized Part d What is the total kinetic energy of the system in terms of the generalized velocities? 1 Part e What is the total potential energy of the system in terms of the generalized coordinates? Part f Show that the Lagrangian of the system is therefore given by 1 L = 1 = 1 ½ Mಠ+ ½ m (i² + (à₁ + 2)² tan²0) + mg (x1+x2) tan ə
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