2. (Further analysis of the nonlinear pendulum ) Consider the nonlinear pendulum equation: d20 g sin (0) dt2 In class we obtained the conservation of energy equation: 2 de L cos (0)) = E g (1 dt 2 (a.) What is the value of total energy E when the pendulum is hanging vertically downwards and is at rest? (b.) What is the value of total energy E when the pendulum is balanced vertically upwards and is at rest? (c.) For what range of total energy values E does the pendulum "loop"? Explain (d.) Show that the "separatrix" (the curve that separates different kinds of behavior) for the nonlinear pendulum is given by (ay) de L = 2g (1 + cos(0) dt

2. (Further analysis of the nonlinear pendulum ) Consider the nonlinear pendulum equation: d20 g sin (0) dt2 In class we obtained the conservation of energy equation: 2 de L cos (0)) = E g (1 dt 2 (a.) What is the value of total energy E when the pendulum is hanging vertically downwards and is at rest? (b.) What is the value of total energy E when the pendulum is balanced vertically upwards and is at rest? (c.) For what range of total energy values E does the pendulum "loop"? Explain (d.) Show that the "separatrix" (the curve that separates different kinds of behavior) for the nonlinear pendulum is given by (ay) de L = 2g (1 + cos(0) dt