Question

Question 3

3. Consider a plane pendulum consisting of a mass m suspended by a massless string of length I. Suppose that that time t-0 the pendulum is put into motion and the length of the string is shortened at a constant rate ot-a (ie. L(t)= Lo-at). Use the angle of the pendulum φ as your generalized coordinate. (a) (2 points) Obtain the Lagrangian and Hamiltonian for this system (b) (0.5 points) Is H conserved? How can you tell? (c) (0.5 points) Compare the Hamiltonian to the total energy of the system (d) (1 point) Using Hamilton's equations, obtain a differential equation that describes φ. (It should not have any momenta in it. Just perhaps and t its time derivatives (e) (0.5 points) Verify that your eqn gives the expected result for a . 0. 4. Extra Credit Review Problem: (1.5 points) Taylor 735 Note that Figure 7 16 is a top view. The hoop and bead are moving in a plane with constant height

Answer 1

x = L sin phi y = -L cos phi x^compositefunction = L^compositefunction sin phi + L phi^compositefunction cos phi y^compositefunction = - (L^compositefunction cos phi - L phi^compositefunction sin phi) Kinetic energy K=1/2 m (x^compositefunction 2 + y^compositefunction 2) potential energy V=mgy = -mg L cos phi Lagrangian L = K -V. alpha = 1/2 m (i^2 + i^2 phi^2) + mg L cos phi L= L_0 - alpha t , alpha constant L^compositefunction = - alpha (constant) alpha = 1/2 m(alpha^2 + i^2 phi^2) + mg L cos phi P_phi = partial differential alpha/partial differential phi = 1/2 m(L^2 2 phi^compositefunction = m L^2 phi^compositefunction phi^compositefunction = Pphi/mL^2 P_L=0 mu = phi^compositefunction - L. mu = Pphi^composite function ^2/mL^2 - 1/2 m aplha ^2 - mg L cos phi Since mu depends on L=L_0 - alpha t that is on t