In Fig. 1, a block of mass m is attached to a wedge of mass M by a spring with spring constant k. The inclined frictionless surface of the wedge makes an angle α to the horizontal. The wedge is free to slide on a horizontal frictionless surface.
(a) Find the Lagrangian for the system as a function of the x coordinate of the wedge and the length of the spring S. Write the equations of motion.
(b) Find the natural frequency of vibration.
2. A simple pendulum of length 2l and mass 2m hangs is attached to a support of 3M mass which is driven horizontally with time as shown in Fig. 2.
(a) Set up the Lagrangian for the system.
(b) Find the equation of motion.
(c) For small oscillations, find the normal modes and their frequencies.
3. Show that the geodesic on the surface of a right circular cylinder is a segment of a helix.
4. Consider a pendulum of mass m suspended by a massless spring with unextended length b and spring constant k. The pendulum's point of support rises vertically with constant acceleration a.
(a) Use the Lagrange method to find the equations of motion.
(b) Determine the Hamiltonian and Hamilton's equations of motion.
(c) What is the period of small oscillations?