Combined string and rod pendulum A simple pendulum (mass my and length l) is suspended from...

 

  

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Combined string and rod pendulum A simple pendulum (mass my and length l) is suspended from a cart (mass m,) that can oscillate on the end of a spring of force constant k, as shown in the figure. (a) Write the Lagrangian in terms of the two generalized coordinates r and o, where I is the extension of the spring from its equilibrium length and o is the angle of the pendulum with respect to the vertical axis. Hint: Starting out in Cartesian coordinates, show that the kinetic energy is given by (2 pts.) (b) Use the Euler-Lagrange equations to find the two (coupled) equations of motion. (c) Simplify the equations in the case that both z and o are small. Make the addi- tional approximation that k x and solve the equation of motion for o(t). What is the physical interpretation of the k+ oc limit?(2 pts.) Combined string and rod pendulum A simple pendulum (mass my and length l) is suspended from a cart (mass m,) that can oscillate on the end of a spring of force constant k, as shown in the figure. (a) Write the Lagrangian in terms of the two generalized coordinates r and o, where I is the extension of the spring from its equilibrium length and o is the angle of the pendulum with respect to the vertical axis. Hint: Starting out in Cartesian coordinates, show that the kinetic energy is given by (2 pts.) (b) Use the Euler-Lagrange equations to find the two (coupled) equations of motion. (c) Simplify the equations in the case that both z and o are small. Make the addi- tional approximation that k x and solve the equation of motion for o(t). What is the physical interpretation of the k+ oc limit?(2 pts.)