Question

In each of the following problems, please show each of the following steps.

- draw the picture, specify the coordinate system, indicate your generalized coordinates

- write the KE as a function of your generalized velocities and coordinate system.

- write the PE as a function of your generalized coordinates and coordinate system.

- write the Lagrangian

- derive the equations of motion, one each for each generalized coordinate

1) Shown is a simple pendulum bob of mass m attached to a block of mass M. The

block can slide frictionlessly on the x-axis. Ignore the mass of the rod joining the masses, m and

M. Gravity points down and there is no air resistance.

Ꮎ \ r . Two masses are connected by a massless string. The ma. initially moving on the table in a circle with radius r - its loc rawn. The mass m hangs straight down from a hole in the ta

2) Two masses are connected by a massless string. The mass M is on a horizontal

table and is initially moving on the table in a circle with radius r – its location at any time given

by (r,q) as drawn. The mass m hangs straight down from a hole in the table.

Answer 1

Using the Lagrangian, we can obtain Equations of Motion

Lagrangian is given by,

L = K − U, which is the difference between the kinetic and potential energy of the system (K = 1/2m$\dot{x}$² and U = 1/2kx²).

i.e, d/dt(∂L/∂$\dot{x}$_i) − ∂L/∂x_i = 0, where i = 1, 2, 3.

This is the lagrangian equation of motion