Problem 1: Suppose you have a pendulum that swings in a vertical 2D plane, but the pivot of the pendulum is attached to an oscillating motor. The motion of the pivot is just in the vertical direction (call it y) , and variation in y of the pivot described by y,(t) =A sin(wt)y. The length of the rod between pivot and bob is L and mass of bob is M. Our goal is to get to the equation of motion of the bob. This is a good problem that illustrates the power of generalized coordinates. The coordinates can be a combination of position, angles and/or time - whatever is most convenient to describe the position of the objects in the system (in this case, the bob of the pendulum). In Example 7.6, the book defined the angle 0 as angle between the vertical line and the line connecting the pivot and the bob. With this angle, the math is simpler than what I did in lecture (I used standard spherical coordinates). Use the books definition for your angle 0. The downside is that you need to figure out the potential energy properly (especially the sign) in the terms of the generalized coordinate. In this case, you need to ensure that the potential energy increases when the angle 0 increases. Unlike an ordinary pendulum problem, the position of the bob requires more than one generalized coordinate, 0. But it seems that 0, and t are sufficient. If you compare the solution to Example 4 in lecture notes to the books version, you see that generalized coordinates do not need to be part of a standard Right Hand Rule coordinate system. a) As suggested in lecture, when one of the coordinates (in this case yp) is itself moving, instead of figuring out the kinetic energy directly in terms of the two generalized coordinates, you begin by writing down the position of the bob in cartesian coordinates in terms of the two generalized coordinates, 0,t. b) Take the time derivative of x(t), y(t) to get vx and vy, and then find the kinetic energy T. c) Determine the potential energy d) Apply Euler-Lagrange for the variable 0 to get the angular acceleration, 0. Do not solve the differential equation