Suppose we have a uniform rod of mass M and length L that can pivot about one end. The other end is attached to a horizontal spring with constant k that is affixed to a wall. The spring is neither stretched nor compressed when the rod hangs straight down. Assume that the rod's angle from the horizontal is always small and that the spring does not bend or bow. k M a) Draw a free body diagram for the pendulum and be sure to explicitly label any angles, forces, locations of forces, center of mass, etc. b) Apply Newton's Torque Law to your free body diagram. c) Derive an equation of motion for the pendulum. See Model 15.1 from your textbook (pg. 406). You may need to look up some trig identities and recall the small angle approximation. d) Deduce the angular frequency w of the pendulum. Again, see Model 15.1 from your textbook. e) Assess the validity of your work. Examine at least 3 limiting cases for your expression in terms of the variables k, g, L, and m. First comment in one sentence what you expect to happen if k, g, L, or m increase/decrease, then show that your expression backs up your intuition.

Suppose we have a uniform rod of mass M and length L that can pivot about one end. The other end is attached to a horizontal spring with constant k that is affixed to a wall. The spring is neither stretched nor compressed when the rod hangs straight down. Assume that the rod's angle from the horizontal is always small and that the spring does not bend or bow. k M a) Draw a free body diagram for the pendulum and be sure to explicitly label any angles, forces, locations of forces, center of mass, etc. b) Apply Newton's Torque Law to your free body diagram. c) Derive an equation of motion for the pendulum. See Model 15.1 from your textbook (pg. 406). You may need to look up some trig identities and recall the small angle approximation. d) Deduce the angular frequency w of the pendulum. Again, see Model 15.1 from your textbook. e) Assess the validity of your work. Examine at least 3 limiting cases for your expression in terms of the variables k, g, L, and m. First comment in one sentence what you expect to happen if k, g, L, or m increase/decrease, then show that your expression backs up your intuition.