[Total: 30 pts] A simple pendulum consists of a point mass m fixed to the end of a massless rod (length l, whose other end is pivoted from the ceiling to let it swing freely in the vertical plane. The pendulum's position can be identified simply by its angle θ from the equilibrium position.
a) [6 pts] Write the equation of motion for θ using Newton's second law. Assuming that the angle θ remains small throughout the motion, solve for θ and show that the motion is periodic. What is the period of oscillation?
b) [6 pts] Show that the pendulum's potential energy (measured from the equilibrium level) is U = A(1 - cosθ). Find A in terms of mg and l. Then write down a formula for the mechanical energy as a function of θ and θ'.
c) [6 pts] Show that by differentiating the energy with respect to t, you can recover the equation of motion you found in part a. What basic principle of physics are you using here? Discuss.
Updated 03/02/22: Previous version had a typo. You need to differentiate the energy with respect to t (not with respect to θ).
d) [6 pts] Picking simple values for parameters (e.g., m = l = 1, g = 10), use your favorite computational environment (e.g., NDSolve in Mathematica, for help see http://youtu.be/zKO6vOwOKaI) to solve for θ given a fairly small starting angle θ₀. Provide output that clearly shows that the period of oscillation is very close to the theoretical prediction from part a. Repeat with θ₀ = 2 rad (which is quite large) and show that the period now deviates from ideal. Does the period get larger or smaller as θ₀ increases?
e) [6 pts] Using your numerical solution from part d and your formulas for energy from part b, generate a single plot which graphs T, U, and total E as a function of time, for one period, in the case where θ₀ = 2 rad is NOT small. Briefly comment: does the system still conserve energy when we cannot use the small angle approximation anymore?
