Chapter 13, Problem 86P

This problem explores the nonlinear pendulum discussed qualitatively in Conceptual Example 13.1. You can tackle this problem if you have experience with your calculator’s differential-equation solving capabilities or if you’ve used a software program like Mathematica or Maple that can solve differential equations numerically. On page 228 we wrote Newton’s law for a pendulum in the form I d²θ/dt² = −mgL sin θ. (a) Rewrite this equation in a form suitable for a simple pendulum, but without making the approximation $\sin \theta \cong \theta$. Although it won’t affect the form of the equation, assume that your pendulum uses a massless rigid rod rather than a string, so it can turn completely upside down without collapsing. (b) Enter your equation into your calculator or software, and produce graphical solutions to the equation for the situation where you specify the initial kinetic energy K₀ when the pendulum is at its bottommost position. In particular, describe solutions for (i) K₀ ≪ U_max, (ii) K₀ ≲ U_max, and (iii) K₀ > U_max. Here U_max is the maximum possible potential energy for the system, which occurs when the pendulum is completely upside down; U₀ = 2Lmg, where L is the pendulum’s length.