How nonlinearity affects a pendulum

The equation of motion for a pendulum is the differential equation where g is the acceleration due to gravity and ℓ is the length of the pendulum. When this is presented in an introductory physics class, the instructor will immediately say something like “we’re only interested in the case where θ is small, so we can rewrite the equation as Questions This raises a lot of questions, or at least it should. Why not leave sin θ alone? What justifies replacing sin θ with just θ? How small does θ have to be for this to be OK? How do the solutions to the exact and approximate equations differ? First, sine is a nonlinear function, making the differential equation nonlinear. The nonlinear pendulum equation cannot be solved using mathematics that students in an introductory physics class have seen. There is a closed-form solution, but only if you extend “closed-form” to mean more than the elementary functions a student would see in a calculus class. Second, the approximation is justified because…