GoKawiilThe 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 sin θ ≈ θ when θ is small. That’s true, but it’s kinda subtle. Here’s a post unpacking that.
The third question doesn’t have a simple answer, though simple answers are often given. An instructor could make up an answer on the spot and say “less than 10 degrees” or something like that. A more thorough answer requires answering the fourth question.
I address how nonlinear affects the solutions in a post a couple years ago. This post will expand a bit on that post.
Longer period
... continue reading