Let's consider a simple pendulum, consisting of a point mass m, fixed to the end of a massless rod of length l, whose other end is fixed so that the mass can swing freely in a vertical plane. The pendulum's position can be specified by its angle Φ from the equilbrium position.Prove that the pendulum's potential energy is U(ϕ)=mgl(1−cosϕ).Write down the total energy E as a function of Φ and ϕ˙.Show that by differentiating E with respect to t you can get the equation of motion for Φ.Solve for Φ(t). If you solve properly, you should find periodic motion. What is the period of the motion?

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College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Let's consider a simple pendulum, consisting of a point mass m, fixed to the end of a massless rod of length l, whose other end is fixed so that the mass can swing freely in a vertical plane. The pendulum's position can be specified by its angle Φ from the equilbrium position.

Prove that the pendulum's potential energy is U(ϕ)=mgl(1−cosϕ).
Write down the total energy E as a function of Φ and ϕ˙.
Show that by differentiating E with respect to t you can get the equation of motion for Φ.
Solve for Φ(t). If you solve properly, you should find periodic motion. What is the period of the motion?

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