Chapter 6.5, Problem 16E

Interpretation Introduction

Interpretation:

To show that when $\text{b = 0}$ and $\text{ω}$ is sufficiently large, the system has asymmetric pair of stable equilibria and find the approximate frequency of small oscillations about these equilibria.

Concept Introduction:

Fixed points occur for the conditions $\dot{\varphi }\text{ = 0}$ and $\dot{\text{y}}\text{ = 0}$ .

Equation of bead’s motion is $\text{mr}\ddot{\varphi }\text{ = - b}\ddot{\varphi }\text{ - mg sin}\left(\varphi \right){\text{+ mrω}}^{\text{2}}\text{sin}\left(\varphi \right)\text{cos}\left(\varphi \right)$.

The angular frequency is given by $\text{ω = 2πf}$.