Chapter 3.6, Problem 5E

Interpretation Introduction

Interpretation:

Determine the equilibrium condition for bead and write this condition in dimensionless form. Do graphical analysis of dimensionless equation for $\text{R<1 and R>1}$. For $\text{r = R - 1}$, write the reduced form of the equilibrium condition. Derive the approximate formula for the saddle-node bifurcation curves. Write the parametric form $\text{h(u) and R(u)}$. Plot the bifurcation curves in (r, h) plane and interpret result physically.

Concept Introduction:

The system remains in equilibrium if all the forces acting on it are zero.

Dimensionless Formulation: The advantage of making equation dimensionless is

The number of parameters in the equation reduces due to lumping them together into dimensionless group

Dimensionless formulation gives the definition of parameter how much small it is $(\ll 1)$

When two bifurcation curves touches each other tangentially at $\left(\text{r, h}\right)\text{=}\left(\text{0,0}\right)$ is known as cusp point.