Chapter 3.5, Problem 4E

Interpretation Introduction

Interpretation:

Newton’s law for the motion of bead is to be derived, for $\text{m=0}$ fixed points is to determine in terms of k, h, m, b, and L₀ and stability of these points classified also bifurcation diagram is to be sketched, for $\text{m }\ne 0$ value of m is to determined that can be negligible.

Concept Introduction:

Newton’s Second Law of motion: The total force acting on a body is directly proportional to its acceleration. $\text{F = ma}$

To determine the fixed points, put $\dot{\text{x}}\text{ = 0}$ in the system equation.

To check the stability of the fixed points, plot the graph $\dot{\text{x}}\text{ vs x}$ or solve mathematically.

Dimensionless Formulation: The advantage of making equations 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 $(\ll 1)$