Chapter 3.5, Problem 5E

Interpretation Introduction

Interpretation:

To estimate the time scale ${\text{T}}_{\text{fast}}$ in terms of original dimensional quantities $\text{m}$, $\text{g}$, $\text{r}$, $\text{ω}$, and $\text{b}$. Rescale the original differential equation using ${\text{T}}_{\text{fast}}$ as the characteristics time, and determine the term which is negligible on this time scale. Show that ${\text{T}}_{\text{fast}}{\text{<< T}}_{\text{slow}}$ if $\text{ε << 1}$.

Concept Introduction:

Express the equation in dimension less form.

Nondimensionalize and simplify the equation to reduce the number of parameters.

Rescale the original equation using characteristic time scale.

Write balance force equation and set the time scale.