# Centrifugal forces

Written by Yuhang Jin, AP225 2011 Fall.

## Introduction

Centrifugal forces are usually referred to as fictitious forces (inertial forces) that arise in a rotating frame of reference. A centrifugal force represents the inertia of a rotating body, and is directed away from the rotating center or rotating axis. This concept can be generalized in Lagrangian mechanics when generalized coordinates are in effect. At times centrifugal forces may also denote the reaction forces in response to centripetal forces.

## General formula of centrifugal forces as inertial forces

The connection between velocity in an inertial frame of reference and that in a frame rotating at an angular velocity of <math>\mathbf\Omega</math> is given as

<math>\mathbf v=\mathbf v'+\mathbf\Omega\times\mathbf r</math>,

where <math>\mathbf r'</math> denotes the displacement in the rotating fram, <math>\mathbf v</math> the absolute velocity (in an inertial frame of reference) and <math>\mathbf v'</math> the velocity in the rotating frame. Similarly, the relation of acceleration is given as

<math>\mathbf a=\mathbf a'+\dot{\mathbf\Omega}\times\mathbf r'+2\mathbf\Omega\times\mathbf v'+\mathbf\Omega\times(\mathbf\Omega\times\mathbf r')</math>.

Applying the Second Law of Newtonian mechanics we have

<math>m\mathbf a=m\mathbf a'+m\dot{\mathbf\Omega}\times\mathbf r'+2m\mathbf\Omega\times\mathbf v'+m\mathbf\Omega\times(\mathbf\Omega\times\mathbf r')

## Keyword in references:

Paper on a disc: balancing the capillary-driven flow with a centrifugal force