# Centrifugal forces

Written by Yuhang Jin, AP225 2011 Fall.

## Contents

## 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 center of axis of rotation. 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

When the motion of a body is studied in a non-inertial frame of reference, fictitious forces are introduced for convenience [1]. 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>\mathbf F=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')</math>,

i.e.

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

Hence from the perspective of this rotating frame of reference, the terms other than <math>\mathbf F</math> on the right hand side of the equation are fictitious forces. Specifically <math>-m\mathbf\Omega\times(\mathbf\Omega\times\mathbf r')</math> is called the centrifugal force, since it points outward perpendicular to <math>\mathbf\Omega</math>. Therefore, centrifugal forces are proportional to the distance to the axis of rotation as well as the square of the angular velocity of the rotating frame.

## Centrifugal forces as reactive forces

In some contexts, centrifugal forces refer to reactive forces [2]. The motion of a rotating body is maintained by a centripetal force provided by another object. According to the Third Law of Newtonian mechanics, the rotating body exerts a reactive force on that object, referred to as a centrifugal force.

## Applications

Many devices make use of centrifugal forces, such as centrifuges and centrifugal pumps, which have found numerous applications in the industry and academia. Centrifugal forces are also an important factor in engineering designs for railways and satellites etc. Recently, in space stations centrifugal forces are used to balance gravity to approximate zero-gravity environments.

## References

[1] Gregory, R. D., "Classical Mechanics", Cambridge University Press, 2006.

[2] Wikipedia on centrifugal forces

## Keyword in references:

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