Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-6bf8c574d5-k2jvg Total loading time: 0 Render date: 2025-03-04T11:09:57.026Z Has data issue: false hasContentIssue false

CHAPTER VI - THE SOLUBLE PROBLEMS OF RIGID DYNAMICS

Published online by Cambridge University Press:  05 August 2012

E. T. Whittaker
Foreword by
Sir William McCrae
Get access

Summary

The motion of systems with one degree of freedom: motion round a fixed axis, etc.

We now proceed to apply the principles which have been developed in the foregoing chapters in order to determine the motion of holonomic systems of rigid bodies in those cases which admit of solution by quadratures.

It is natural to consider first those systems which have only one degree of freedom. We have seen (§ 42) that such a system is immediately soluble by quadratures when it possesses an integral of energy: and this principle is sufficient for the integration in most cases. Sometimes, however (e.g. when we are dealing with systems in which one of the surfaces or curves of constraint is forced to move in a given manner), the problem as originally formulated does not possess an integral of energy, but can be reduced (e.g. by the theorem of § 29) to another problem for which the integral of energy holds; when this reduction has been performed, the problem can be integrated as before.

The following examples will illustrate the application of these principles.

(i) Motion of a rigid body round a fixed axis.

Consider the motion of a single rigid body which is free to turn about an axis, fixed in the body and in space. Let I be the moment of inertia of the body about the axis, so that its kinetic energy is ½Iθ2 where θ is the angle made by a moveable plane, passing through the axis and fixed in the body, with a plane passing through the axis and fixed in space. […]

Type
Chapter
Information
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies , pp. 131 - 176
Publisher: Cambridge University Press
Print publication year: 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×