The Isochronous Problem inside the Spherically Uniform Earth

Document Type


Publication Date



The isochronous problem is worked out assuming that a particle oscillates along a constraining curve inside the spherically uniform Earth. This modification of the usual isochronous problem becomes complicated due to (1) the force acting on the particle is not unidirectional; (2) the curve of constraint must have its endpoints on the Earth’s surface. A set of parametric equations satisfying the condition of isochronous motion are obtained. Then the values of these equations that are permitted by the second condition mentioned above are calculated and presented in a graph. The calculation shows that, in addition to the familiar straight line along which the particle oscillates isochronously inside the Earth, there are many curves that satisfy the conditions. The frequencies of the isochronous oscillation along these curves are simply related to the frequency along the straight line. Thus the problem treated here also has application to the Brachistochrone problem that is similarly modified. © 1972., American Association of Physics Teachers. All rights reserved.

Publication Title

American Journal of Physics