The Top With a Blunted Vertex Slipping With Light Friction at a High Rate of Spin

Abstract

The problem of determining the motion of a top is a classic example of a complex analysis in analytical dynamics. Adding a blunt tip to the top and setting it spinning on a surface with sliding friction might be thought to render it intractable for simple analysis. However, if it is set in motion with a high rate of spin it is possible to find a simple approximate solution for the case of approximate steady precession. For this pseudo-steady motion it will be noted that the rate of diminution of the nutation will also be almost constant. Further, the ratio of these rates (latter over former) will be equal to the negative of the coefficient of friction for the top slipping on the surface. As a consequence the mass center of the top will tend to proceed around a steady circle above the plane. These results will first be observed by writing the full Lagrange’s equations for the problem and reducing them prior to integration by observing appropriate approximations by deleting relatively smaller terms. The above results will then follow directly. Further, the full Lagrange’s equations will be numerically integrated to show that the analytically developed approximate results are appropriate. Once these results are known, it is observed that a subsequent intuitive analysis based on time rate of change of angular momentum leads to the same results, if only the angular momentum about the spin axis is considered with other relevant assumptions. [S0021-8936(00)00203-8]