Spherical Tippe Tops

Abstract

A tippe top (see Fig. 1) is usually constructed as a truncated sphere with a cylindrical peg on top, as indicated in Fig. 2(a). When spun rapidly on a horizontal surface, a tippe top spins about a vertical axis while rotating slowly about a horizontal axis until the peg touches the surface. At that point, weight is transferred to the peg, the truncated sphere rises off the surface, and the top spins on the peg until it is upright. A feature of a tippe top is that its center of mass, labeled G in Fig. 2, is below the geometric center of the sphere, C, when the top is at rest. That is where it will return if the top is tilted sideways and released since that is the stable equilibrium position. The fact that a tippe top turns upside down when it spins is therefore astonishing. The behavior of a tippe top is quite unlike that of a regular top since the spin axis remains closely vertical the whole time. The center of mass of a regular top can also rise, but the spin axis tilts upward as the top rises and enters a "sleeping" position.An interesting question is whether a tippe top can be constructed from an ordinary sphere with a small mass attached, as indicated in Fig. 2 Fig. 2(c). The small mass will shift the center of mass ofthe top away from the geometric center, so in theory the tops in Figs. 2(b) and (c) might also be expected to work as tippe tops. There is a practical difficulty with top (b) in that the mass m can't be located at the bottom, but the top can be started from an inclined position with m near but not touching the horizontal surface. A practical difficulty with top (c) is inserting a mass inside a hollow sphere, but a suitable sphere should be readily available at a toy shop or office supply shop. It needs to be constructed from two halves that push or screw together neatly.The author constructed tops (b) and (c) and discovered that top (b) doesn't work as a fully inverting tippe top, but top (c) does. If the mass m in top (b) is started near the horizontal surface, it rises to about the equator but Fig. 1. Wolfgang Pauli and Niels Bohr ponder the mystery of an inverting tippe top. The photo was taken in 1951. Others solved the mystery a few years later. (Ptiotograph by Erik Gustatson, courtesy AtP Emitió Segrè Visual Archives, tVtargrethe Botir Cotlection)doesn't rise any further. If m is started at the top, as drawn in Fig. 2(b), then it falls to a point near the equator but doesn't fall any further undl the top almost stops spinning. If the mass m in top (c) starts at the bottom, it ends up at the top. If m in top (c) starts at the top, it stays at the top.Others have studied spherical tops and have even constructed spherical tippe tops by 3-D printing.' The critical factors are the distance a between G and G, the radius R ofthe sphere, the moment of inertia /j ofthe top for rotation about an axis passing through G and G, and the moment of inertia ¡2 for rotation about a perpendicular axis through G. In order to function as a tippe top, it is necessary' thatFor a solid sphere of mass M, /j = 2M ...

Dynamics of a spherical tippe top

“…A tippe top is a well known physics toy that has been studied for many years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The most familiar type is a hollow, truncated sphere with a short peg attached so that it can be spun between the thumb and index finger.…”

“…A question therefore arises as to whether the inversion of a tippe top can also be explained in terms of precession about the horizontal axis. To investigate the problem, measurements were undertaken using a spherical tippe top [13,14] constructed from a hollow sphere with a small mass attached to the inside surface to shift the centre of mass away from the centre of the sphere. The results were compared with theoretical calculations, showing that the inversion is indeed due to precession about the horizontal axis.…”