 has a low centre of
gravity. When you give it a push, it readily restores itself to the
upright position. The top can be made to spin by twisting the stem between
the finger and thumb. The top then precesses with the axle getting lower
and lower until it flips over and continues to spin in an upturned
position. When the spinning stops the top returns to its original stable
state. Can anyone explain why this happens?
_______________________________________________________ a. This question can only be answered rigorously by a quite complex set of mathematical equations, but a simplified answer in broad terms may suffice. The top is a tippe top, a perverse little device which, when spinning, refuses to sit on its rounded end, but flips over and rotates on its stem. This inversion is remarkable for two reasons: the centre of mass is raised in the process of inversion, and the direction of rotation in respect to the fixed-body coordinates is reversed as the top turns over.Analysis of this unexpected behaviour has, over the years, attracted the attention of a number of very famous scientists, ranging from William Thomson in the late 19th century to Niels Bohr and Wolfgang Pauli in this century. Indeed, there is a wonderful photograph in the American Institute of Physics's Niels Bohr Library of Pauli and Bohr watching an inverting tippe top. The first accurate modern explanations of the top's behaviour date from the 1950s, when they were put forward independently by Braams, Hughenholz and Pliskin. But possibly the most rigorous analysis of the top's mechanics is that by the American physicist Richard Cohen at the Massachusetts Institute of Technology in 1977. He also developed computer-generated solutions of the equations of motion. The behaviour of the tippe top can be described for the non-mathematician in fairly simple terms by reference to the diagram below. When the top is set spinning the low centre of mass causes the centre to be centrifugally displaced from the spin axis (Z), which remains perpendicular to the surface on which the top is spinning. The angular momentum component along Z remains dominant before and after inversion, although in respect of the solid-body coordinates of the top, the direction of rotation has been reversed. During the inversion the centre of mass of the top is raised and its rotational kinetic energy is reduced, providing the potential energy to raise the centre of  mass. Thus
the total angular velocity and the total angular momentum are reduced
during the inversion process. This process requires the action of a
torque, but this torque cannot be provided by gravity or the normal forces
exerted at the point of contact with the surface (T). The explanation lies in the presence of sliding frictional forces between the round bottom of the top and the surface on which it is spinning. These forces arise through the centrifugal displacement of the centre of mass when the top is set spinning. If the initial spin velocity imparted to the top when it is set in motion is sufficiently high, nutation and oscillations of the tippe top's motion eventually result in the edge of the stem making contact with the surface and the top then rising to the inverted position. This behaviour of raising the centre of mass also occurs in a normal whip-top. It is not normally appreciated that this occurs because the whip-top does not perform the spectacular inversion that is so obvious in the case of the tippe top. It merely rises up from lying on its side into the vertical position. |
