October 31, 2007

Fig. 1: The tippe top inverting. The black and red
dot signifies the center of sphere and center of mass,
respectively. |

The motion of gyroscopic bodies often baffle intuition. One famous example is the tippe top, which consists of a truncated sphere, flat on one surface, upon which a short rod is mounted. The tippe top is known for its counterintuitive behavior that after being spun rapidly enough with the stem upwards, the top refuses to sit on its rounded end, and proceeds to turn upside-down to rotate on its elongated stem, as in Fig. 1. Surprisingly, the center of mass is lifted after the inversion. This happens irrespective of the type of contacting surface, and seems to be independent of initial conditions. Moreover, this inversion phenomenon is a property of many spherical objects whose center of mass is displaced from the geometrical center of the sphere. I will now give a brief introduction to the theory of the top.

Del Campo (1955) demonstrated from a simple physical
analysis that friction must be important in the behavior of the tippe
top, which is summarized here. Let *z* be the axis perpendicular
to the table surface. One observes that during inversion, the *z*
component of the angular momentum is dominant, both before and after the
inversion. Hence, the direction of rotation actually reverses with
respect to coordinates fixed in the body of top. More importantly,
after the inversion, the center of mass of the top actually elevates.
This gain of gravitational potential energy necessarily comes from the
loss of rotational kinetic energy, which implies the total angular
velocity and total angular momentum in the *z* direction decrease
during inversion. A reduction in the angular momentum
*L _{z}* requires torques along the

Fig. 2: Tippe top and coordinate axes. |

We now present a simplified quantitative argument on
why the tippe top tips over; this argument was first given by Pliskin
(1954). As in Fig. 2, coordinates defined by the unit vectors
*e _{x}, e_{y}, e_{z}* are fixed in the
laboratory frame. Making straightforward the argument of Cohen (1977),
and keeping consistency with his notation, we define the new coordinate
system

Using the Euler angles defined in Fig. 2, we can transform the two coordinate systems into each other:

(3) | |||

(4) | |||

(5) |

Hence, we see from above that

(6) |

The angular velocity α of the coordinate system
*e _{n}, e_{n}', e_{3}* is then

(7) |

Note that since we spin the tippe top rapidly,
dφ/dt >> dθ/dt, and the top precesses rapidly as it slowly
falls over. Hence, α points very closely to the positive *z*
axis.

If we ignore the translation motion of the top, the
frictional force *F _{f}* exerted at the contact point
opposes the sliding motion of the top, hence

(8) |

where, as in Fig. 2, **r** is the vector from the
center of mass to the contact point, α is the distance between the
center of mass and the geometrical center, and *R* is the distance
between the geometrical center and the contact point. Since the origin
of our coordinate system *e _{n}, e_{n}',
e_{3}* is the center of mass, gravity exerts no torque.
Also, the torque from the normal force points in the

In the beginning, 0 ≅ θ <
cos^{-1}(*a/R*), the torque about *e _{n}'* is
positive and the torque about

(c) 2007 Shih-Arng Pan. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] A.R. Del Campo, Am. J. Phys. **23**, 544
(1955).

[2] W.A. Pliskin, Am. J. Phys. **22**, 28
(1954).

[3] R. J. Cohen, Am. J. Phys. **45**, 1
(1977).