The rise and fall of spinning tops
(a) Gyroscopic precession when the spin axis is horizontal and the axle is supported at the left end. (b) Precession of a gyroscope or a top when the spin axis is inclined at an angle θ to the vertical. The top precesses by pivoting about point O, rotating into the page about an axis in the xz-plane that is perpendicular to the spin axis.
Motion of a spinning top with a spherical bottom end. The top initially rolls along a path that spirals inward as shown in (a). The top leans in toward the center of the path and precesses slowly about a vertical axis through the center of the path. As the top slows down, the radius of the pathdecreases and the top can assume a sleeping position as in (b) or it canprecess as shown in (c). The circular disk has been omitted from part (c) for clarity.
Details of a spinning top with a rounded bottom peg of radius A. Here, G denotes the center-of-mass of the top. The normal reaction force N and the centripetal force F both act through the contact point P at the bottom end of the top.
Experimental top constructed from a 76-mm diameter, 8-mm thick aluminum disk with a 60-mm long threaded rod through the center of the disk. The bottom end of the rod was tapered to a sharp point. Alternatively, a spherical ball could be screwed onto the bottom end.
Motion of the upper end of Top 1, viewed from above, recorded over four different time intervals during a single spin of the top. Each time interval corresponds to one low-frequency precession cycle. The (x, y) coordinates of the upper end are plotted at intervals of 0.01 s, as indicated by the dots in the outer two trajectories. As time passes the angle of inclination of the top increases until it eventually falls at t = 31 s after completing 430 revolutions.
Motion of the upper end of Top 1, viewed from above, when the spin is initially small, showing the first precession cycle. The (x, y) coordinates of the upper end are plotted at intervals of 1/150 s, as indicated by the dots. The top fell at t = 7.2 s after completing 50 spin revolutions. A 300-fps video of this motion can be viewed in the online version of the paper or downloaded from the online supplement (enhanced online). 21 [URL: http://dx.doi.org/10.1119/1.4776195.1] [URL: http://dx.doi.org/10.1119/1.4776195.2] [URL: http://dx.doi.org/10.1119/1.4776195.3] [URL: http://dx.doi.org/10.1119/1.4776195.4] [URL: http://dx.doi.org/10.1119/1.4776195.5] [URL: http://dx.doi.org/10.1119/1.4776195.6]doi: 10.1119/1.4776195.1.
Precession data obtained with Top 1 (solid dots) and Top 2 (open squares). The solid and dashed curves are solutions of Eq. (1) for these tops, assuming .
Precession data obtained with Top 3. The solid dots (open circles) are the experimental data for the spin ω (precession frequency Ω) vs. time. Also shown are best fit curves to the experimental data for θ and R 0 vs. time. The curve passing through the Ω data is the solution given by Eq. (4) . Videos of the motion, taken at 300 fps, can be viewed in the online version of the paper or downloaded from the online supplement. 21
Observed trajectory of the tippe top peg when the top was spun at low speed. Observed from above, the top spins counter-clockwise in the laboratory reference frame, precesses slowly along a 16-mm-radius circular path in a clockwise direction, and precesses rapidly around a small radius path in a counter-clockwise direction. The center of the peg is shown by dots at intervals of 0.02 s. The peg rotated slowly away from a vertical position but the top did not invert. Videos of the motion can be viewed in the online version of the paper or downloaded from the online supplement. 21
Typical result showing (a) the precession frequency Ω and tilt angle θ, and (b) the spin ω vs. time for the tippe top. The spin reversed direction when . The top became airborne and bounced several times soon after the peg touched the surface. A 300-fps video of the jumping tippe top can be viewed in the online version of the paper or downloaded from the online supplement. 21
The forces on a rolling ball include a horizontal friction force F and the normal reaction force N acting a distance S ahead of the center of the ball.
Parameters of the four tops.
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