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The beauty of mathematics: the spinning tops

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The top is supposed to be simmetrical about its axis, and spinning with its point in a small, smooth cup $O$, like the Maxwell top; as his apparatus is no longer procurable, a bicycle wheel will be found effective for experimental demonstration.

The physical constants of the top are given in C. G. S. units by

(i) the weight W in grammes (g), as weighed in a balance;

(ii) the distance $h$ in centimetres (cm) between the point $O$ and the centre of gravity, and then $Wh$ (g-cm) may be called the *preponderance*;

(iii) $C_1$ and $A_1$, the moment of inertia (g-cm^{2}) about the axis of figure $OC'$ and about any axis through $O$ at right angles to $OC'$.

The moment of inertia $A_1$ can be measured experimentally by swinging the top, without revolution about $OC'$, as a plane or conical pendulum, and observing the length $l$ (cm) of the equivalent simple pendulum, or the angular velocity $n$ (radians/second), or period $2\pi/n$ (seconds), when swung without rotation as a conical pendulum of small angular aperture ; then

(1) \[l = \frac{A_1}{Wh}, \qquad A_1 = Whl\]
(2) \[n^2 = \frac{g}{l}= \frac{Wgh}{A_1}\]

(from

*The Mathematical Theory of the Top* by

**A. G. Greenhill**)

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