Slings and Spirals

Spirals whose curvatures are exponential functions of the arclength are denser near their centers than logarithmic spirals (see Figure 1). I stumbled upon these exponential spirals thanks to an inspiring conversation with mathematician Tadashi Tokieda.

In a previous SIAM News article, I discussed the following parody of the sling [1]. If we want to spin a stone (held by a rope) in a circle with a hand that is spinning in a smaller concentric circle—where the rope forms a fixed angle \(\alpha > \pi /2\) with the hand’s velocity—then we must accelerate the hand in direct proportion to its speed: \(\dot v = c v ^2\) (with some \(c= \textrm{const.}\) that depends on the circles’ radii and on \(\alpha\)). Such \(v=v(t)\) blows up in finite time, meaning that maintaining constant \(\alpha\) is possible only for that long.

Here is a reasonable modification of the problem: What path must my hand take if I no longer insist on moving it in a circle, but instead move it with constant speed (say, \(v=1\)) while still keeping \(\alpha = \textrm{const.}\)?¹ An exponential spiral turns out to be the answer, and the following is a precise statement:

Tracking at a Constant Angle

Consider a pendulum (i.e., a point mass on a massless rod) in a plane, with no gravity (see Figure 2). If the pivot \(P\) moves with constant speed \(v=1\) in such a way as to keep the angle \(\alpha\) in Figure 2 constant, then the path of \(P\) is an exponential spiral with curvature

\[k = \omega_0 e^{-as}, \quad \textrm{where} \quad  a = \frac{\cos \alpha}{r}. \tag1\]

Here, \(s=t\) is the arclength \(=\) time (recall that \(v=1\)), \(r\) is the rod length, and \(\omega_0\) is the initial angular velocity of the pendulum.

I omit the proof, which available upon request.

Some Observations

1. If \(\alpha\) is obtuse (as in Figure 2) so that the pendulum lags, then \(k(s) \rightarrow \infty\) as \(s \rightarrow \infty\) according to \((1)\). The pivot moves in a tightening spiral to keep \(\alpha = \textrm{const.}\) 
2. The curvature \(k\) plays a double role in that it is also the angular velocity of the rod. Indeed, the angle between the rod and the velocity \(\mathbf{v}_P\) is constant. For obtuse \(\alpha\), \(k\) grows exponentially — as does the angular velocity of the rod. The bob then spirals towards a circle of radius \(r\), while \(P\) spirals towards a point. And since the bob’s speed approaches infinity, so does the force with which the hand at \(P\) pulls on the rod.

3. 

   The hand that holds \(P\) expends power (work per unit of time) given by \(- T \cos \alpha\), where \(T\) is the tension of the rod. Note that the hand does work thanks to the fact that the velocity of \(P\) has nonzero projection onto the direction of the force, i.e., the direction of the rod. When swinging an axe overhead while splitting wood, we give the axe its kinetic energy by exactly the same mechanism.² This is also how one can throw a stick with great speed.

Some Questions

1. At what rate do the coils of an exponential spiral accumulate? (Providing a precise meaning of this rate is part of the question.)
2. Does an exponential spiral have an asymptote? More generally, consider a curve that is defined by its curvature \(k=k(s)\). How fast must \(k\) approach \(0\) as \(s \rightarrow \infty\) (or \(- \infty\)) for the curve to have an asymptote?

Do exponential spirals arise in any other settings? It would seem likely that such natural objects exist somewhere in nature, but I couldn’t find any other examples.

¹ As before, \(\alpha\) is the angle between the hand’s velocity vector and the rope.

² Except, perhaps, for the initial part of the swing when one applies torque to the handle to begin the swing.

The figures in this article were provided by the author.

References
[1] Levi, M. (2018, January 29). Slings, bullets, blow-up, and linearity. SIAM News, 51(1), p. 6.