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Swinging balls

Oscillations are a mathematically very interesting topic. In Mathematics Adventure Land, the exhibit “Swinging Balls” makes it possible to experience the behavior of swinging mathematical pendulums. As shown in the following figure 1, 13 metal spheres hang from thread pendulums of different lengths, which are attached to a curvilinear suspension.

Figure 1: Oscillating metal balls on thread pendulums

In this experiment, lowering the blue rail (see Figure 1) simultaneously triggers the oscillation of all 13 spheres. They leave their respective maximum deflections at the same time. Thus, for the observer, they initially oscillate back and forth seemingly without any rules.

If the oscillations of the spheres are observed over a longer period of time, the following behavior is discovered:

At the start, the oscillations of all pendulums begin simultaneously and then change to a discrete sinusoidal form whose amplitudes continuously increase. When the largest number of amplitudes of this “oscillating phenomenon” is reached, the direction of oscillation of each pendulum is opposite to the direction of oscillation of its neighbors (the first and the last pendulum, of course, have only one neighbor). So every second pendulum swings in the same direction. This gives the optical impression that the pendulums are temporarily swinging in two “fronts” running towards each other and then “interlock” again. Finally, one can observe a discrete sinusoidal oscillation of the spheres again, whose amplitude decreases until they reach again together (approximately) their initial position (in the experiment in the Mathematics Adventure Land after 40 seconds). Of course, the process described in this way decays with time, since friction slows down the pendulum movements (i.e. the maximum deflections of the individual pendulums gradually become smaller until they come to a complete stop).

And now … the mathematics of it:

Let A(n) denote the number of oscillations of the n-th pendulum (n=1,\ldots,13) in the time interval T=40\mathrm s. Then, given an oscillation period of two seconds for the longest pendulum and assuming that the number of oscillations of neighboring pendulums differ by exactly one in the time interval T, A(n+1)=A(n)+1, i.e. A(n)=n+19 for n=1,\ldots,13.

Thus, if the n-th pendulum has period of oscillation T(n), T(n)=\frac{40}{19+n}\mathrm s holds. Because of the relationship between the period of oscillation T(n) and length l(n) of the mathematical (string) pendulum T(n)=2\pi\sqrt{l(n)/g} (where g denotes the acceleration due to gravity), we now obtain for the lengths l(n) of the individual pendulums:

    \[l(n)=\left(\frac{T(n)}{2\pi}\right)^2 g\quad(\ast).\]

The specific values for l(n), n=1,\ldots,13 for the exhibit in Mathematics Adventure Land are shown in Table 1 below:

nl(n) (in meters)
10,9940
20,9016
30,8215
40,7516
50,6903
60,6361
70,5881
80,5454
90,5071
100,4728
110,4418
120,4137
130,3883
Table 1: Length of the individual thread pendulums in meters

Thus, for the frequency f(n) and the angular frequency \omega(n) of the n-th pendulum, it follows from f(n)=1/T(n) and \omega(n)=2\pi f(n)=2\pi/T(n) that f(n)=\sqrt{g/l(n)}/(2\pi) and \omega(n)=\sqrt{g/l(n)}. The individual values for f(n) and \omega(n) can be read in Table 2 below (given in Hertz):

nFrequency f(n)Circular frequency \omega(n)
10,53,1416
20,5253,2987
30,553,4558
40,5753,6128
50,63,7700
60,6253,9270
70,654,0841
80,6754,2412
90,74,3982
100,7254,5553
110,754,7124
120,7754,8695
130,85,0265
Table 2: Frequency and angular frequency of the n-th pendulum

The instantaneous angles \alpha_n(t) at time t of the thirteen mathematical pendulums (n=1,\ldots, 13) have the following equations of motion

    \[\alpha_n(t)=\alpha_{\max}(n)\sin(\omega(n)t+\pi/2)=\alpha_{\max}(n)\sin\left(\frac{\pi(19+n)t}{20\mathrm s}+\frac{\pi}{2}\right).\]

At the sinusoidal term one can see the reason for the apparent appearance of a discretized sinusoid along the pendulum. Here \alpha_{\max}(n)=\arcsin(L/l(n)) with L=0.33\mathrm m as the horizontal distance of the spheres in the delivery at the start and the respective vertical under their suspension. For these values \alpha_{\max}(n) (n=1,\ldots,13) one obtains for the considered experiment in radians (and in radians) the values of the following table 3:

n\alpha_{\max}(n) (Radian measure)\alpha_{\max} (Radian)
10,338419,39^\circ
20,374721,47^\circ
30,413423,69^\circ
40,454826,04^\circ
50,498528,56^\circ
60,545431,25^\circ
70,595734,13^\circ
80,649937,23^\circ
90,708540,60^\circ
100,772744,27^\circ
110,843648,33^\circ
120,923452,91^\circ
131,015958,20^\circ
Table 3: Deflections of the individual pendulums

For the selected pendulums n=1,6,11, the instantaneous deflection angle (at time t) shown in Figure 2 is then obtained in radians in a graphical representation for t=0\mathrm s to t=10\mathrm s:

Figure 2

Remark: From equation (\ast) it follows that the upper bounding line of the suspensions satisfies an equation of the form

    \[y=f(x) =g\left(\frac{20\mathrm s}{\pi(19+13-x)}\right)^2\]

with 0\leq x\leq 13 suffices, where one unit length of x corresponds to the distance of each two adjacent spheres at rest. In the following graphical representation, the value 1 was chosen for this length unit:

Figure 3: Suspension curve of the balls

Remark:

The idea for the exhibit came from a study published in February 1991 by American physicist Richard E. Berg (University of Maryland), which appeared in the American Journal of Physics and dealt with swinging mathematical pendulums. The impetus came from a video recorded by C. Alley (also University of Maryland) at Moscow State University, where he stayed during a study visit in 1987.

Literature

[1] Berg, Richard E.: Pendulum waves: A demonstration of wave motion using pendula. in: American Journal of Physics 59 (2), S. 186–187, 1991.

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