# 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.

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 denote the number of oscillations of the -th pendulum ( ) in the time interval . 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 , , i.e. for .

Thus, if the -th pendulum has period of oscillation , holds. Because of the relationship between the period of oscillation and length of the mathematical (string) pendulum (where denotes the acceleration due to gravity), we now obtain for the lengths of the individual pendulums: The specific values for , for the exhibit in Mathematics Adventure Land are shown in Table 1 below:

Thus, for the frequency and the angular frequency of the -th pendulum, it follows from and that and . The individual values for and can be read in Table 2 below (given in Hertz):

The instantaneous angles at time of the thirteen mathematical pendulums ( ) have the following equations of motion At the sinusoidal term one can see the reason for the apparent appearance of a discretized sinusoid along the pendulum. Here with as the horizontal distance of the spheres in the delivery at the start and the respective vertical under their suspension. For these values ( ) one obtains for the considered experiment in radians (and in radians) the values of the following table 3:

For the selected pendulums , the instantaneous deflection angle (at time ) shown in Figure 2 is then obtained in radians in a graphical representation for to :

Remark: From equation it follows that the upper bounding line of the suspensions satisfies an equation of the form with suffices, where one unit length of 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: