Galilei tub

As shown in Figure 1 below, the so-called Galilean trough consists of a rectilinear channel inclined by an angle \alpha and having an approximately semicircular cross-section.

Figure 1: Galilean tub

In the MATHEMATICS ADVENTURE LAND, the trough, i.e. the Galilean trough, is a blue tread of a length of 3.77\mathrm m, which is inclined at the angle \alpha=7^\circ with respect to the horizontal. The approximately semicircular cross-section of the trough, which is constant over the entire length, is “bounded upward” by an “imaginary” straight line. The section it produces has length d= 430\mathrm{mm} as shown in Figure 2 below:

Figure 2: Experimental setup in the mathematics adventure land

In the “bottom” of the Galilei trough there are small obstacles so that a ball starting at a right angle (to the trough) and at the top of the right edge (in the direction of travel) of the Galilei trough does not touch it during its run “downwards” if a suitable start of the ball is carried out. For this purpose, there are 10 different starting possibilities as “small” rectilinear depressions perpendicular to the Galilei trough (see Figure 2). The experimenter now has the task to find out the start possibility which allows the ball an “undisturbed” run downwards — i.e. without touching the obstacles. The path traversed by the ball then represents approximately a “distorted” cosinusoidal motion.


And now … the mathematics of it:

The motion of the sphere is approximately the motion of a point mass on a two-dimensional surface, represented by a Cartesian xy coordinate system. The (positive) x-direction describes the direction of the valley bottom running from top to bottom and the (positive) y-direction describes the direction running perpendicular to it from the valley bottom to the right (in running direction) upper edge of the Galilei trough (see following figure 3):

Figure 3: Introduction of the coordinates x and y

The motion of the sphere in x-direction is then — neglecting friction — described by the displacement-time law

    \[x(t) =\frac{g}{2}t^2\sin(\alpha) \quad(1)\]

for t>0 (where t denotes the time in seconds). Here g= 9.81\mathrm{m}/\mathrm{s}^2 is the acceleration due to gravity and \alpha=7^\circ is the angle of inclination of the Galilean trough. In y-direction there is approximately a harmonic oscillation (“cosine oscillation”) of the form

    \[y(t)=A\cos(\omega t)\quad (2)\]

exists. Here \omega denotes the angular frequency of this oscillation and A the amplitude with A=\pi d/4.

If we rearrange the equation (1) according to the time, we get:

    \[(1)\Rightarrow t^2=\frac{2x(t)}{g\sin(\alpha)}\Rightarrow t=\sqrt{\frac{2x(t)}{g\sin(\alpha)}}\quad (3).\]

Consequently, for all time points t, the path-time law y(t) in y-direction is obtained as a function of the path-time law x(t) in x-direction:

    \[(2),\,(3)\Rightarrow y(t)=A\cos\left(\omega\sqrt{\frac{2x(t)}{g\sin(\alpha)}\right)\]

for t>0, so

    \[y=A\cos\left(\omega\sqrt{\frac{2x}{g\sin(\alpha)}\right)\]

and thus

    \[y=A\cos(\omega'\sqrt{x})\]

with \omega'=\omega\sqrt{\frac{2}{g\sin(\alpha)}} and x\geq 0. So y with the concrete values for the inclination angle \alpha=7^\circ, the amplitude A=\pi d/4=337.7\mathrm{mm} and the angular frequency \omega=6.9\mathrm{Hz} is:

    \[y= 337.7\mathrm{mm}\cos(0.2822\sqrt{x/\mathrm{mm}}).\]

Then the trajectory of the ball rolling “down” is approximately described by the curve shown in Figure 4 below:

Figure 4: xy diagram of the path curve