# Galilei tub

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

In the MATHEMATICS ADVENTURE LAND, the trough, i.e. the Galilean trough, is a blue tread of a length of , which is inclined at the angle 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 as shown in Figure 2 below:

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 coordinate system. The (positive) -direction describes the direction of the valley bottom running from top to bottom and the (positive) -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):

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

for (where denotes the time in seconds). Here is the acceleration due to *gravity* and is the angle of* inclination* of the Galilean trough. In -direction there is approximately a *harmonic oscillation (“cosine oscillation”)* of the form

exists. Here denotes the *angular frequency* of this oscillation and the *amplitude* with .

If we rearrange the equation according to the time, we get:

Consequently, for all time points , the path-time law in -direction is obtained as a function of the path-time law in -direction:

for , so

and thus

with and . So with the concrete values for the inclination angle , the amplitude and the angular frequency is:

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