Experimental validation of the brachistochrone curve

We built a simple, cheap and accessible experiment to study the motion of spherical beads through three different curves, one of them being the curve of fastest descent, or brachistochrone. Despite the absence of a no-friction constraint and the rolling of the beads, our experimental findings consistently support the conclusion that the brachistochrone path remains the fastest. We perform a numerical simulation to confront with experiments and obtain that the values are in agreement, with ttheory=0.237 s, and t‾braq=0.25±0.03 s. This simple system not only serves as a practical demonstration of minimization principles but can also inspire and engage students in high schools and colleges interested in physics.


The brachistochrone problem
Galileo was the first to start working on the problem of the fastest descent.He suggested that it Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. was a cycloid, but he was not able to prove it formally.Christiaan Huygens, another important historical figure in physics and mathematics, is the founder of the tautochrone curve problem.In this curve, no matter at what height it is placed, any mass will reach the lowest point in equal time, again given a constant acceleration of gravity.The proof the tautochrone curve was done using geometric arguments, while Jakob Bernoulli was the first who derived it with calculus in a more efficient way, 30 years later.In 1696, Johann Bernoulli proposed the problem [1], challenging the greatest mathematicians thinking at the time, saying 'I, Johann Bernoulli, address the most brilliant mathematicians in the world.Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument.Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect.If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise' [2].Four more solutions are known to have been part of the competition, by Isaac Newton, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l'Hôpital.
The problem of the brachistochrone, is a simple physical system that undergraduate students in physics study in introductory variational calculus or analytical mechanics classes.The statement of the problem is simple: consider two points in space, A(x 1 , y 1 ), and B(x 2 , y 2 ), with y 1 = y 2 and x 1 = x 2 .What is the fastest curve that could take a frictionless sliding bead from point A to point B, considering purely the influence of the gravitational potential?In other words, what is the curve that minimizes the time of the trajectory between A and B? This problem, depicted in figure 1(a), inaugurated the field of the calculus of variations.

Theoretical solution
We follow the solution presented in [3].Using conservation of energy, we consider that first the bead is at rest at point A. As the bead slides down the path, the gravitational potential energy is converted into kinetic energy, and it reaches point B with velocity v = √ gy.The infinitesimal path that the bead will follow ds, still with unknown shape, can be described in Cartesian coordinates as ds = √ dx 2 + dy 2 = √ 1 + y ′2 dx, with y ′ being the derivative of y(x) with respect to x.From the definition of velocity, v = ds dt , we can isolate the time differential as, When integrating dt, we recognize the that the integrand is a function L = L(y, y ′ , x), that obeys the Euler-Lagrange equation, d dx The non-dependence of L explicitly in x allows us to parametrize the equation, yielding, as a solution the parametric equation for x and y, which represent the curve of fastest descent.This curve, known as the brachistochrone, has the shape of a cycloid, of radius r, and angle θ.
The geometrical shape of this curve is shown in figure 1(b).

Experimental setup
The experimental setup consists of cardboard pieces, small spherical beads, tape, toothpicks, and a camera to record the motion of the bead along the different paths.We printed the figure shown in figure 2(a), taken from the Instructables repository4 on many A4 sheet, with height y = 11.4 ± 0.05 cm, and length x = 17.5 ± 0.05 cm.The sheets were positioned on top of cardboard and cut to size using scissors.We built a retainer at the top of the curves to hold the beads in place before each experiment, and a retainer at the end to hold the beads in place.The setup is shown in figure 2(b).The spheres used were smooth, however the friction with the surface was not negligible.This was also pointed out to the students as a possible source of error during the experiments, but since propagation of uncertainties requires slightly higher understanding of calculus, it was not covered in great detail.

Results and discussion
Three identical beads were released at the same time by lifting the retainer at the top.The motion of the three beads was captured by a cellphone  camera, at rates of 30 frames per second, and 60 frames per second (slow-motion option).A video analysis software, Tracker Physics, was introduced to the students as a way to analyse the data obtained from the experiments, and compare the data to the theoretical predictions and expectations for the motion of the beads.An example of the experimental measurements is presented in figure 3, with the reference length in yellow, and the paths followed by the beads in the three curves: the parabola, in green, the brachistochrone curve in blue, and the straight line in orange.
The total descent time is considered to be the time between when the beads start the motion on the cardboard shape and the end of the shape.Ten different trials were conducted for each path, resulting in t par = 0.27 ± 0.03s, t braq = 0.25 ± 0.03s, t line = 0.34 ± 0.03s, where the uncertainties correspond to the duration of at least one frame, since not always it was possible to correctly assign a position to the beads, and would appear blurred.Noticeably, the brachistochrone curve was the one with the smallest time for all trials.In figure 4(a), we present the evolution of δy = y i − y 0 , where i represents the different time steps.We present the results for the motion on the three different paths constructed.In this plot, the origin of the system was taken as the centre of each bead at the instant where the descent motion starts.The last point in the time axis corresponds to the time in which the beads arrived to the end of the cardboard path.The offset between each measurement is justified by parallax when taking the experimental videos.However, for a trajectory comparison between different curves and time measurement difference, we considered the effects of parallax negligible.
We use a numerical simulation [4] to get an estimate for the time of fastest descent through the brachistochrone.This curve connects points A = (0, 0) and B = (x f , y f ).We solve numerically using the Newton-Raphson method, to find the angle θ of the cycloid that connects both points, from θ 1 = 0 to θ 2 = π/2.The radius of the cycloid is given by, and the minimized descent time can be approximated by We used the distances constructed experimentally in the numerical simulation solver, and obtained that t theory = 0.237s.In figure 4(b), we see the comparison between the experimental data and the numerical simulation of the path of the cycloid curve that connects the distances used experimentally in x and y, in a measurement where the effect of parallax was minimized.In this plot, we took the initial reference point (0, 0) at the initial motion of the bead during the video analysis and included the radius of the beads used, r bead = ± 0.05cm.The experimental path and theoretical path are consistent, and the time of descent is noticeably very similar for both, even with disregards to the friction force that is present on the system.We believe that an optimization of the setup using even smoother beads can lead to better results, as well as a better imaging system with increased time resolution.
It is also crucial to mention the role of rolling in the system.The theoretical considerations regarding the brachistochrone curve consider that the sphere solely slides without friction.In our experiment, the rolling of the beads makes part of the initial kinetic energy be transferred to the rolling motion, leading to smaller velocity, and therefore increasing the time of descent of the bead.

Conclusions
We constructed a simple system that can be used to study the path of fastest descent between two points in space.The setup is composed only by cardboard pieces carefully cut to the shape of the different curves which are taped together.With such a simple system, it was verified that for all experimental realizations, the measured time for descent of the beads through the brachistochrone was smaller than in all other curves.The analysis of descent times also show excellent agreement with the theoretical predictions, even though disregarding the presence of a friction force.The addition of a friction force could retard the motion of the bead, and account for the small time differences between theory and experiment.In the experimental scenario, the rolling of the beads however can be understood as the predominant mechanism that explains theoretical and experimental results for total time of descent.For further optimizations of the setup, we would recommend the usage of a camera with a better imaging resolution, so that the uncertainty in position and time could be minimized, and the usage of different materials to diminish friction between the beads and the surface.In this optimized scenario, the employment of a theory that also accounts for rotation could then be considered.
This experiment is very simple to build and very accessible, and serves as an excellent tool for the introduction of one of the most interesting theoretical results concepts in physics to high schools and colleges: the minimization principle.

Figure 1 .
Figure 1.In (a), a depiction of the system.Consider a bead sliding without friction from point A(x 1 , y 1 ) to point B(x 2 , y 2 ), under the action of the gravitational potential.What is the curve that minimizes the time of descent?In (b), the geometrical depiction of the cycloid curve, between points A and B.

Figure 2 .
Figure 2. (a) Figure used to build the experimental setup, containing a linear path, the brachistochrone, and a parabola.(b) The experimental setup built using cardboard.

Figure 3 .
Figure 3. Example of experimental analysis using the Tracker Physics Software.We used a reference length of 11.4 cm highlighted in yellow, to perform the appropriate conversions from pixels to centimetres.