# Energy, Christiaan Huygens, and the Wonderful Cycloid—Theory versus Experiment

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## Abstract

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## 1. Introduction

#### 1.1. The Cycloid: Coordinates, Equations of Motion, And Approximations

#### 1.2. Cycloid Coordinates

#### 1.3. Equations of Motion for a Cycloid

**V**is given by

#### 1.4. "Get the Ball Rolling"—Correcting for Angular Kinetic Energy

#### 1.5. The Influence of Slipping While Rolling in Motion on Cycloid Pathways

## 2. Methods

#### The Experimental System

## 3. Results and Discussion

^{2}and r = 16 cm into Equation (16)). This means that a relative deviation of 10.45% was recorded.

#### The Energy Consideration

_{r}= 0.044, which is a fairly reasonable value for the rolling friction coefficient.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Huygens’ Proof That a Cycloid Is an Isochronic Curve 2

**Figure A1.**The tangent to a cycloid at a point ${A}_{t}$ passes through the point $F$ that is the highest point of the circle and creates an angle $\alpha $ with the diameter $F{B}_{t}$. The proof that the tangent passes at the highest point is that the direction of the tangent ${A}_{t}F$ is the same as the velocity of the point on the cycloid, and that velocity is equal to two equal velocities; one is the right velocity parallel to the x axis and the other is the tangential velocity resulting from the circular motion. Using geometric considerations, it can be shown that $F{B}_{t}$ is indeed the diameter of the circle and, therefore, ${A}_{t}{B}_{t}$ is a normal for the cycloid. This feature can also be demonstrated by writing the tangent equation to the cycloid and finding its points of intersection with the circle.

**Figure A2.**Huygens’ proof is based on the idea that the velocity vector along the cycloid can be presented as a sum of two speeds: the vertical axis is a circular motion at angular velocity independent of the starting height along the semicircle ${C}_{0}{C}_{t}^{\u2019\u2019}B\u2019$ and the horizontal axis is a variable speed movement that increases with time.

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**Figure 1.**A cycloid is the curve that is created by the pathway of the point that is on the perimeter of the wheel without sliding. In this sketch, the parametric equations of a cycloid are presented, where a is the radius of the wheel and θ is the angle that is defined in the illustration. When the wheel completes a full circle, the angle changes from 0 to 2π.

**Figure 2.**Rolling motion of a ball on a cycloid path, with slippage. $\mathsf{\alpha}$ is the relative slipping factor, which represents how much of the movement is slipping and how much is slipless rolling. ${\mathsf{\mu}}_{\mathrm{k}}$ was set to 0.4, which is approximately the kinetic friction coefficient for steel on steel.

**Figure 3.**The experimental system: the cycloid, B; the inclined plane, C; and the flexible railing. In the inset, one can see the electromagnets that were used in a controlled release of the balls on railings A and B.

**Figure 4.**A schematic description of the experimental system for measuring the duration of the ball’s motion on one of the rails [4].

**Figure 5.**Duration times of the small ball’s movement on the cycloid (black crosses) and on the inclined plane (red squares) as a function of the starting height as it was measured in the experiment.

**Figure 6.**The motion time of the ball along the sloped plain versus the initial height. The blue dots indicate the measured time, the black line indicates the theoretical time of the plot without rolling, and the red line describes the sliding time according to Equation (19); the correspondence between the theory and the experiment is very good.

**Figure 7.**The total mechanical energy (in units of erg) as a function of the initial height $h$ (in cm). The blue line describes the potential energy $mgh$, and the light-blue line indicates the total energy calculated using Equations (28), (31) and (32). The remaining lines indicate kinetic energy, rotational energy, and the work of rolling friction force as a function of the initial height. The correspondence between the calculation and the theory is very good and indicates that the motion of the ball down the sloping plane is a rolling without smoothing.

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**MDPI and ACS Style**

Ben-Abu, Y.; Wolfson, I.; Eshach, H.; Yizhaq, H.
Energy, Christiaan Huygens, and the Wonderful Cycloid—Theory versus Experiment. *Symmetry* **2018**, *10*, 111.
https://doi.org/10.3390/sym10040111

**AMA Style**

Ben-Abu Y, Wolfson I, Eshach H, Yizhaq H.
Energy, Christiaan Huygens, and the Wonderful Cycloid—Theory versus Experiment. *Symmetry*. 2018; 10(4):111.
https://doi.org/10.3390/sym10040111

**Chicago/Turabian Style**

Ben-Abu, Yuval, Ira Wolfson, Haim Eshach, and Hezi Yizhaq.
2018. "Energy, Christiaan Huygens, and the Wonderful Cycloid—Theory versus Experiment" *Symmetry* 10, no. 4: 111.
https://doi.org/10.3390/sym10040111