[color=#0000ff][i][color=#0000ff][i][color=#999999]This activity belongs to the GeoGebra book [url=https://www.geogebra.org/m/mes4bgft]The Domain of the Time[/url].[/color][/i][/color][/i][/color][br][br]This animation simulates the motion of a mass falling along a cycloid in [b]real time[/b], neglecting friction. The animation [b]does not use formula[/b]s (neither trigonometry, equations, nor differential calculus), but simply makes the necessary adjustments to the vectors that guide the movement.[br][br]Similar to what we did with the pendulum, the animation continuously adjusts both the velocity vector [b][color=#cc0000]v[/color][/b] (in red) and the position [color=#0000ff]M[/color] of the mass [i]m[/i], due to the action of gravity, whose constant acceleration is represented by the vector [b][color=#6aa84f]g[/color][/b] (shown as a dashed green line). This vector can be decomposed into two components: one tangent to the path (in green, [color=#6aa84f][b]gt[/b][/color]) and the other perpendicular to it (this perpendicular vector does not affect the motion, as its effect is canceled out by the resistance of the material, shaped like a cycloid, that supports the mass).[br][br]Press the [img]https://www.geogebra.org/resource/hwdawgnn/MmhoDfF5M6lNH9D4/material-hwdawgnn.png[/img] button to bring [color=#0000ff]M[/color] to position H, then press the button [img]https://www.geogebra.org/resource/yxbcmb2f/CZJZaLQBirTUHVXU/material-yxbcmb2f.png[/img] to start the motion.[br][list][*][color=#999999]Note: We previously saw in the inclined plane activity that the [i]free fall[/i] time of[/color] [color=#0000ff]M[/color] [color=#999999](from H to O) was [/color][math]t_0=\sqrt{\frac{2HO}{\left|g\right|}}[/math][color=#999999], and that if[/color] [color=#0000ff]M[/color] [color=#999999]followed the inclined plane (from H to S), we needed to multiply that time by the factor:[/color][br][center][math]\frac{HS}{HO}=\frac{\sqrt{\left(2r\right)^2+\left(\pi r\right)^2}}{2r}=\sqrt{1+\left(\frac{\pi}{2}\right)^2}[/math][/center][color=#999999][/color][color=#999999]Huygens demonstrated that if, instead of following the inclined plane[/color][color=#999999], [/color][color=#0000ff]M[/color] [color=#999999]follows the cycloid, then the factor to multiply the free fall time is smaller, exactly π/2:[/color][br][center][math]t=\frac{\pi}{2}\sqrt{\frac{2HO}{\left|g\right|}}=\frac{\pi}{2}\sqrt{\frac{2\cdot2r}{\left|g\right|}}=\pi\sqrt{\frac{r}{\left|g\right|}}[/math][br][/center][color=#999999]As this path represents one-quarter of a full oscillation, the theoretical period of a complete oscillation (round trip) of[/color] [color=#0000ff]M[/color] [color=#999999]along the cycloid is:[/color][br][center][math]T=4\pi\sqrt{\frac{r}{\left|g\right|}}[/math][/center][color=#999999]Let us remember that this calculation is not necessary to observe the movement of [color=#0000ff]M[/color] in the animation, but is only needed to display the theoretical period.[/color][br][/*][/list]We deduce that [b][color=#cc0000]the period of the fall along the cycloid does not depend on the mass[/color][/b], only on the radius of the wheel that generates the cycloid and gravity. Any mass will always take the same time to complete a full oscillation. As we have already seen, this property is called [i]isochronism[/i].

[b]SCRIPT FOR SLIDER anima[/b][br][br][color=#cc0000][color=#cc0000]# Calculate the elapsed seconds dt; add one second if t1(1) < tt[/color][/color][br][color=#999999]SetValue(tt, t1(1))[br]SetValue(t1, First(GetTime(), 3))[br]SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)[/color][br][br][color=#cc0000]# Move M[/color][br][color=#999999][color=#999999]SetValue[/color](aux, vt)[br][color=#999999]SetValue[/color](v, vt + dt gt)[/color][br][color=#999999]V[color=#999999]SetValue[/color]alor(M, M + dt v)[/color][br][br][color=#cc0000]# Record the period time and the number of complete oscillations[/color][br][color=#999999][color=#999999]SetValue[/color](reg, If(x(aux) < 0 ∧ x(vt) > 0, Append(t, reg), reg))[br][color=#999999]SetValue[/color](osci, If(x(aux) < 0 ∧ x(vt) > 0, osci + 1, osci))[/color][br][br][br][br][color=#999999][color=#999999][color=#0000ff][color=#0000ff][color=#999999][color=#999999]Author of the activity and GeoGebra construction: [/color][/color][/color][color=#0000ff][color=#999999][color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color][/color][/color][/color][/color][/color]