[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 fall of several masses along a cycloid in [b]real time[/b], neglecting friction. The animation [b]does not use formulas[/b] (neither trigonometry, equations, nor differential calculus), but only makes the necessary variations in the vectors that guide the movement.[br][br]Observe the figure that appears when the construction starts. The masses at points [color=#0000ff]M[/color], [color=#ff7700]A[/color] and [color=#6aa84f]B[/color] are released to fall by their own weight, all onto the cycloid. You might assume that [color=#6aa84f]B[/color] will reach the lowest point of the cycloid before [color=#ff7700]A[/color], and [color=#ff7700]A[/color] will reach it before [color=#0000ff]M[/color]. But that's not the case! All three masses arrive at the same time.[br][br]Press the [img]https://www.geogebra.org/resource/hwdawgnn/MmhoDfF5M6lNH9D4/material-hwdawgnn.png[/img] button. You can reposition points [color=#ff7700]A[/color] and [color=#6aa84f]B[/color] anywhere on the arc of the cycloid. You'll see that they all cross the lowest point of the cycloid at the same time as [color=#0000ff]M[/color].[br][br]The cycloid is the [b]only curve[/b] with the property of being a [i]tautochrone[/i], meaning the time it takes for a mass to slide without friction in uniform gravity to its lowest point is independent of its starting point. As we have seen, Huygens discovered that this time is π/2 times the free fall time from H(0, 2[i]r[/i]):[br][center][math]t=\pi\sqrt{\frac{r}{\left|g\right|}}[/math][/center]That is, the oscillation period of the three points is always the same.

[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, A and B[/color][br][color=#999999][color=#999999]SetValue[/color](aux, vt)[br][color=#999999]SetValue[/color](v, vt + dt gt)[/color][br][color=#0000ff]SetValue(vA, vtA + dt gtA)[br][color=#0000ff]SetValue[/color](vB, vtB + dt gtB)[/color][br][color=#999999][color=#999999]SetValue[/color](M, M + dt v)[/color][br][color=#0000ff][color=#0000ff]SetValue[/color](A, A + dt vA)[br][color=#0000ff]SetValue[/color](B, B + dt vB)[/color][br][br][color=#cc0000]# Record the period time and the number of complete oscillations[/color][br][color=#999999][color=#999999][color=#999999]SetValue[/color][/color](reg, If(x(aux) < 0 ∧ x(vt) > 0, Append(t, reg), reg))[br][color=#999999][color=#999999]SetValue[/color][/color](osci, If(x(aux) < 0 ∧ x(vt) > 0, osci + 1, osci))[/color][br][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]