[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]After creating, as we have seen, a time record, we place a point [color=#3d85c6][color=#0000ff]M[/color][/color] (representing a mass [i]m[/i]) at a distance [i]r[/i] from point O. If we call [color=#ff3366][i]ω[/i][/color] a constant [i]angular velocity[/i] (in radians) and substitute in the slider [b]anima [/b]the instruction:[br][br] SetValue([color=#3d85c6][color=#0000ff]M[/color][/color], [color=#3d85c6][color=#0000ff]M[/color][/color] + [i]dt[/i] [b][color=#6aa84f][color=#cc0000][b]v[/b][/color][/color][/b])[br][br]with this other one:[br][br] SetValue([color=#3d85c6][color=#0000ff]M[/color][/color], Rotate([color=#3d85c6][color=#0000ff]M[/color][/color], [i]dt[/i] [i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i], O))[br][br]then [color=#3d85c6][color=#0000ff]M[/color][/color] will move in [i]uniform circular motion[/i] around O.[br][br]To keep track of the time taken for each lap and the number of laps completed, we create the lists [i]reg [/i]and [i]regDif[/i]:[br][br] [b]reg [/b]= {0}[br] [b]regDif [/b]= If(Length(reg) ≟ 1, {reg(1)}, Sequence(reg(k) - reg(k + 1), k, 1, Length(reg) - 1))[br][br]Thus, the [b]average animation time[/b] after a complete lap, that is, the [i]period of the animation[/i], will be given by:[br][br][b] T[/b] = mean(regDif)[br][br][b][color=#cc0000]We find the mean of all the recorded periods because, although theoretically all of them should be equal, as the animation does not follow continuous motion but rather at intervals [i]dt[/i], small deviations may occur in each lap. [/color][/b]Now we just need to add the following instructions to the slider [b]anima [/b]script:[br][br] SetValue(reg, If(y([color=#333333][color=#3d85c6][color=#0000ff]M[/color][/color][/color]) < 0 ∧ y(Rotate([color=#333333][color=#3d85c6][color=#0000ff]M[/color][/color][/color], [i]dt[/i] [i][i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i][/i], O)) ≥ 0, Append([i]t[/i], reg), reg))[br] SetValue(laps, If(y([color=#333333][color=#3d85c6][color=#0000ff]M[/color][/color][/color]) < 0 ∧ y(Rotate([color=#333333][color=#3d85c6][color=#0000ff]M[/color][/color][/color], [i]dt[/i] [i][i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i][/i], O)) ≥ 0, laps + 1, laps))[br][list][*][color=#333333]Note: [color=#333333]Since any circumference measures[/color] 2π [color=#333333]radians, the [i]theoretical period[/i] should be[/color] [color=#333333]2π[/color][/color][i]/[/i][i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i][color=#333333]. [color=#333333]In that time[/color], [color=#3d85c6][color=#0000ff]M[/color][/color] [color=#333333]should cover the circumference[/color] [color=#333333]2π[i]r[/i], [color=#333333]so the magnitude of the tangential velocity[/color] [/color][/color][color=#cc0000][b]v[/b][/color][color=#333333][color=#333333] ([color=#333333]which appears in the construction[/color]) [color=#333333]should be equal to[/color] 2π[color=#333333][color=#333333][i]r[/i][/color][/color] divided by 2π/[i][i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i][/i], [color=#333333]that is[/color], [/color][/color]|[color=#cc0000][b]v[/b][/color][color=#333333][color=#333333]| [/color][/color]=[i] [i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i][/i][color=#333333][color=#333333] [i]r[/i][/color][/color][color=#333333].[/color][/*][/list]

[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]# Register the lap time and the number of laps completed[/color][br][color=#0000ff]SetValue(reg, Si(y(M) < 0 ∧ y(Rotate(M, dt ω, O)) ≥ 0, Append(t, reg), reg))[br]SetValue(laps, Si(y(M) < 0 ∧ y([color=#0000ff]Rotate[/color](M, dt ω, O)) ≥ 0, laps + 1, laps))[/color][br][br][color=#cc0000]# Move M[/color][br][color=#0000ff][color=#0000ff]SetValue[/color](M, Rotate(M, dt ω, O))[/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]