[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][br]In the UCM activity, we saw how the mass m, represented by point [color=#0000ff]M[/color], moved in uniform circular motion (UCM) around point O, that is, at a distance [i]r [/i]with a constant angular velocity [color=#ff3366][i]ω[/i][/color]. There was also a tangential velocity [color=#cc0000][b]v[/b][/color], whose magnitude is the constant [i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i] [i]r[/i].[br][br]However, the fact that [color=#cc0000][b]v[/b][/color] has a constant magnitude does not mean that velocity [color=#cc0000][b]v[/b][/color] is constant, since its direction is not. This means that there must be a force (a rigid body, the tension of a rope, gravity, a magnetic force...) that forces mass [i]m[/i] to maintain the circular motion. Otherwise, as we've seen, it would follow a uniform rectilinear motion due to inertia.[br][br]This force is known as [i][url=https://en.wikipedia.org/wiki/Bucket_argument]centripetal force[/url][/i], because its direction are toward the center of the circle. This force causes a [i]centripetal acceleration[/i] [b][color=#6aa84f]c[/color][/b], represented by the green vector. The magnitude of this acceleration is exactly what is needed to keep the mass in circular motion and prevent it from continuing in a straight line by inertia.[br][br]If we place ourselves at point [color=#0000ff]M[/color], we will feel a force that seems to pull us away from the circular path, as if we were "get off on a tangent". This apparent (i.e., fictitious) force is called the "centrifugal force", but it is simply our perception of the [b]resistance inertia offers[/b] to the real centripetal force.[br][br]To better observe the relationship between acceleration [b][color=#6aa84f]c[/color][/b] and velocity [color=#cc0000][b]v[/b][/color], activate the "View variation of v" box (a diagram known as the [i]hodograph [/i]of the motion).[br][list][*]Note: In the hodograph, point [i]A[/i] travels 2π|[color=#cc0000][b]v[/b][/color]| with each lap, that is, every [i]T[/i] = 2π/[color=#ff3366][i]ω[/i][/color] segundos. seconds. As [i]A[/i] moves at speed [b][color=#6aa84f]c[/color][/b][color=#cc0000] [/color](acceleration is the [i]rate of change[/i] of velocity), we have |[b][color=#6aa84f]c[/color][/b]| = 2π|[color=#cc0000][b]v[/b][/color]|/[i]T[/i] = [i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i] |[color=#cc0000][b]v[/b][/color]| = [i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i][sup]2[/sup] [i]r = [/i][color=#cc0000][b]v[/b][/color][sup]2[/sup]/[i][i]r[/i][/i]. [/*][/list]Notice that the vectors [b][color=#6aa84f]c[/color][/b] and [color=#cc0000][b]v[/b][/color] completely determine the motion of [color=#0000ff]M[/color]. In the animation, each time the time advances "a little bit" ([i]dt[/i]), the velocity becomes [color=#cc0000][b]v[/b][/color] + dt [b][color=#6aa84f]c[/color][/b] ([i]Newton's 2nd law[/i]), so the position of [color=#0000ff]M[/color] becomes [color=#0000ff]M[/color] + dt [color=#cc0000][b]v[/b][/color].

[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][color=#0000ff]SetValue[/color](reg, If(x(v)>0 ∧ x(v + dt c) ≤ 0, Append(t, reg), reg))[br][color=#0000ff]SetValue[/color](laps, If(x(v) > 0 ∧ x(v + dt c) ≤ 0, [color=#0000ff]laps[/color] + 1, [color=#0000ff]laps[/color]))[/color][br][br][color=#cc0000]# Move M[/color][br][color=#0000ff]SetValue(v, v + dt c)[br][color=#0000ff]SetValue[/color](M, M + dt v)[/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]