[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]In UCM, as we have seen, a mass [i]m[/i], represented by point [color=#0000ff]M[/color], moves in Uniform Circular Motion around a central point, i.e., at a distance [i]r[/i] with a constant angular velocity [color=#ff3366][i]ω[/i][/color] (and therefore a theoretical period of 2π/[i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i]). We have also seen the appearance of a tangential velocity [color=#cc0000][b]v[/b][/color] (red vector), whose magnitude is the constant [i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i] [i]r[/i] resulting from the action of a centripetal acceleration [b][color=#6aa84f]c[/color][/b] (green vector) with a magnitude of [color=#ff3366][i]ω[/i][/color][sup]2[/sup][i]r[/i].[br][br]In the case of the [i]conical pendulum[/i], the mass [i]m[/i], represented by point [color=#0000ff]M[/color], hangs from a string (of negligible mass) from point [color=#ff7700]A[/color]. The centripetal acceleration [b][color=#6aa84f]c[/color][/b] is given by the horizontal component of the string's tension, whose vertical component compensates for gravity [color=#6aa84f][b]g[/b][/color]. To observe this, activate the animation in the construction. Toggle the checkboxes "Show cone," "Show circle," and "Show vectors" as you wish.[br][br]The angular velocity [i][i][color=#6aa84f][color=#ff3366][i]ω[/i][/color][/color][/i][/i] is determined solely by the height [i][color=#ff7700]h[/color][/i] of the cone. You can vary this height by moving point [color=#ff7700]A[/color]. You'll notice that the lower the height, the greater the angular velocity (and therefore, the shorter the period). However, changing the radius of the circle does not affect the angular velocity (or the period); it only affects the magnitude of the tangential velocity [color=#cc0000][b]v[/b][/color].[br][br]Both [color=#cc0000][b]v[/b][/color] and [b][color=#009900]c[/color][/b] have constant magnitudes that depend only on the angular velocity [color=#ff3366][i]ω[/i][/color] and the radius [i]r[/i]. Now, the centripetal acceleration [b][color=#009900]c[/color][/b] is given by the horizontal component of the string's tension, whose vertical component compensates for gravity [color=#009900][b]g[/b][/color].[br][list][*]Note: Observe that the right triangle with the radius [i]r[/i] and height [i][color=#ff7700]h[/color][/i] of the cone as legs is similar to the right triangle with the magnitudes of [color=#6aa84f][b]c[/b][/color] and [color=#6aa84f][b]g [/b][/color]as legs[b][/b]. Since we already know (UCM) that |[color=#6aa84f][b]c[/b][/color]| = [color=#ff3366][i]ω[/i][/color][sup]2[/sup][i][color=#6aa84f] [/color]r[/i], then [color=#ff3366][i]ω[/i][/color][sup]2[/sup] [i]= |[/i][color=#6aa84f][b]c[/b][/color][i]|[/i]/[i]r = [i]|[/i][/i][color=#6aa84f][b]g[/b][/color][i][i]|[/i]/[i][i][color=#ff7700]h[/color][/i][/i][/i].[i][i] [/i][/i][/*][/list]It is impossible to achieve zero height, no matter how strong the string is, because its tension must always include a vertical component to compensate for gravity (if the height were zero, the period would be zero, and both angular and tangential velocities would be infinite).

[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][color=#999999][br][color=#999999][color=#999999]SetValue[/color](reg, If(arg(M − O) < 0 ∧ arg(M − O) + dt ω ≥ 0, Append(t, reg), reg)) [/color][br][color=#999999][color=#999999]SetValue[/color](laps, If(arg(M − O) < 0 ∧ arg(M − O) + dt ω ≥ 0, laps + 1, laps)) [/color][br][br][/color][color=#cc0000]# Move M[/color][br][color=#999999][color=#999999][color=#999999][color=#999999]SetValue[/color][/color][/color](M, O + (r; t ω))[/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]