Tautochrone ("equal time")
This activity belongs to the GeoGebra book The Domain of the Time.
This animation simulates the fall of several masses along a cycloid in real time, neglecting friction. The animation does not use formulas (neither trigonometry, equations, nor differential calculus), but only makes the necessary variations in the vectors that guide the movement.
Observe the figure that appears when the construction starts. The masses at points M, A and B are released to fall by their own weight, all onto the cycloid. You might assume that B will reach the lowest point of the cycloid before A, and A will reach it before M. But that's not the case! All three masses arrive at the same time.
Press the 
button. You can reposition points A and B 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 M.
The cycloid is the only curve with the property of being a tautochrone, 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, 2r):
That is, the oscillation period of the three points is always the same.
button. You can reposition points A and B 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 M.
The cycloid is the only curve with the property of being a tautochrone, 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, 2r):
That is, the oscillation period of the three points is always the same.SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Move M, A and B
SetValue(aux, vt)
SetValue(v, vt + dt gt)
SetValue(vA, vtA + dt gtA)
SetValue(vB, vtB + dt gtB)
SetValue(M, M + dt v)
SetValue(A, A + dt vA)
SetValue(B, B + dt vB)
# Record the period time and the number of complete oscillations
SetValue(reg, If(x(aux) < 0 ∧ x(vt) > 0, Append(t, reg), reg))
SetValue(osci, If(x(aux) < 0 ∧ x(vt) > 0, osci + 1, osci))
Author of the activity and GeoGebra construction: Rafael Losada.