Fall along a cycloid
This activity belongs to the GeoGebra book The Domain of the Time.
This animation simulates the motion of a mass falling along a cycloid in real time, neglecting friction. The animation does not use formulas (neither trigonometry, equations, nor differential calculus), but simply makes the necessary adjustments to the vectors that guide the movement.
Similar to what we did with the pendulum, the animation continuously adjusts both the velocity vector v (in red) and the position M of the mass m, due to the action of gravity, whose constant acceleration is represented by the vector g (shown as a dashed green line). This vector can be decomposed into two components: one tangent to the path (in green, gt) and the other perpendicular to it (this perpendicular vector does not affect the motion, as its effect is canceled out by the resistance of the material, shaped like a cycloid, that supports the mass).
Press the 
button to bring M to position H, then press the button 
to start the motion.
button to bring M to position H, then press the button to start the motion.
- Note: We previously saw in the inclined plane activity that the free fall time of M (from H to O) was , and that if M followed the inclined plane (from H to S), we needed to multiply that time by the factor:
Huygens demonstrated that if, instead of following the inclined plane, M follows the cycloid, then the factor to multiply the free fall time is smaller, exactly π/2:
As this path represents one-quarter of a full oscillation, the theoretical period of a complete oscillation (round trip) of M along the cycloid is: Let us remember that this calculation is not necessary to observe the movement of M in the animation, but is only needed to display the theoretical period.
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
SetValue(aux, vt)
SetValue(v, vt + dt gt)
VSetValuealor(M, M + dt v)
# 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.