Zip Line
This activity belongs to the GeoGebra book The Domain of the Time.
The load on an ideal zip line (that is, without friction) behaves similarly to that of a double pendulum. The difference lies in the fact that now the pulley (the first pendulum) does not trace a circular arc, but rather an catenary arc. When a load is suspended, the pulley divides the cable into two different catenary arcs. If the weight of the load is large, we can assume that these arcs are practically straight, meaning the pulley will travel an elliptical arc, since its path is determined by the length of the cable, which is the sum of the distances from the pulley to ends A and B. Since an ellipse is a much simpler curve than a catenary, we choose this option.
We have utilized the slider script to record the maximum speed reached (with or without load). Thus, we can observe that the load (red point) can move faster than the pulley connected to the cable (blue point). In reality, the load is usually very close to the pulley, which minimizes, along with friction, the oscillations caused by the load.
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 M1 and M2 and record the maximum speed
SetValue(v1, vt1 + dt gt1)
SetValue(v2, vt2 + dt gt2)
SetValue(M1, M1 + dt v1)
SetValue(M2, M2 + dt v2)
SetValue(vMax1, Max(abs(v1), vMax1))
SetValue(vMax2, Max(abs(v1 + v2), vMax2))
# Add the positions of M1 and M2 to the record for the polylines traces
SetValue(reg1, Append(reg1, M1))
SetValue(reg2, Append(reg2, M2))
Author of the activity and GeoGebra construction: Rafael Losada.