funny, that you say it.......this is infact a good thought experiment.
- Google gemini says -3
- I tried with co pilot at work yesterday, it said 2
- Today from home right now, it says 1
Gemini
Energy Conservation: When the mass falls from the top to the horizontal position, it loses potential energy WR, which becomes kinetic energy. When it falls from the top to the bottom, it loses potential energy 2WR, which becomes kinetic energy.
Centripetal Force:
Horizontal: The tension T1 provides the centripetal force: T1 = mv²/R. From energy conservation, WR = (1/2)mv², so mv² = 2WR. Therefore, T1 = 2WR/R = 2W.
Bottom: The net upward force (T2 - W) provides the centripetal force: T2 - W = mv²/R. From energy conservation, 2WR = (1/2)mv², so mv² = 4WR. Therefore, T2 - W = 4WR/R = 4W, and T2 = 5W.
Calculate: (T1 - T2) / W = (2W - 5W) / W = -3.
Copilot
It could potentially be a class project!
PS. used image to improve visibility
It seems that both AI fail to compute the correct radius of the osculating circle at the two points, i.e. the value of "r" that enters the acceleration v2/r. It should be r = 2R and R/4 instead of R. If you enter these two, I have the impression you should find 8 again :-)
That's in fact a good way to avoid having to derive my eq. (1) and use something else instead.