Advanced — Tension vs Height in Vertical CircleHardQuestion 30673
Question
For a ball in a vertical circle (string, radius r, speed v₀ at bottom), the tension as function of angle θ from the bottom is:
- AT = mv₀²/r − mg(2−3cosθ)
- BT = mv₀²/r − mg(3−2cosθ)Correct
- CT = mv₀²/r + mg(3cosθ−2)
- DT = mv₀²/r + mg cosθ
Correct answer
T = mv₀²/r − mg(3−2cosθ)
Explanation
At angle θ: height = r(1−cosθ). v² = v₀²−2gr(1−cosθ). Centripetal: T − mg cosθ' = mv²/r (where θ' is angle from vertical at that point). Full derivation gives T = mv₀²/r − mg(3−2cosθ) where θ is measured from the bottom.