Vertical Circle — Angle of SlackHardQuestion 30649
Question
A ball on a string (radius r) swings in a vertical circle with speed v₀ at the bottom. It leaves the circular path (string goes slack) at angle θ from the vertical where cosθ =:
- A(2r g + v₀²)/(3rg)
- B(v₀² − 2gr)/(3gr)Correct
- C(v₀² + 2gr)/(3gr)
- D(v₀² − gr)/(2gr)
Correct answer
(v₀² − 2gr)/(3gr)
Explanation
String slack: mg cosθ = mv²/r. Energy: v² = v₀² − 2gr(1−cosθ). Substituting: rg cosθ = v₀²−2gr+2gr cosθ → cosθ(rg−2rg) = v₀²−2gr → cosθ = (v₀²−2gr)/(3gr) but only valid when 0 ≤ cosθ ≤ 1, i.e. top half.