🎡 A ball on a vertical circle — energy sets the speed, gravity sets the limit

Energy conservation fixes the ball's speed at every angle. The string tension T is largest at the bottom and smallest at the top; a full loop needs v_top ≥ √(gR).

Walkthrough 1 / 4

Step 1 — the ball is launched from the bottom.

Launch speed v_bottom 9.0 m/s

Radius R 3.0 m

Mass m 1.0 kg

θ = 0° · v = 0 m/s · T = 0 N
v_top = 0 · √(gR) = 0 m/s · —

T = m v²/R + m g cosθ
loop needs v_top ≥ √(gR)

Animation speed 1.0×