🌀 Vertical Circular Motion

📖 The Physics

Swinging a bucket of water over your head in a full loop, why doesn't the water fall out at the top? In a vertical circle gravity is sometimes with the motion and sometimes against it, so the ball is slowest at the top and fastest at the bottom.

The string tension follows suit: it is largest at the bottom (tension and weight together supply the centripetal force) and smallest at the top, where gravity is already helping pull inward.

T = m·v²/r + m·g·cosθ (θ from the bottom) Bottom: T = mv²/r + mg Top: T = mv²/r − mg Full loop needs v_top ≥ √(gr), i.e. v_bottom ≥ √(5gr)

Below the minimum speed the string goes slack near the top (red) and the ball can't complete the circle — it falls inward. Give it enough speed and it whirls all the way round with the tension pulsing each lap.

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