Vertical Circles, Banked Curves and Conical Pendulums

Circular motion never needs a new force — it needs the forces already present to add up to mv²/r pointed at the center. The skill in this topic is picking apart which forces (or which components of them) do that job in each geometry. At the top of a vertical loop, gravity and any string tension both point toward the center (downward), so mg + T = mv²/r. The minimum speed happens when tension drops to zero, leaving gravity alone to supply the centripetal force: mg = mv²/r, so v_min = √gr. Swing a bucket of water fast enough and the water needs that same inward force; the bottom of the bucket pushes it, and as long as you're above v_min, the push stays positive instead of needing to pull.

A banked curve tilts the normal force itself inward. On a flat road, friction alone turns the car, and it can run out. Tilt the road by angle theta, and the normal force now has a horizontal component, N sin(theta), pointing toward the center, plus a vertical component N cos(theta) balancing gravity. With zero friction needed, dividing those two equations cancels N and mass, leaving tan(theta) = v²/(gr) — the same shape every banking problem reduces to.

A conical pendulum is the same split applied to a string. The bob moves in a horizontal circle while the string sweeps out a cone. Tension's vertical component, T cos(theta), holds the bob up against gravity; its horizontal component, T sin(theta), supplies the centripetal force. These two equations divide the same way banked-curve equations do, linking the cone angle to speed and radius, and connect directly to the next topic's treatment of oscillating and rotating constraint forces.