Circular Motion in Practice

The method is always the same: identify the real forces, resolve towards the centre, and set their resultant equal to mv²/r.

Vertical circles — the weight acts downwards throughout, so its contribution to the centripetal force changes around the loop.

At the top, weight acts towards the centre and helps:

T + mg = mv² / r

so the tension is at its smallest here.

At the bottom, weight acts away from the centre and opposes:

T − mg = mv² / r

so the tension is at its greatest. The string, if it breaks, breaks at the bottom.

The minimum speed at the top — as the speed falls, the required tension falls. At the critical speed the tension reaches zero and gravity alone provides the centripetal force:

mg = mv² / r, giving v = √(gr)

Below this the object cannot maintain the circle and leaves the path. This is why a bucket of water swung fast enough stays full at the top, and why a rollercoaster loop has a minimum speed.

Example: notice the mass cancels in v = √(gr). The minimum speed at the top of a loop is the same for a heavy car and a light one, which is the same cancellation that appears in free fall.

Banked tracks — tilting the track means the normal contact force is no longer vertical, so its horizontal component can supply some or all of the centripetal force. This lets vehicles corner faster than friction alone would allow, and is why velodromes and motorway slip roads are banked.

The conical pendulum — a mass swung in a horizontal circle on a string at an angle. The tension has a vertical component balancing the weight and a horizontal component providing the centripetal force. Resolving in two directions and dividing one equation by the other eliminates the tension.

Centrifuges — spinning a mixture at high ω requires a large centripetal force to keep the denser particles moving in a circle. The surrounding fluid cannot supply it, so the denser material moves to the outside, separating the mixture.