What keeps things moving in circles?

Whirl a ball on a string. Its speed can stay the same, but its direction changes all the time. A change of direction is a change of velocity, so the ball is accelerating, and by Newton's second law (see NewtonClear) something must push or pull it. That something is the string, pulling towards the centre. We call a pull like this a centripetal force: "centre-seeking".

How big must it be? For mass m going at speed v round a circle of radius r:

F = m v² ÷ r, and the acceleration is a = v² ÷ r.

Force is in newtons, acceleration in m/s². The speed is squared, so double the speed and the pull goes up four times. A tighter circle, smaller r, also needs a bigger pull. You can measure it by putting a spring balance in the string, as here.

Spinning things are often described by how fast they turn instead: the angular speed ω in radians per second, or rpm, turns per minute. Every point on a spinning thing turns at the same ω, but points further out move faster: v = ω r. Watch the bead halfway along the string: it goes round with the ball but at half the speed.

Now cut the string. The ball does not fly outwards. It goes straight on, along the line touching the circle, the tangent. Nothing was pushing it out; the string was only bending its path in.

Every spinning machine is the ball on a string again. A point at radius r turning at ω needs an inward acceleration a = ω² r. Engineers compare it with gravity and call it g-force: a ÷ 9.81.

Spin-dry. In a washing machine (see WasherClear) the 50 cm drum turns at up to 1,200 rpm. The wall then moves at 31 m/s and must push the clothes inwards about 400 times harder than gravity. The clothes are held by the wall, but the water in them has nothing to hold it, so it carries straight on, out through the holes. Below about 60 rpm the wall can't even beat gravity at the top, and the clothes tumble instead.

Blades. A mixer grinder (see MixerClear) at 18,000 rpm moves its blade tips, only 3 cm out, at about 200 km/h: over 10,000 g. A ceiling fan (see FanClear) turns far slower, 350 rpm, but its tips are 60 cm out, so they still reach 80 km/h, and each blade's root must hold a pull of about 170 N.

Centrifuges. A salad spinner flings water off lettuce at about 20 g. A hospital centrifuge spins blood at about 1,000 g for ten minutes: the heavy red cells pack at the bottom and yellow plasma floats on top, a job gravity alone takes hours to do, and does less neatly. The same trick separates cream from milk in every dairy.

A car going round a bend is the ball on the string again, but there is no string. On a flat road the only thing that can push the car sideways is friction between the tyres and the road (see FrictionClear and CarClear). The most it can give is μ m g, so:

top speed on a flat bend: v = √(μ g r)

The mass cancels out: a truck and a scooter have the same limit on the same road. But the limit falls fast on a wet road (μ 0.5) and collapses on ice (μ 0.1). Go faster than √(μ g r) and the tyres can't bend your path enough: you slide off the outside of the curve, along a wider circle.

That is why tight bends have speed limits. Indian road engineers use v² ÷ (127 R) = e + f, allowing only f = 0.15 of friction to leave a safety margin: a 100 m bend gets about 50 km/h.

Banked roads. Tilt the road inwards and the road's push leans inwards too, doing part of the turning. At one speed, √(g r tan θ), no friction is needed at all. Race tracks and cycling velodromes bank steeply, up to about 45°.

Leaning. A bicycle or motorbike (see CycleClear and MotorcycleClear) leans into the bend until the road's push points through the rider's centre of mass: tan θ = v² ÷ (g r). Faster or tighter means more lean, and more grip needed.

Fairground rides are circular motion you can feel. Your seat, or the wall, must push you towards the centre with m v² ÷ r, on top of holding you up against gravity. You feel that push as your weight changing.

Loop-the-loop. At the top of a loop the track and gravity both point to the centre. If the cart is fast enough, v ≥ √(g r) at the top, the track still pushes on it and it stays on, upside down. Too slow and gravity alone is more than the turn needs: the cart falls off the track. To get over, a cart must enter a round loop at √(5 g r), and then riders feel 6 g at the bottom. That is why real coasters use tear-drop loops, tight at the top and wide at the bottom.

Giant wheel. Going over the top, the seat pushes up with less than your weight, so you feel lighter; at the bottom you feel heavier. A fast mela wheel can make you 20% lighter at the top.

The wall of death. In India's maut ka kuan riders circle a wooden well about 9 m across. The wall pushes them inwards, and that big push lets friction hold them up. Below about 30 km/h friction can't carry their weight and they slide down.

In space. A spinning ring could make artificial gravity: the floor pushes you inwards with ω² R. For 1 g at a comfortable 2 rpm the ring needs to be 224 m in radius. The astronauts on the ISS float for the opposite reason: gravity is supplying exactly the pull their orbit needs, so they fall round the Earth with the station (see GravityClear).

Myth: "centrifugal force throws you outwards." In a car taking a sharp bend you feel shoved towards the door. But nothing is pushing you out. Your body simply tries to keep going straight (inertia, Newton's first law), and the car turns away underneath you until the door pushes you in. Watch the ball on the cart: from the road its path is a straight line. Only from inside the turning cart does it seem to slide outwards.

Physicists call centrifugal force a fictitious force: it appears when you describe motion from a rotating frame. That can be a handy trick, which is how Huygens first worked it out in the 1600s, but there is no outward push acting on the ball. Cut the string (chapter 1) and the ball flies off along the tangent, never straight out.

Coriolis. Slide a ball straight out across a smooth, turning roundabout. From the ground it goes straight; to someone riding the roundabout its path curves sideways. The Earth is a slow roundabout too, turning once a day, so winds and ocean currents curve: that is why cyclones spin anticlockwise north of the equator and clockwise south of it.

Myth: sinks drain the other way in Australia. The Earth's Coriolis effect on a sink is about a millionth of g. The shape of the basin and the swirl left from filling it matter thousands of times more. An MIT professor, Ascher Shapiro, needed a still 2 m tank, left for a whole day, to see it in 1962.

Very fast. Near the speed of light, F = m v² ÷ r needs Einstein's correction: the protons in the Large Hadron Collider need thousands of times more bending force than Newton's formula alone says.