Circular Motion and Orbits

Going around in circles is harder than it looks

A car rounding a bend, a conker whirling on a string, the Moon sweeping around the Earth, an electron looping in a particle accelerator — all of these are examples of circular motion. It seems simple, but it hides a surprising and important idea: an object going round a circle at a steady speed is constantly accelerating. Understanding why unlocks how satellites, planets, and even theme-park rides work.

Why circular motion is always accelerating

In everyday speech, "accelerating" means speeding up. In physics it means something broader: acceleration is any change in velocity. And velocity is not just how fast you are going — it includes the direction you are going.

This is the crucial point. An object travelling in a circle may keep a perfectly constant speed, but its direction of travel changes at every instant. Because direction is part of velocity, the velocity is always changing — and a changing velocity means acceleration. So circular motion is accelerated motion, even when the speedometer reads a steady value.

Centripetal force: the inward pull

Newton's first law tells us that an object will travel in a straight line unless a force acts on it. So to bend a straight path into a circle, there must be a force constantly pulling the object away from the straight line and toward the centre. We call this the centripetal force — Latin for "centre-seeking".

The centripetal force always points toward the centre of the circle.

Take away that inward force and the object instantly flies off in a straight line, tangent to the circle (not outward, but in the direction it was heading at that moment). Let go of a whirling conker and it shoots off sideways — proof that the string was pulling it inward, not outward.

Where does the centripetal force come from in real situations? It is always some ordinary force, redirected to do the job:

• A ball on a string → the tension in the string.
• A car going round a bend → friction between the tyres and the road.
• The Moon orbiting Earth → gravity.

The equations

The acceleration of an object in circular motion is called centripetal acceleration, and it points toward the centre. Its size is:

a = v² / r

where v is the speed (m/s) and r is the radius of the circle (m). Notice the v²: doubling the speed quadruples the acceleration. And the smaller the radius, the larger the acceleration — tight turns are far more demanding than gentle ones.

Combining this with Newton's second law (F = ma) gives the centripetal force:

F = m v² / r

Worked example. A car of mass 1000 kg goes round a bend of radius 50 m at 15 m/s. What centripetal force must friction supply?

F = mv²/r = 1000 × 15² ÷ 50 = 1000 × 225 ÷ 50 = 4500 N

If the car went twice as fast (30 m/s), the v² term would make the required force four times larger — 18,000 N. If friction cannot supply that much force, the tyres slip and the car skids straight on. This is exactly why speeding into a sharp corner is so dangerous.

What about "centrifugal force"?

When a car turns sharply, you feel flung outward against the door. So isn't there an outward force? Not really. The only real force on you is the inward push from the door and seat. Your body, obeying Newton's first law, "wants" to keep moving in a straight line, so it presses against the door as the car curves inward beneath you. The outward "centrifugal force" is an apparent effect of sitting inside a turning object — useful for everyday talk, but not a true force.

Orbits: falling forever

Now for the beautiful part. An orbit is simply circular motion where the centripetal force is gravity.

Newton imagined firing a cannonball horizontally from a very tall mountain. Fire it slowly and it falls to the ground nearby. Fire it faster and it lands further away, its path curving as it falls. Fire it fast enough, and something remarkable happens: the ground curves away beneath the falling ball just as fast as the ball falls. The cannonball keeps falling toward Earth but never reaches it — it has gone into orbit.

This is the key insight: a satellite in orbit is in continuous free-fall. Astronauts float not because there is no gravity in space (there is plenty), but because they and their spacecraft are falling together around the Earth. Gravity provides exactly the centripetal force needed to keep the satellite curving around the planet. To explore how gravity holds the whole Solar System together, see the Solar System and gravity, and for the rule that makes orbits possible at all, revisit Newton's laws of motion.

For a circular orbit, setting gravity equal to the centripetal force shows that the required orbital speed depends only on the height: lower satellites must move faster. A satellite skimming low above Earth races round at about 7.8 km/s, completing an orbit in roughly 90 minutes. A satellite far out at 36,000 km moves slowly enough to take exactly 24 hours, so it appears to hover over one spot on the equator — a geostationary orbit, ideal for TV and weather satellites.

Try it yourself! 🧪

Experiment 1 — The whirling water bucket. Half-fill a sturdy bucket with water, hold the handle, and swing it in a fast vertical circle. At the top, the bucket is upside-down, yet the water stays in! The water is being pushed toward the centre (downward, at the top) fast enough that it is effectively "falling" along the circular path — the bucket bottom provides the centripetal force, and the water never gets a chance to fall out. Swing too slowly and gravity wins: you get wet. (Do this outside.)

Experiment 2 — Feel the force. Tie a soft object to about a metre of string and whirl it gently in a horizontal circle above your head. Notice how hard you must pull inward on the string — that is the centripetal force. Now whirl it faster: the pull becomes much stronger, just as F = mv²/r predicts. Finally, let go at a chosen moment and watch the object fly off in a straight line, tangent to the circle — the instant the inward force vanished, straight-line motion took over.