a = v²/r · F = mv²/r · the tangent · why there is no outward force
Whirl a stone on a string and cut it. It does not fly outwards — it goes straight, along the tangent, exactly where it was already heading. Nothing was ever pushing it out. Something was always pulling it in, and the moment that stopped, so did the circle.
This page covers centripetal acceleration v²/r, the force that supplies it, where that force actually comes from in each situation, and what happens when it stops. By the end you'll be able to:
This is the chapter where students first meet an acceleration that changes direction rather than size, and it is the foundation of orbits, banking and rotation:
Searched as: centripetal force, v squared over r, why no centrifugal force, banking of roads, tangential velocity, circular motion formula.
Predict the force in the string, lock it, then spin it and cut the string. Watch which way it actually goes. 🟢 real circular-motion engine
Guess, then lock — the bench waits until you commit.
Set the speed and the radius — then cut the string and watch which way it leaves.
Centripetal force is a job, not a substance. In every circle, some real force has taken the job. Six situations — name it, and your answers build the table underneath.
A stone whirled on a string. What supplies the inward force?
Read the picture, then name the real force doing the pulling.
There is never a force labelled "centripetal" on an object. There is always a real one taking the role.
The mass and radius are fixed. Dial the speed until the string tension reaches the target — 3 attempts.
Set the speed, then spin. F = mv²/r decides it, and v is squared.
Each one breaks the physics somewhere. Tap the broken idea.
A net outward force would make the object spiral away, not circle. The net force in uniform circular motion always points at the centre.
At the moment of release the stone is moving along the tangent, and with no force to bend it, that is the line it continues on.
Velocity includes direction. Going round a bend changes direction continuously, so there is an acceleration of v²/r even at a perfectly steady speed.
Thinking beats memorising. Sort every circular-motion claim into three boxes.
An object is released at the marked point. Choose the path it takes. This one question catches almost everybody once.
Ask which way it was moving at that instant, and whether anything is still pushing it.
Pick a design speed, then choose a radius and a banking angle that let a car take the bend with no help from friction at all.
A bend is right for one speed only: tanθ = v²/rg.
Read the scene. Every one of these is v²/r wearing different clothes.
The drum supplies an inward force to the clothes. Water reaching a hole has nothing pulling it inwards, so it continues in a straight line and leaves. Nothing threw it.
At the top the centre is directly below, so gravity points the right way and does the job. Below a minimum speed gravity is MORE than the circle needs and the car falls away from the track.
Tilting the surface turns part of the push from the road into an inward force. At the design speed tanθ = v²/rg and the bend works even on ice.
In plain terms: anything moving in a circle is being pulled towards the centre, continuously, by something real. Stop the pull and the circle stops — but the motion does not.
Velocity has a size and a direction. Going round a bend at a steady 40 changes the direction every instant, so the velocity is changing and there is an acceleration — of size v²/r, pointing at the centre. This is the idea the whole chapter rests on, and it is the one that feels wrong at first.
Nothing is ever labelled "centripetal force" on a free-body diagram. Ask instead: what real force is pointing at the centre? A string's tension. The Earth's gravity. Friction from the road. A wall pushing. Whichever it is, that force IS the centripetal force for that situation, and writing F = mv²/r tells you how big it has to be.
On a roundabout you feel flung outwards, and nothing is flinging you. You are travelling in a straight line and the floor is curving away beneath you; the wall then pushes you inwards to make you turn with it. The outward feeling is real; the outward force is not, unless you choose to do the physics in the spinning frame, where it is a bookkeeping term rather than a push from anything.
At the top of a vertical loop there is a minimum speed, √(gr), below which gravity exceeds what the circle needs and contact is lost. In non-uniform circular motion the speed changes too, so there is a tangential acceleration alongside the radial one. And a satellite is not weightless: gravity there is most of its surface value, and it is doing all of the pulling.
Exams reward the method. Work it one step at a time — read the thought, predict the line, then reveal it.
Three small labs, each one native to this motion.
Six wrong ideas that survive because they match how it feels. Tap a card to bust the myth.
Christiaan Huygens worked out the force needed to keep something moving in a circle, years before Newton's Principia, while designing pendulum clocks.
Newton imagined a cannon fired ever faster from a mountain top. Fast enough, and the ground curves away as quickly as the ball falls — which is an orbit, and it made gravity and circular motion one subject.
His pendulum in Paris swung in a fixed plane while the building turned beneath it, showing that rotating frames are not the same as still ones — which is where centrifugal terms come from.
The first satellite turned Newton's thought experiment into engineering, and v²/r became something you have to get right or lose the spacecraft.
Banked roads, roundabouts, centrifuges, hard drives and washing machines all run on the same line of algebra a student meets in Class 11.
Friction at four contact patches supplies mv²/r. Ask for more than it has and the car understeers straight on.
Gravity is the whole force. The satellite falls continuously and moves sideways fast enough never to arrive.
The drum holds the clothes in. The water, with nothing to hold it, keeps going straight and leaves through the holes.
One sentence each: what is pulling, and what happens when it cannot.
Drag the slider. Six circles, from metres to hundreds of thousands of kilometres.
Twelve orders of magnitude in radius, and the same three letters underneath.
Each one is a build and a measurement — the measuring is what turns a demo into a project.
Build: a ball swung on a string over a large sheet of paper, released at a marked point.
Measure: where it lands, several times. Draw the line back — it meets the circle as a tangent, not a radius.
Build: a bung on a string whirled at a comfortable steady rate, at several string lengths.
Measure: time twenty turns each time and work out the speed. Longer strings turn more slowly.
Build: a bung on a string through a smooth tube, with a known weight hanging below to set the tension.
Measure: the speed needed to keep the radius steady, for several hanging weights. Plot F against v².
Build: a bucket of water swung in a vertical circle, outdoors, starting fast and slowing gradually.
Measure: the slowest turn that stays dry, and compare with √(gr) for your arm length.
Build: a curved ramp whose banking angle you can change, with a marble launched at a measured speed.
Measure: the angle that keeps the marble on its line without drifting up or down the bank.
Build: a phone's accelerometer app, used as a passenger on a known roundabout.
Measure: the sideways acceleration, and compare with v²/r from your speed and the map radius.
Straight answers, each one complete on its own:
Acceleration is the rate of change of velocity, and velocity has a direction as well as a size. Going round a bend changes the direction continuously, so the velocity is changing even though the speedometer does not move. The size of that acceleration is v²/r and it points towards the centre.
Not as a push from anything, in the ordinary ground frame. What you feel on a ride is your own body continuing in a straight line while the wall curves into you and pushes you inwards. If you choose to do the physics in the rotating frame, a centrifugal term appears as bookkeeping — useful, but not a force exerted by any object.
Straight along the tangent, in the direction it was travelling at that instant, and then it falls under gravity as it goes. It does not move outwards along the radius, because nothing was ever pushing it outwards — the string was pulling it in.
A real force, different in each case: the tension in a string, gravity for a satellite or the Moon, friction for a car on a flat bend, the normal force from a wall or a banked road, or a combination. Centripetal is the job description, not the name of a new force.
Because the force needed is mv²/r. Doubling the speed asks for four times the force, and the only way to bring that back down without more grip is to increase r. That is why motorway curves are gentle and why a tight bend carries a low speed limit.
At the top, the centre of the circle is directly below, so gravity points the right way and can supply the force by itself. If the car is slower than √(gr), gravity is MORE than the circle needs, so the car cannot follow the track and falls away from it. The track can push but it cannot pull.
No — gravity at the space station is about ninety per cent of its value at the ground. They float because they and the station are both falling at the same rate, continuously, around the Earth. Weightlessness here means free fall, not the absence of gravity.
On a flat bend, no. Friction available is μmg and the force needed is mv²/r, so the mass cancels and the limit v = √(μgr) is the same for a loaded lorry and an empty car. Mass matters for stopping, for tyre wear and for how the load shifts, but not for this limit.
Spot these before the exam does.
Seven question formats, the way LLOS.ai serves them — multiple choice, multiple-correct, fill-in-the-blank, match, sequence, read-think-connect, and write-your-own. Every question has layered hints: a quick nudge, the reasoning, then a deeper connection — so a wrong answer opens a door, never a dead end. 🟢 received from a board-tagged question bank · seed toward 2,000
Pick your board — the set re-tunes to its wording and emphasis. Competitive draws the JEE / NEET / Olympiad lane.
Four circles, no numbers yet. Tap them in order, biggest inward force first. Then read what mv²/r gives — and see what fooled you.
A 4 kg hammer swung slowly at 3 m/s on a 1 m rope — by far the heaviest.
A 0.2 kg ball whipped round at 20 m/s on the same 1 m rope.
A 1 kg mass at 6 m/s, but on a very short 0.2 m rope.
A 2 kg mass at 8 m/s on a long 8 m rope.
Tap the four in the order you believe, biggest force first.
Nothing goes round on its own. Every circle in the universe is something being pulled inwards, continuously, by something real — and every straight line is what happens the moment that stops.