# Centripetal Force Laplace’s Mini World · Plate 80 · Space > To keep something going round, you have to pull it toward the centre the whole time. Let go and it flies off along the tangent, straight on, not outward. - Interactive model: https://world.lapla.si/en/centripetal/ - Embed: https://world.lapla.si/embed/en/centripetal/ - 中文: https://world.lapla.si/zh/centripetal/ - Updated: 2026-11-02 ## 01 Pulled In, All the Time A brass post stands on the whirling table’s spindle, with a silk thread tied to its top and a 100 g brass ball at the other end. The spindle turns 2.5 times a second; the ball runs round a circle of radius 12 cm at about 1.9 m a second. The ball’s speed stays the same, but its direction changes all the time. Changing a velocity’s direction takes a force, and this one points at the centre of the circle all the time: the centripetal force. Here the thread supplies it. How big it must be is set by F = m v²/r, with m the mass, v the speed and r the radius. This ball needs a pull of about 3.0 N, three times its own weight. The thread also holds the ball up, so it slopes down a little, by about 18°. *Red arrow: the centripetal force · black arrow: the velocity · 100 g, radius 12 cm, 2.5 turns a second* ## 02 Why v²/r In a very short time Δt the ball runs along a short arc, and its velocity turns through a small angle vΔt/r. Its size has not changed; the small change Δv is about v times that angle, v²Δt/r, and it points at the centre. Divide by Δt and you have the centripetal acceleration, a = v²/r. The v appears twice: the faster the ball goes, the further its direction turns in the same time, and each turn changes the velocity by more. So twice the speed needs four times the pull; at the same speed, half the radius needs twice the pull. The Dutch physicist Christiaan Huygens wrote down these proportions in 1659 and published the results in 1673 without proofs; his full manuscript was printed only in 1703. In the Principia of 1687, Isaac Newton called a force that draws a body toward a centre a centripetal force, from the Latin for “seeking the centre”. *Grey arrows: the velocity a moment before and after · their difference points at the centre · 8 × slower* ## 03 Letting Go Once the thread breaks, nothing pulls the ball in any more. It keeps the velocity it had at that moment and flies off straight along the tangent at that point, falling under gravity as it goes. It does not fly outward along the radius. Seen from above, its path is a straight line that touches the circle only where the thread broke; the ball gets further from the centre only because a straight line leaves a circle. A hand holding the thread feels the ball pulling outward. That force is real, but it acts on the hand: the thread pulls the ball in, and the ball pulls the thread out just as hard. On the ball, the only pull is inward. *Red dashes: the path after the break (computed here from Newton’s laws) · 6 × slower* ## 04 A Car’s “Centrifugal Force” A car turns left, and its passengers feel pushed toward the right-⁠hand door. Seen from the roadside there is no such push: a passenger’s body is carrying on straight, the car is turning, and the door pushes the passenger round with it. The so-⁠called centrifugal force is needed only from inside, turning with the car; physicists call it an inertial force. Seen from the ground, it is not there. On a single-⁠lane town roundabout, a car circling on a radius of about 16 m at 25 km/h needs a centripetal acceleration of about 3.0 m/s², about 0.31 times the acceleration of gravity, all of it from the grip of the tyres on the road. *Roundabout path radius about 16 m · 25 km/h · about 0.31 g* ## 05 The Whirling Table Physics lectures of the 18th and 19th centuries showed these rules on a hand-⁠turned whirling table. It is already in Hauksbee and Whiston’s course of lectures printed in London in 1713; in 1762 the instrument maker George Adams built one for King George III. By the late 19th century, American colleges and schools ordered them complete from instrument makers in Philadelphia and Boston. The one on the board follows the American school pattern of the late 19th century: a mahogany board, at the right a flat cast-⁠iron wheel with a wooden handle standing on it, and a gut cord from the wheel’s groove to a small pulley in the iron bracket at the left. The wheel is six times the pulley, so one turn of the handle spins the spindle six times. The chuck on top of the spindle takes accessories. Some sets also had a pair of hinged brass balls that swing out, the same idea as the flyball governor on a steam engine (see the page Watt’s Steam Engine). The next three chapters show three others; the ball on a thread before them is the page’s own. *Wheel to pulley 6 : 1 · one turn of the handle, six of the spindle* ## 06 Two Balls on a Rod A big and a small brass ball ride on a smooth steel rod, joined by a brass tube so they can only slide together. The big ball has twice the small one’s mass. Once they turn, each ball needs an inward pull equal to its mass times its distance from the axis times the square of the turning rate. The tube can only pull both with one and the same force, so the pair stays put only where m₁r₁ = m₂r₂: the big ball 4 cm from the axis, the small one 8 cm. There the balls’ centre of mass is on the axis. That place does not depend on how fast they turn. Push the pair a little to one side and, once it turns, it slides all the way to one end of the rod. *2 parts of mass × 4 cm = 1 part × 8 cm · one division of the scale is 1 cm* ## 07 The Flattening Hoops Two springy brass bands are bent into two crossed hoops, fixed to the spindle at the bottom and free to slide up and down the rod at the top. Once they turn, their middles bulge out, the top slides down and the hoops flatten. Every bit of a hoop needs an inward pull, more the further it is from the axis; the bands only manage it by bending outward a little. Lecturers of the 18th century compared it to the Earth, which is a little flattened by its own spin. Today’s figures: an equatorial radius of 6,⁠378.137 km and a polar radius of 6,⁠356.752 km, about 21 km less, a flattening of about 1⁠/⁠298. *The hoops’ shape is a drawing · the Earth’s figures: WGS 84* ## 08 Mercury Out, Cork In Two sloping glass tubes hold water: one has a little mercury at its lower end, the other a cork floating at its upper end. Once they turn fast enough, the mercury rises through the water to the outer end of its tube, and the cork goes back down to the inner end of the other. Every bit of the liquid needs an inward force, which comes from the difference in pressure between the inside and the outside. At any one place that difference pushes every bit of volume equally hard: just enough for water, not enough for much heavier mercury, which moves out; more than enough for the lighter cork, which is pushed in. A washing machine’s spin works the same way. An 11 kg commercial washer with a drum 595 mm across spins at 1,⁠165 turns a minute; at the drum’s wall that is a centripetal acceleration of about 450 times the acceleration of gravity. The wall pushes the clothes round, but the water in them cannot hold on with anything like that force, so it leaves through the drum’s holes along the tangent. *Tubes at 40° · above about 1.7 turns a second the mercury moves out (computed here)* ## 09 Banked Curves On a flat road, the whole centripetal force comes from the grip of the tyres. Tilt the road toward the inside of the curve and the road’s push on the car leans inward a little; at the right angle a car can take the curve without any grip at all: tan θ = v²/(r g). The tubes on the table, sloped at 40°, follow the same rule: at about 1.7 turns a second the mercury just stays where it is, and any faster it moves out. Japan’s Road Structure Ordinance sets the smallest curve radius for roads designed for 60 km/h at 150 m, as a rule. Taking such a curve with no grip at all would need a bank of about 10.7°; the ordinance allows at most 10 %, about 5.7°, and only 6 to 8 % where snow and cold are common, and the tyres do the rest. Roads cannot be banked too steeply: on ice, slow or stopped cars would slide to the inside of the curve. Racing tracks bank far more. Daytona’s turns are banked at 31° on a radius of about 305 m, which suits about 153 km/h with no grip at all. *60 km/h on 150 m: 10.7° · the ordinance’s limit 10 % (5.7°) · Daytona 31°* ## 10 Gravity from Spinning On a station turning in space, a person standing on the outer floor is pushed round by the floor. That push points at the centre, and the feet feel it as weight. To match the ground takes ω²r = g, with ω the angle turned per second: at 2 turns a minute the radius must be about 224 m; at 1 turn a minute, about 894 m. In 1975 a summer study by NASA and Stanford University designed a ring-⁠shaped settlement 1,⁠790 m across, turning once a minute, its outer floor at about 0.93 to 1 times the ground’s gravity. The study kept the rate to one turn a minute at most, because faster spinning makes people dizzy when they turn their heads. A station orbiting the Earth has no such gravity: it falls round the Earth with everyone in it, and the Earth’s pull is exactly the centripetal force. That is on the page The Space Station Is Falling. *1 g: 894 m at 1 turn a minute · 224 m at 2 · the Stanford torus 1,⁠790 m across* ## 11 Taking the Table Apart The whirling table comes apart in seven pieces: the two-⁠ball cradle, the chuck and spindle, the small pulley, the bracket, the gut cord, the handle and the wheel. Rest the pointer on one and it is lifted out and named; left alone, all seven are named beside it. *Two-⁠ball cradle · chuck and spindle · small pulley · bracket · gut cord · handle · wheel* ## 12 The Ball on a Thread This is the page’s model: the ball’s mass, the circle’s radius and the turns a second can be set, and the pull it needs comes from F = m v²/r. The thread’s slope is computed too: the faster the spin, the flatter the thread. Press “Let go” and the thread breaks at that instant; the ball flies off along the tangent and falls by Newton’s laws, with no air. This chapter runs 6 times slower. *The page’s model · 6 × slower* **Try this**: In the lab, take the 100 g ball on a 12 cm radius at 2.5 turns a second: how big a pull does it need? Double the speed: how many times the pull? Halve the radius? ## What the model leaves out The whirling table on the board is a model after American school whirling tables of the late 19th century (like the one in James W. Queen of Philadelphia’s catalogue of 1888, kept at Kenyon College), its sizes and proportions estimated from photographs; it is not a copy of any one instrument, and its 6 : 1 between wheel and pulley is the page’s choice. The brass post, silk thread and ball on the spindle are the page’s own addition for the demonstration, not a historical accessory. The ball’s motion, the pull it needs, the thread’s slope and the path after the break are all computed from Newton’s laws, without air and with a massless thread. The flattening hoops are only a drawing: they are drawn as ellipses that flatten with the speed, not computed from the springiness of brass. The speed at which the mercury and the cork change places comes from the tubes’ slope; how fast they move is a drawing. “Why v²/r” runs 8 times slower, “Letting Go” and “The Ball on a Thread” 6 times slower; the rest turn at their real speed. ## Easy to get wrong No force pushes a circling body outward. The net force on it points inward; let go, and it flies off along the tangent, not outward along the radius. The “centrifugal force” appears only from a point of view that turns along, as an inertial force. The outward pull felt in the hand is real, but it acts on the hand, not on the ball. Nor is the centripetal force another kind of force: it is the job done here by a string’s pull, friction, a surface’s push or gravity, so when listing the forces on a body, do not add a “centripetal force” as an extra one. Motion in a circle at a steady speed is still accelerated, because the direction changes. The water in a washing machine is not pushed out; it is simply not held in. ## Common questions ### What is the centripetal force? Changing a velocity’s direction takes a force, and this one points at the centre of the circle all the time: the centripetal force. See: [01 Pulled In, All the Time](https://world.lapla.si/en/centripetal/#pull) ### Where does a whirled ball go when it is let go? Once the thread breaks, nothing pulls the ball in any more. It keeps the velocity it had at that moment and flies off straight along the tangent at that point, falling under gravity as it goes. See: [03 Letting Go](https://world.lapla.si/en/centripetal/#letgo) ### Is the centrifugal force real? The so-⁠called centrifugal force is needed only from inside, turning with the car; physicists call it an inertial force. Seen from the ground, it is not there. See: [04 A Car’s “Centrifugal Force”](https://world.lapla.si/en/centripetal/#car) ### Why are curves banked? Tilt the road toward the inside of the curve and the road’s push on the car leans inward a little; at the right angle a car can take the curve without any grip at all: tan θ = v²/(r g). See: [09 Banked Curves](https://world.lapla.si/en/centripetal/#bank) ### How can a spinning station make gravity? On a station turning in space, a person standing on the outer floor is pushed round by the floor. That push points at the centre, and the feet feel it as weight. See: [10 Gravity from Spinning](https://world.lapla.si/en/centripetal/#space) ## Sources 1. [OpenStax, University Physics Volume 1, §4.4: Uniform Circular Motion (deriving a = v²/r, toward the centre)](https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-circular-motion) 2. [OpenStax, University Physics Volume 1, §6.3: Centripetal Force (banked curves, tan θ = v²/(r g); inertial forces and the centrifugal force)](https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force) 3. [C. Huygens: De vi centrifuga (written 1659, printed 1703), Œuvres complètes vol. XVI (1929)](https://www.dbnl.org/tekst/huyg003oeuv16_01/huyg003oeuv16_01_0014.php) 4. [C. Huygens: the thirteen theorems printed in 1673 at the end of Horologium oscillatorium (same volume, appendix III)](https://www.dbnl.org/tekst/huyg003oeuv16_01/huyg003oeuv16_01_0017.php) 5. [I. Newton: Principia (1687), Definition V, the centripetal force (English translation of 1729)](https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Definitions) 6. [J. Ferguson: Lectures on Select Subjects (1760), “Of central forces” (the whirling table and its experiments: the two balls, the tubes with mercury and cork, the flattening hoops)](https://archive.org/details/b30501003) 7. [Science Museum Group: King George III’s central forces machine, George Adams, 1762](https://collection.sciencemuseumgroup.org.uk/objects/co1827/king-george-iiis-central-forces-machine) 8. [T. B. Greenslade Jr., Instruments for Natural Philosophy (Kenyon College): whirling tables and accessories](https://physics.kenyon.edu/EarlyApparatus/Mechanics/Whirling_Table_Apparatus/Whirling_Table.html) 9. [The same: the figure of the Earth (the flattening hoops)](https://physics.kenyon.edu/EarlyApparatus/Mechanics/Figure_of_the_Earth/Figure_of_the_Earth.html) 10. [US Federal Highway Administration: Roundabouts: An Informational Guide (2000), chapter 6 (roundabout radii and speeds)](https://www.fhwa.dot.gov/publications/research/safety/00067/000676.pdf) 11. [e-⁠Gov: Japan’s Road Structure Ordinance (article 15, curve radius; article 16, superelevation)](https://laws.e-gov.go.jp/law/345CO0000000320) 12. [Washington State DOT Design Manual, chapter 1250: cross slope and superelevation (side friction factors)](https://wsdot.wa.gov/publications/manuals/fulltext/M22-01/1250.pdf) 13. [Daytona International Speedway: track facts (turns banked 31°, radius 1,⁠000 ft)](https://web.archive.org/web/20121016222618/http://www.daytonainternationalspeedway.com/News/Track-Facts.aspx) 14. [Electrolux Professional: front load washer 11 kg, WH6-11 (drum diameter, extraction speed, G-⁠factor)](https://www.electroluxprofessional.com/en-my/pd/front-load-washers/soft-mount-high-spin-450g-530g/front-load-washer-11-kg-wh6-11/front-load-washer-11-kg-wh6-11/WH6-11/) 15. [NASA SP-⁠413: Space Settlements: A Design Study (1977), chapter 1 (the Stanford torus)](https://nss.org/settlement/nasa/75SummerStudy/Chapt.1.html) 16. [The same, chapter 3 (physiological criteria for gravity and rotation)](https://nss.org/settlement/nasa/75SummerStudy/Chapt3.html) 17. [The same, chapter 4 (1 g below 1 rpm needs a radius above 895 m)](https://nss.org/settlement/nasa/75SummerStudy/Chapt4.html) 18. [NGA: WGS 84 defining parameters (submitted to the UN ICG, 2012; the Earth’s equatorial radius and flattening)](https://www.unoosa.org/pdf/icg/2012/template/WGS_84.pdf) ## Data The law of circular motion after C. Huygens (De vi centrifuga, written 1659, printed 1703; Œuvres complètes XVI, 1929) and I. Newton (Principia, 1687); the whirling table and its accessories after J. Ferguson (Lectures on Select Subjects, 1760), the Science Museum Group (King George III’s central forces machine, 1762) and T. B. Greenslade Jr.’s Instruments for Natural Philosophy (Kenyon College). Road values from Japan’s Road Structure Ordinance (e-⁠Gov) and the FHWA roundabout guide (2000); the Daytona track facts from the speedway; the washer from Electrolux Professional’s product data; the spinning settlement from NASA SP-⁠413 (1977); the Earth’s radii from WGS 84. The whirling table on the board is this page’s model, after American school whirling tables of the late 19th century (James W. Queen, Philadelphia, catalogue of 1888, the example at Kenyon College), in proportions estimated from photographs; it is not a copy of one instrument. The ball on a thread is the page’s own accessory; its motion, the pull and the flight after the release are computed from Newton’s laws in the page’s script (no air). Material presets and the model kit as on the site’s other model plates (public/looks/CREDITS.md). ## Licence CC BY-NC 4.0 The text, pictures and models on this page are licensed CC BY-NC 4.0: free to use and adapt for teaching and other non-commercial purposes, with credit to world.lapla.si. Third-party material listed above keeps its own licence. https://creativecommons.org/licenses/by-nc/4.0/deed.en ## Cite this plate Laplace’s Mini World, Plate 80, “Centripetal Force”, https://world.lapla.si/en/centripetal/, CC BY-NC 4.0, updated 2026-11-02.