Top view: a ball on a string whirling on a frictionless table
Controls
Scene
Readings
- Speed, v
- 3.00m/s
- Angular speed, ω
- 3.00rad/s
- Time period, T
- 2.09s
- Frequency, f
- 0.477Hz
- Centripetal acceleration, a = v²/r
- 9.00m/s²
- Centripetal force, F = mv²/r
- 4.50N
How to use this simulation
- In "Ball on a string", change the radius and speed and watch the red arrow (centripetal force) and the amber arrow (velocity).
- Double the speed and read the force in the readings panel: how many times bigger is it?
- Switch on "Cut the string": the ball goes straight along the tangent, not outward.
- In "Banked curve", slide the banking angle to the ideal value and watch the friction needed fall to zero.
The fair ride and the whirling stone
Have you been on a spinning fair ride? As it turns, you feel pressed against the wall, as if something is pulling you outward. Or take a sharp bend in a bus and slide into the person beside you. Both feelings come from the same short story.
Many of us tied a stone to a string as kids and whirled it overhead. Now tell me: if the string snaps, where does the stone go? A lot of people say it flies outward. In fact it shoots straight off in the direction it was moving at that instant, along the tangent to the circle. This is the biggest misconception in the whole chapter, and in the simulation you will break it with your own eyes.
Ceiling fans, wheels, planets, race-track bends, the drum of a washing machine: every one rests on a single idea. Something is pulling the object away from its straight path toward the inside of the circle at every moment. We call that pull the centripetal force.
Starting from zero: why does it turn instead of going straight?
Newton's first law says that with no net force an object moves in a straight line at constant velocity. So for anything to turn, something must be pushing or pulling it sideways. For a ball on a string it is the tension; on a road bend it is tyre friction; for a planet it is gravity.
Now look at the words "uniform speed". It means the speed, the number on the speedometer, stays the same. But velocity is a vector, with a size and a direction. When the direction changes, the velocity changes. When the velocity changes, there is acceleration. So even at constant speed the object accelerates, and that acceleration is perpendicular to the motion.
An acceleration at right angles to the motion cannot speed the object up or slow it down; it only turns it. That is why the centripetal force does no work (the force and the displacement are always perpendicular) and why whirling a ball on a string does not add to its energy.
A tiny experiment: tie a key to a string and whirl it in a horizontal circle. Your hand holds the string inward, and the string holds the key. You feel the pull on your finger. Whirl faster and the pull grows; shorten the string at the same speed and it grows again. The formulas below say exactly how.
Words you need to know
These terms turn up in every circular-motion question. Each has a plain meaning and an everyday example.
| Term | Plain meaning | Everyday example |
|---|---|---|
| Circular motion | Moving along a circular path | The tip of a clock hand, a Ferris-wheel seat |
| Radius vector | A vector from the centre to the object; it turns with the object | An arrow along the string, hand to ball |
| Angular velocity (ω) | Angle the radius vector sweeps each second; unit rad/s | How fast a fan blade turns |
| Time period (T) | Time for one full revolution, in seconds | The minute hand: T = 3600 s |
| Frequency (f) | Revolutions per second; f = 1/T, in hertz | How many turns a fan makes each second |
| Linear speed (v) | Speed along the tangent; v = rω | The speed the ball has when the string snaps |
| Centripetal acceleration | Acceleration toward the centre, of size v²/r | A car taking a bend |
| Centripetal force | The net force toward the centre, of size mv²/r | Tension, friction or gravity |
Angular velocity, period and linear speed
Say the ball takes T seconds for one full turn. One turn sweeps an angle of 2π radians and covers a path of 2πr. Counting angle gives the angular velocity; counting path gives the linear speed. The two are tied by v = rω.
ω = 2π/T = 2πfangular velocity
T = 2πr/vtime period
f = 1/Tfrequency
v = rωlink between linear and angular speed
Two points on the same record
Picture a spinning record. Take one point near the rim and one near the middle. They go round once in the same time, so T, f and ω are identical. But the outer point covers a bigger circle in that same time, so its linear speed v = rω is larger.
That is why angular velocity is the natural way to describe how fast a whole body rotates: it is the same for every point, while linear speed depends on distance from the axis.
Radians, not degrees
The relation v = rω works only when ω is in radians per second. A full circle is 360° = 2π radians. If you are given revolutions per minute (rpm), multiply by 2π and divide by 60. For example 120 rpm is 2 revolutions each second, so f = 2 Hz and ω = 4π rad/s.
Centripetal acceleration and force
Now the key results. Centripetal acceleration grows with the square of the speed and shrinks with the radius. The force comes from Newton's second law, F = ma. Notice that doubling the speed quadruples the force, which you can test in the simulation.
a = v²/r = ω²rcentripetal acceleration
F = ma = mv²/r = mω²rcentripetal force
Where the formula comes from (a picture argument)
In a short time Δt the ball turns through Δφ = ωΔt. The velocity arrow turns through the same angle but keeps its length v. The starting and ending velocities form an isosceles triangle whose base is the change in velocity Δv. For a small angle the base is about v·Δφ, and it points inward, toward the centre.
Acceleration is change in velocity divided by time, so a = vΔφ/Δt = vω. Put ω = v/r and you get a = v²/r; put v = rω instead and you get a = ω²r. Same thing, two clothes.
|Δv| ≈ vΔφ ⇒ a = vω = v²/r
Three lessons from the formula
Double the speed, get four times the force: with m = 0.5 kg, r = 1 m and v = 3 m/s the force is 4.5 N; at twice the speed it is 18.0 N.
Double the radius at the same speed and the force halves: 4.5 N becomes 2.25 N.
Double the mass and the force doubles, but the acceleration stays the same, because mass is not in the acceleration.
What happens when the string is cut: the big misconception
Here is the famous question. If you cut the string of a whirling ball, which way does it go? Switch on "Cut the string" and see for yourself. The ball moves off along the tangent, in a straight line, at the speed it had.
The reason is simple. The centripetal force was the string tension. The moment the string is cut that force is zero. With zero force, the first law says the object keeps its velocity and travels in a straight line. Its velocity was along the tangent, so it travels along the tangent. There was never any outward force to push it away.
Check it with numbers: after 2 s the ball has travelled 6 m along the tangent, and its distance from the centre is √(1² + 6²) = 6.08 m. It has moved away from the circle, certainly, but in a straight line, not along a curve.
So why do you feel "thrown outward" on a ride? That is your inertia. When a car turns, your body wants to go straight on, and the door pushes you inward. You notice the push from the door and it feels as if something pulls you outward. Some books call this a centrifugal or fictitious force. It is useful for calculations from inside a rotating frame, but to an observer standing on the ground only the centripetal force exists.
Centripetal force is not a separate force
Students often draw an extra arrow called "centripetal force" on a free-body diagram. That is a mistake. Centripetal force is not a new kind of force; it is a job description. Whatever forces add up to a net force toward the centre are doing that job, and their sum is what we name the centripetal force.
For a ball on a string the string tension does the job. On a flat road bend, tyre friction does it. On a banked bend it is the horizontal part of the normal reaction. For the Moon it is the Earth's gravity. So draw the real forces, then say that their sum points to the centre.
Banking of roads
On a flat bend only friction keeps a car on the curve. When it rains the friction drops and the car slides outward. So engineers raise the outer edge of the road and call it banking. Because the road is tilted, the normal reaction N leans inward, and its horizontal part pushes the car toward the centre.
With no friction, write the vertical and horizontal equations: N·cosθ = mg and N·sinθ = mv²/r. Dividing gives tanθ = v²/(rg). For a bend of radius 50 m and a speed of 15 m/s the ideal angle is 24.7°. At that angle the car needs no friction at all, even on ice.
N cosθ = mgvertical balance
N sinθ = mv²/rhorizontal, toward the centre
tanθ = v²/(rg)ideal banking angle
v = √(rg·tanθ)safe speed for a given angle
When the angle is not ideal
If the car goes faster than the ideal speed it needs a bit more force toward the centre, and friction helps by acting down the slope. If it goes slower, friction acts up the slope so the car does not slide down. The green arrow in the simulation is this friction. A positive "friction" reading means it points toward the centre and a negative one means it points the other way.
Change the car's mass and the ideal angle does not move: m cancels from both sides of the equations. Try it with the slider.
Try it yourself in the simulation
Reading a formula is not the same as seeing it work. Before each experiment, say out loud what you expect, and only then move the sliders.
- Keep r = 1 m and raise v from 1 to 6 m/s: how many times does the centripetal force grow? (Six times the speed should give thirty-six times the force.)
- Keep v = 3 m/s and change r from 0.5 m to 2 m: what happens to ω and to the force?
- Change the mass from 0.1 to 2 kg: does the acceleration change? Does the force?
- Cut the string at any instant and compare the ball's path with the amber arrow.
- Press Reset, shrink the radius and keep the speed high: how large does the force get? Could a real string cope?
- In "Banked curve", set v = 15 m/s and r = 50 m and slide the angle to the ideal one: what does the friction reading show?
- Set the angle to 0°: now friction has to supply all the centripetal force.
Step-by-step solved problems
For each problem, list what you know, convert to SI units, then substitute. Use g = 9.8 m/s² unless told otherwise.
Problem 1: the default ball in the simulation
A 0.5 kg ball moves on a circle of radius 1 m at 3 m/s. The angular velocity is ω = v/r = 3/1 = 3.00 rad/s. The period is T = 2π/ω = 2.094 s and the frequency f = 1/T = 0.477 Hz.
Centripetal acceleration a = v²/r = 3²/1 = 9.0 m/s². Centripetal force F = ma = 0.5 × 9.0 = 4.5 N. The readings panel shows the same numbers.
Problem 2: a turntable
A disc turns at 120 rpm. A point on its edge is 15 cm from the centre. ω = 120 × 2π/60 = 12.57 rad/s. Linear speed v = rω = 1.88 m/s. Centripetal acceleration a = ω²r = 23.7 m/s².
Problem 3: the fastest a string can whirl a stone
A string can bear at most 20 N. It is used to whirl a 0.4 kg stone in a horizontal circle of radius 0.8 m. The tension is the centripetal force, so T_max = mv²/r gives v_max = √(T_max·r/m) = √(20 × 0.8/0.4) = 6.32 m/s, with ω = 7.91 rad/s. Go faster and the string snaps; the stone then flies off along the tangent.
Problem 4: safe speed on a flat bend
A flat road bend has radius 50 m and the coefficient of friction between tyres and road is μ = 0.4. Friction f = μmg supplies the centripetal force: μmg = mv²/r. The mass cancels, so v_max = √(μrg) = √(0.4 × 50 × 9.8) = 14.0 m/s, about 50 km/h. A heavy truck and a light car share the same limit.
Problem 5: banking angle and normal reaction
For a bend of radius 50 m taken at 15 m/s, the frictionless ideal angle is θ = tan⁻¹(v²/rg) = tan⁻¹(4.5/9.8) = 24.7°. For a 1,000 kg car the centripetal force is 4,500 N and N = m√(g² + a²) = 10,784 N, larger than the weight mg.
Problem 6: the same bend with a 15° slope
Suppose the banking is only 15° while the speed is still 15 m/s. That is less than the ideal angle. The friction still needed is f = m(a·cosθ − g·sinθ) = 1,810 N, toward the centre. With no friction the car would skid outward. At this angle the frictionless safe speed is v = √(rg·tanθ) = 11.5 m/s, about 41 km/h.
Problem 7: the Moon's centripetal acceleration
Take the textbook values: the Moon is 384,000 km from Earth and goes round once in 27.3 days. v = 2πr/T = 1,023 m/s. Centripetal acceleration a = v²/r = 2.72 mm/s², about 1/3,597 of g at the Earth's surface. This is how Newton saw that the pull that drops an apple also steers the Moon, only weakened by distance.
Speed, radius and force side by side
Every number in this table is computed (mass 0.5 kg). Notice that at the same radius, going from 1 to 2 m/s multiplies the force by 4, and going to 6 m/s multiplies it by 36.
| r (m) | v (m/s) | ω (rad/s) | T (s) | a (m/s²) | F (N) |
|---|---|---|---|---|---|
| 1.0 | 1 | 1.00 | 6.28 | 1.0 | 0.50 |
| 1.0 | 2 | 2.00 | 3.14 | 4.0 | 2.00 |
| 1.0 | 3 | 3.00 | 2.09 | 9.0 | 4.50 |
| 1.0 | 6 | 6.00 | 1.05 | 36.0 | 18.00 |
| 0.5 | 3 | 6.00 | 1.05 | 18.0 | 9.00 |
| 2.0 | 3 | 1.50 | 4.19 | 4.5 | 2.25 |
How much banking for how much speed
The frictionless ideal banking angle for bends of radius 50 m and 100 m at several speeds. Faster traffic needs a steeper road, and a wider bend needs less.
| Speed (km/h) | v (m/s) | a (m/s²), r = 50 m | Ideal θ, r = 50 m | Ideal θ, r = 100 m |
|---|---|---|---|---|
| 36 | 10.0 | 2.00 | 11.5° | 5.8° |
| 54 | 15.0 | 4.50 | 24.7° | 12.9° |
| 72 | 20.0 | 8.00 | 39.2° | 22.2° |
| 90 | 25.0 | 12.50 | 51.9° | 32.5° |
Mistakes everyone makes
These slips turn up again and again in answer sheets and classrooms.
- "The ball flies outward when the string is cut": no, it goes straight along the tangent.
- "At constant speed the acceleration is zero": no, the direction changes, so there is an acceleration v²/r toward the centre.
- "Centripetal force is an extra force": no, it is the sum of real forces such as tension, friction or gravity.
- Putting ω in degrees per second: v = rω needs radians.
- Doubling the speed and doubling the force: the force quadruples because of v².
- Thinking the centripetal force does work: it is perpendicular to the motion, so it does none.
- Thinking the banking angle depends on the mass of the car: it does not.
Circular motion in real life
Washing-machine spin dry: the drum spins fast. The cloth is held by the drum wall, but water droplets pass through the holes and fly off along tangents. It is the cut-string experiment at scale.
Bicycles and motorbikes: a rider leans into a bend so that the ground reaction and weight together provide the centripetal force. A faster rider leans more, as tanθ = v²/rg predicts.
Satellites and the Moon: gravity is the centripetal force. A satellite keeps falling toward Earth but moves sideways so fast that the Earth's curve drops away beneath it.
Centrifuges: sample tubes are spun quickly. The walls give the centripetal force; heavy particles settle at the bottom of the tubes.
Roller coasters and loops: the seat pushes you toward the centre of the loop, which is why you stay in place. What feels like extra weight is the seat pressing on you.
Railways and race tracks: the outer rail or edge is raised on bends, which cuts the sideways load on wheels and tyres.
Exam corner: how questions are asked
Circular motion questions come as short definitions, derivations (centripetal acceleration, banking angle), numerical problems and "explain why" items. Here are typical examples of each, with short answers.
Definitions
Q: Define centripetal force. A: The net force acting toward the centre of the circle on an object in circular motion, which keeps it on that path, is the centripetal force.
Q: Define angular velocity. A: The angle swept by the radius vector per unit time; unit rad/s.
Explain why
Q: Why is uniform circular motion accelerated motion? A: The speed is constant but the velocity changes direction at every instant, so there is an acceleration toward the centre.
Q: Why does centripetal force do no work? A: It is always perpendicular to the displacement, so W = Fs·cos90° = 0.
Numerical and reasoning questions
Numerical: find the centripetal force on a 0.5 kg body going round a circle of radius 1 m at 3 m/s. F = mv²/r = 4.5 N.
Reasoning: "A car takes a bend at 15 m/s with radius 50 m. What banking removes the need for friction, and what happens at higher speed?" The ideal angle is 24.7°; at a higher speed friction acts down the slope, toward the centre.
Quick revision
The night before the exam, this list is enough to bring everything back.
- Uniform circular motion: speed constant, velocity not; acceleration points to the centre.
- ω = 2π/T = 2πf and v = rω.
- a = v²/r = ω²r, toward the centre.
- F = mv²/r = mω²r; F ∝ v², F ∝ 1/r, F ∝ m.
- Centripetal force is the sum of real forces, not a separate force; it does no work.
- If the string is cut, the object moves along the tangent, never outward.
- Frictionless banking: tanθ = v²/(rg), independent of mass.
- Flat bend safe speed: v = √(μrg).
Frequently asked questions
What is centripetal force?
Centripetal force is the net force that acts toward the centre of a circle on an object moving round it, keeping the object on its circular path. Its size is F = mv²/r = mω²r. It is not a separate kind of force: tension, friction or gravity usually plays this role.
Why is there acceleration in uniform circular motion?
Acceleration is the rate of change of velocity, and velocity is a vector. The speed is constant, but the direction keeps changing. That change points toward the centre, so the acceleration has size v²/r and is directed to the centre.
Which way does the ball go when the string is cut?
The ball moves in a straight line along the tangent to the circle at the point of release, not outward. The centripetal force was the string tension; with it gone, the net force is zero and Newton's first law keeps the ball moving at its last velocity.
How are angular velocity and linear speed related?
They are related by v = rω, with ω in radians per second. Every point of a rotating body has the same ω, but points farther from the axis have a larger linear speed.
Does centripetal force do any work?
No. The force is always at right angles to the velocity, so it has no component along the displacement, and W = F·s·cos90° = 0. That is why the speed and kinetic energy stay constant.
What is the formula for the banking angle?
For a frictionless bend tanθ = v²/(rg). With r = 50 m and v = 15 m/s, θ = 24.7°. The angle does not depend on the mass of the vehicle.
What is the maximum safe speed on a flat curve?
With friction coefficient μ the limit is v_max = √(μrg). For μ = 0.4, r = 50 m and g = 9.8 m/s², v_max = 14.0 m/s, about 50 km/h. Beyond that the car slides outward.
Is centrifugal force real?
Centrifugal force is a fictitious force that an observer in the rotating frame feels because of inertia. An observer standing on the ground finds no outward force; only the inward centripetal force acts on the object.
Keep studying this topic
The animation made the idea click; now turn it into marks. Syllabus, suggestions, textbooks and admission-test guides are below.
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Grade 12
The semiconductor diode: p–n junction, I–V curve and rectifier
Bias a p–n junction and watch the depletion layer shrink or grow, carriers cross, and the Shockley I–V curve trace live. Compare silicon and germanium, see breakdown, then build a half-wave rectifier.
Grade 10–12
Radioactive decay: half-life, the decay law and penetration
Watch 240 nuclei decay at random, see the count follow N = N₀e^(−λt), measure the half-life from the graph, then test how alpha, beta and gamma radiation pass through paper, aluminium and lead.
Grade 11–12
The Bohr atom: orbits, energy levels and the hydrogen spectrum
Make a hydrogen electron jump between orbits n = 1 to 6, watch the photon colour come from 1/λ = R(1/n₁² − 1/n₂²), and see the Lyman, Balmer and Paschen series on an energy diagram and spectrum strip.
Animations in other subjects
Chemistry
Structure of the atom
Chemistry
Acids, Bases and the pH Scale
Chemistry
Electron configuration
Chemistry
Gas laws: Boyle, Charles, Gay-Lussac, Avogadro and PV = nRT
Chemistry
How an Ionic Bond Forms
Chemistry
Electrolysis and Faraday’s Laws
Chemistry
Covalent Bond
Chemistry
Periodic Trends
Chemistry
Reaction rate and collision theory
Chemistry
States of matter: solid, liquid, gas and the heating curve
Chemistry
Atomic orbitals: the shapes of s, p and d, quantum numbers and nodes
Chemistry
Molecular shapes and VSEPR theory: bond pairs, lone pairs and angles
Chemistry
The mole concept: molar mass, Avogadro’s number and molar volume
Chemistry
Diffusion and effusion: Graham’s law of gases
Chemistry
Balancing chemical equations: method, examples and practice
Chemistry
Chemical equilibrium: the equilibrium constant Kc and Le Chatelier’s principle
Chemistry
Titration: finding an unknown concentration with a burette and an indicator
Chemistry
Galvanic cell: the Daniell cell, EMF and the Nernst equation
Chemistry
Redox reactions: oxidation numbers, electron transfer and half-equations
Chemistry
Hydrocarbons: alkanes, alkenes, alkynes and structural isomers in 3D
Chemistry
Functional groups: alcohols, aldehydes, ketones, acids, esters and amines
Chemistry
Polymerisation: addition and condensation polymers, polythene and nylon
Chemistry
Separation techniques: filtration, distillation, chromatography, crystallisation
Chemistry
Crystal lattice: SC, BCC and FCC unit cells, packing efficiency and density
Chemistry
Radioactivity: Alpha, Beta, Gamma Decay and Half-Life
Biology
Mitosis cell division
Biology
Diffusion and Osmosis
Biology
Photosynthesis
Biology
The human heart: structure and function
Biology
Mendelian Genetics
Biology
DNA Replication
Biology
Cellular Respiration
Biology
Enzyme Action
Biology
The Nerve Impulse and the Action Potential
Biology
The Nephron and Urine Formation
Biology
Meiosis Cell Division
Biology
Digestive System
Biology
Breathing Mechanism: Inhalation and Exhalation
Biology
Protein synthesis
Biology
Blood circulation: double circulation and cardiac output
Biology
Cell structure and organelles: plant cell vs animal cell
Biology
Reflex action and the reflex arc
Biology
The human eye: parts, image formation and accommodation
Biology
Muscle contraction and the sliding-filament model
Biology
Transport in plants: xylem, phloem and transpiration
Biology
Food chain and food web: energy flow and the 10 % rule
Biology
Population growth: exponential and logistic curves
Biology
Natural selection: peppered moths and the sooty tree
Biology
Immune response: primary and secondary antibody response
Biology
Pollination and fertilisation
Mathematics
Pythagorean Theorem
Mathematics
Trigonometric Ratios and the Unit Circle
Mathematics
Quadratic functions and their graphs
Mathematics
Circle theorems
Mathematics
Heights and distances: elevation and depression angles
Mathematics
Probability: experimental vs theoretical probability
Mathematics
Arithmetic and Geometric Sequences
Mathematics
Mean, Median and Mode
Mathematics
Sets and Venn Diagrams
Mathematics
Linear equations: graphs, slope and simultaneous pairs
Mathematics
Graphs of sin, cos and tan
Mathematics
Completing the Square and the Quadratic Formula
Mathematics
Area of a Circle: Proof, Formula and Solved Problems
Mathematics
Volume and Surface Area of Solids: Cube, Cuboid, Cylinder, Cone, Sphere
Mathematics
Similar Triangles: Scale Factor, AA, SSS, SAS and Shadow Heights
Mathematics
Coordinate Geometry: Distance, Midpoint, Slope, Section Formula and Locus
Mathematics
Conic Sections: Circle, Ellipse, Parabola and Hyperbola
Mathematics
Function Transformations: Shifts, Stretches and Reflections
Mathematics
Derivative as the Slope of a Tangent: Secant to Tangent
Mathematics
Riemann Sums and Integration: Area Under a Curve
Mathematics
Limits and the ε–δ Definition: A Visual Guide
Mathematics
3D Vectors: Sum, Dot Product and Cross Product
Mathematics
Matrix Transformations: Rotation, Reflection, Shear and the Determinant
Mathematics
Complex Numbers: Sum, Product, Powers and Roots on the Argand Plane
Mathematics