2The Idea, Step by Step
Sit on a fast merry-go-round and you feel flung toward the edge. Take a corner too quickly in a car and you slide across the seat into the door. It feels like something is shoving you outward — but nothing is. Your body simply wants to keep going in a straight line, and the spinning floor (or the turning car) keeps swinging in underneath you. The push you feel is the seat or the railing shoving you inward.
That inward shove is the only real force here, and it has a name: the centripetal force. It is whatever actually grips you — friction, a seatbelt, your hands on the bar — and bends your straight-line motion into a circle. Its size depends on how fast the ride spins and how far out you sit. Call the spin rate $\Omega$ (in rad/s), your distance from the centre $r$, and your mass $m$. Then $F_{\text{cp}} = m\,\Omega^2 r$. Picture a 30 kg child sitting $r=2$ m out on a merry-go-round turning once every 4 s, so $\Omega = 2\pi/4 \approx 1.6$ rad/s. The inward force needed is $F_{\text{cp}} = 30 \times 1.6^2 \times 2 \approx 150$ N — that is exactly how hard their arms must pull on the bar.
Now ride with the merry-go-round and do physics from your spinning seat. Newton's law $\vec F = m\vec a$ only works in a non-spinning (inertial) frame, so to use it while rotating you must invent two make-believe "pseudo-forces." The first is centrifugal force, $\vec F_{\text{cf}} = +m\Omega^2\vec r$, pointing outward — same size as the centripetal pull, which is why, in your frame, you sit still. The second only acts on things that move within the spinning frame: the Coriolis force, $\vec F_{\text{Cor}} = -2m\,\vec\Omega\times\vec v_{\text{rot}}$, which always pushes sideways, perpendicular to the velocity. On the spinning Earth ($\Omega = 7.3\times10^{-5}$ rad/s) this sideways kick, of size $2m\Omega v\sin\varphi$, nudges every wind to the right in the Northern Hemisphere and left in the Southern — which is why hurricanes swirl. The sliders map straight onto this: $\Omega$ sets the spin (and both pseudo-forces), $r$ the starting distance, $v$ the launch speed (which powers Coriolis), and the latitude $\varphi$ controls the Foucault period $T = 2\pi/(\Omega\sin\varphi)$.
Try this in the sim above. (1) In Rotating Frame, push $\Omega$ high and watch the straight red inertial path curl into a blue spiral — same motion, two frames. (2) Switch to Coriolis and raise $v$: the sideways deflection grows because the kick scales as $2\Omega v$. (3) In Foucault, drag $\varphi$ from 45° down toward 0° and watch the plane-rotation period blow up — at the equator it never turns at all.
3Equation Derivation
Centripetal Acceleration (real, in inertial frame)
$$\vec{a}_{\text{cp}} = -\Omega^2 \vec{r}\quad\Rightarrow\quad \vec{F}_{\text{cp}} = -m\Omega^2\vec{r}$$
Centrifugal Force (apparent, rotating frame)
$$\vec{F}_{\text{cf}} = +m\Omega^2\vec{r}$$
Coriolis Force (apparent, rotating frame)
$$\vec{F}_{\text{Cor}} = -2m\,\vec{\Omega}\times\vec{v}_{\text{rot}}$$
Symbol Definitions
| Symbol | Meaning | SI Unit |
|---|
| $\vec\Omega$ | Angular velocity of the rotating frame | rad s⁻¹ |
| $\vec r$ | Position vector from the rotation axis | m |
| $\vec v_{\text{rot}}$ | Velocity in the rotating frame | m s⁻¹ |
| $F_{\text{cp}}$ | Real centripetal force (e.g., tension, gravity, friction) | N |
| $F_{\text{cf}}$ | Pseudo-centrifugal force (only in rotating frame) | N |
| $F_{\text{Cor}}$ | Coriolis pseudo-force (only in rotating frame) | N |
| $\varphi$ | Latitude on Earth | rad or ° |
1
Two frames, two stories. Newton's 2nd law $\vec F=m\vec a$ holds in inertial frames. In a rotating (non-inertial) frame, two extra apparent forces must be added so that $\vec F+\vec F_{\text{cf}}+\vec F_{\text{Cor}}=m\vec a_{\text{rot}}$.
2
Velocity transformation. If $\vec r$ is fixed in the rotating frame, then in the inertial frame $\vec v_{\text{in}}=\vec v_{\text{rot}}+\vec\Omega\times\vec r$. Differentiating again: $\vec a_{\text{in}}=\vec a_{\text{rot}}+2\vec\Omega\times\vec v_{\text{rot}}+\vec\Omega\times(\vec\Omega\times\vec r)$.
3
Identifying the pseudo-forces. Multiplying by $m$ and rearranging $m\vec a_{\text{rot}}=\vec F_{\text{real}}-2m\vec\Omega\times\vec v_{\text{rot}}-m\vec\Omega\times(\vec\Omega\times\vec r)$. The last term simplifies to $+m\Omega^2\vec r$ for $\vec\Omega$ perpendicular to $\vec r$ — that is the centrifugal force, directed outward.
4
Centripetal vs centrifugal — they are not opposites of each other. Centripetal force is real and points inward; it is whatever physical force (tension, gravity, friction) keeps the body in circular motion. Centrifugal is fictitious and exists only in the rotating frame; it appears outward because the rotating frame itself is accelerating.
5
Coriolis on Earth. $\vec\Omega_\oplus=7.27\times10^{-5}$ rad s$^{-1}$ along the rotation axis. The horizontal component of $-2\vec\Omega\times\vec v$ deflects moving objects to the right in the Northern hemisphere, left in the Southern, with magnitude $2v\Omega\sin\varphi$. This drives cyclonic weather patterns and was first measured directly by Foucault's 1851 pendulum.
Primary references: Kleppner & Kolenkow 2nd Ed., Ch. 9; Marion & Thornton 5th Ed., Ch. 10; HyperPhysics — Coriolis & Foucault.
4Frequently Asked Questions
💡 Concept Is centrifugal force 'fake'? Then why do I feel it?▼
Centrifugal force is fictitious in the sense that no actual interaction (gravitational, electromagnetic) produces it. You feel something pushing you outward in a turning car because your body is trying to continue in a straight line (Newton's 1st law) while the car turns. The car wall pushes you inward — that inward push is real and is the centripetal force. You feel the contact force, but interpret it as 'something pushing me outward'.
Key: What you feel is the inward contact force; centrifugal is a useful frame-dependent label, not a real interaction.
🎯 Simulation What exactly is the simulation showing?▼
Two trajectories are drawn simultaneously: red is the path in the inertial (lab) frame — typically a straight line or a circle if a real centripetal force acts. Blue is the same motion as seen by someone rotating with $\Omega$ — looped, curved, or even closed depending on the situation. The Coriolis mode shows a freely launched ball deflecting in the rotating frame even though no real force acts on it.
Key: Side-by-side inertial vs. rotating-frame views of the same motion.
🌍 Real Life Where do these effects matter in everyday life?▼
Coriolis force determines the direction of cyclones (counterclockwise in NH, clockwise in SH), affects long-range artillery and missile aiming, deflects ocean currents (gyres), and explains the Foucault pendulum's daily rotation. Centrifugal effects matter for planetary oblateness (Earth bulges at equator), washing-machine spin cycles, and centrifuges in laboratories.
Key: Cyclones, ocean gyres, projectile aim, satellite orbits, washing machines, centrifuges.
🔬 Non-Obvious Why doesn't water in a draining sink rotate due to Coriolis?▼
On the scale of a bathtub, Coriolis acceleration is $a_{\text{Cor}}\sim2\Omega v\approx10^{-5}$ m/s² for $v=0.1$ m/s — utterly negligible compared to perturbations from the geometry of the drain, residual circulation, and surface tension. Coriolis becomes important only for slow, large-scale flows where the residence time is comparable to $1/\Omega\approx14000$ s.
Key: Coriolis matters only for slow, large-scale flows — not for sinks; that is a popular myth.
📐 Mathematical Derive the Coriolis deflection for a falling object.▼
An object dropped from height $h$ at the equator falls for $t=\sqrt{2h/g}$. The eastward deflection is $\Delta x=\tfrac{1}{3}\Omega g t^3 \cos\varphi$ — about 22 mm for $h=100$ m at the equator. This was first measured by Reich in a deep mine in 1831 and used as a confirmation of Earth's rotation.
Key: Falling object deflects east by $\Delta x=(1/3)\Omega g t^3\cos\varphi$ — small but measurable.
🧠 Deep What is the connection between centrifugal force and gravity?▼
Einstein's equivalence principle says no local experiment can distinguish acceleration from gravity. In a rotating frame, the centrifugal pseudo-force is mathematically indistinguishable from a 'gravitational' field $g_{\text{eff}}=\Omega^2 r$ pointing outward. This is why on a rotating space station, residents experience artificial gravity — the centrifugal pseudo-force is, locally, indistinguishable from real weight.
Key: Equivalence principle: rotating frame's centrifugal force is locally equivalent to a 'gravitational' field — basis of artificial gravity.
Explanatory resources: Kleppner & Kolenkow 2nd Ed., Ch. 9; Marion & Thornton 5th Ed., Ch. 10; HyperPhysics — Coriolis & Foucault.
5Common Misconceptions
❌ Misconception: Centrifugal force is the reaction (Newton's 3rd law) to centripetal force.
✅ Correction: They are not a Newton's 3rd-law pair. The reaction to centripetal force on the orbiting body is the equal-and-opposite force the body exerts on the centre (e.g., the planet pulls on the Sun). Centrifugal force is a frame-dependent pseudo-force that exists only in the rotating frame, not a reaction force at all.
📖 Reference: Halliday, Resnick & Walker 10th Ed., §6-6; Kleppner & Kolenkow 2nd Ed., §9.2.
❌ Misconception: Coriolis force is what makes water spiral down a drain in opposite directions in the two hemispheres.
✅ Correction: On bathtub scales, Coriolis acceleration is roughly $10^{-5}$ m/s² — drowned out by initial conditions, geometry, and turbulence. Direction in your sink is essentially random. Coriolis effects only become dominant for large, slow flows (cyclones, ocean gyres) where the time scale is comparable to a day.
📖 Reference: Marion & Thornton — Classical Dynamics, 5th Ed., Ch. 10; Trefethen et al. (1965), Nature 207, 1084.
❌ Misconception: Pseudo-forces are useless mathematical tricks — real physics happens only in the inertial frame.
✅ Correction: Pseudo-forces are essential for working in the rotating frame, which is often the natural choice. Earth itself is a rotating frame; meteorology, geophysics, and astronomy routinely use the Coriolis and centrifugal terms. The 'real' inertial-frame description is sometimes more cumbersome (try doing global atmospheric dynamics from outside Earth!).
📖 Reference: Kleppner & Kolenkow 2nd Ed., §9.4–9.6; Greiner — Classical Mechanics, §38.
❌ Misconception: On a rotating space station the artificial 'gravity' is centripetal force.
✅ Correction: From outside the station (inertial frame), the floor pushes residents inward — a real centripetal contact force. From inside the station (rotating frame), residents feel pulled outward by the centrifugal pseudo-force, balanced by the normal force from the floor. Calling this 'centripetal gravity' confuses the two frames.
📖 Reference: HRW 10th Ed., §6-5, Sample Problem 6.07; Kleppner & Kolenkow §9.5.
❌ Misconception: Foucault pendulum proves Earth is rotating about its axis 24 hours per day.
✅ Correction: Foucault pendulum's plane rotates with period $T=24\text{h}/\sin\varphi$ — at the equator ($\varphi=0$) it does not rotate at all, and at the pole it rotates in 24 hours. The sidereal day (23h 56m) is the relevant period, not solar day. The pendulum directly demonstrates Earth's rotation in the inertial frame.
📖 Reference: Foucault (1851); HyperPhysics — Foucault Pendulum; Marion & Thornton §10.5.
Misconception research: Persson (1998), Bull. Amer. Meteor. Soc. 79, 1373; Trefethen et al. (1965), Nature 207, 1084; Arons — A Guide to Introductory Physics Teaching, Wiley 1990.