Centrifugal force is not a real force: it is an inertial force, a corrective term added to the balance of forces when working in a rotating, and therefore non-inertial, reference frame. No object exerts it, and it does not appear among the interactions of physics.
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How to measure it in class
The accelerometer of a smartphone measures proper acceleration: placed on a turntable, it indicates an acceleration directed toward the center, the centripetal acceleration a = ω²R. This is the best way to show experimentally that the vector points inward, not outward.
Steps:
- Firmly attach the smartphone to a turntable, a record player or the inside of a salad spinner, at a known distance R from the axis
- Note the orientation of the phone’s axes relative to the radius
- Start recording acceleration (linear or absolute depending on the experiment) in FizziQ
- Set the device rotating at a speed as constant as possible and measure the period T over several turns, hence ω = 2π/T
- Repeat for three or four rotation speeds, then plot a as a function of ω²
- Check that the graph is a straight line through the origin, with slope R. Make sure the plane of rotation is horizontal, otherwise gravity adds a sinusoidal component to the signal at each turn
Scientific activities on this topic
Several FizziQ activities allow you to measure the effects of a circular trajectory and verify that the acceleration is indeed centripetal:
- Could an astronaut survive in a salad spinner? - estimate the acceleration experienced in a salad spinner and compare it to physiological limits
- Centripetal acceleration: the relation a = ω²R - establish the law experimentally by varying the angular velocity
- Variation of g with latitude - measure the effect of the Earth’s rotation on apparent gravity
- Normal acceleration in rotation - show the relation between speed and centripetal acceleration
- Pendulum: the relation a = 2gh/r - measure the centripetal acceleration at the lowest point of a pendulum’s swing
Learn more
The passenger in a turning car: what really happens
This is the example that settles everything. A car enters a left turn; the passenger is “pressed” against the right-hand door. They conclude that a force pushed them outward. This is wrong.
Seen from the ground, an inertial reference frame: the passenger was going straight, and by inertia they keep going straight, in accordance with the principle of inertia. It is the car that turns underneath them. The door therefore comes to meet them, and once contact is established, it is the door that exerts a real force on them - directed toward the inside of the turn - forcing them to turn with the vehicle. The sensation of being pushed out is in reality the sensation of being pushed in by the door: what our body perceives is the compression of the contact, not its absolute direction.
The same reasoning applies to the seat belt, to laundry pressed against the drum of a washing machine (it is the drum that pushes the laundry toward the axis), and to rotating-wall fairground rides.
Why the word “centrifugal” remains useful
The term should not be banned; it should be put in its place. In a rotating reference frame - and an engineer designing a centrifuge naturally works in the frame of the machine - writing the centrifugal force greatly simplifies the calculations. The rule is simple: either you work in an inertial reference frame and speak only of centripetal force, or you work in the rotating frame and add the inertial forces. What is forbidden is mixing the two, for example writing that a rotating object is subject to both a centripetal force and a centrifugal force that cancel out: it would not accelerate, and would therefore have no reason to turn.
Inertial force of transport and Coriolis force
In a rotating reference frame, strictly speaking, two corrective terms are needed. The inertial force of transport (of which the centrifugal force is the circular-motion case) equals mω²r and points outward. The Coriolis force, −2m ω⃗ ∧ v⃗, appears only if the object moves within the rotating frame. It is this force that makes atmospheric depressions rotate and deflects air masses, not the centrifugal force. These two forces belong to higher-education programs, but their existence sheds light on phenomena seen in high school.
The Earth is a rotating reference frame
The terrestrial reference frame is not strictly inertial: the Earth rotates on itself in 23 h 56 min, that is ω ≈ 7.29 × 10⁻⁵ rad·s⁻¹. At the equator, the centrifugal term equals ω²R ≈ 0.034 m·s⁻², approximately 0.3% of g. It is zero at the poles. This is one of the two reasons why we measure g ≈ 9.78 m·s⁻² at the equator and g ≈ 9.83 m·s⁻² at the pole - the other being the flattening of the globe, itself a consequence of this rotation. The weight we measure is therefore an apparent weight, which already includes this inertial term.
Artificial gravity: what the movies get right
A rotating space habitat can indeed simulate gravity: the floor (the outer wall) exerts a real centripetal force on the occupant, exactly as the ground on Earth exerts a normal reaction. The sensation is identical. To obtain 1 g with acceptable comfort, a large-radius station and slow rotation are needed: with R = 224 m and two revolutions per minute (T = 30 s, ω ≈ 0.209 rad·s⁻¹), ω²R ≈ 9.8 m·s⁻². At only one revolution per minute, a radius of about 895 m would be needed to reach the same value. If the radius is too small, the variation of acceleration between head and feet, and the Coriolis effects during movements, cause nausea.
Formula
Centripetal acceleration, in an inertial reference frame:
a = v²/R = ω²R
Centripetal force, real, directed toward the center:
F = m v²/R = m ω²R
Centrifugal force, fictitious, in the rotating frame, directed outward:
F_centrifugal = −m ω²R (same magnitude, opposite direction)
Relation between angular velocity and period:
ω = 2π/T = 2πf
Number of “g” experienced:
n = a/g = ω²R/g
where:
- a: centripetal acceleration (m·s⁻²)
- v: linear velocity (m·s⁻¹)
- R: radius of the trajectory (m)
- ω: angular velocity (rad·s⁻¹)
- T: rotation period (s)
- m: mass (kg)
- g: gravitational field strength, approximately 9.81 m·s⁻²
Application examples
- A 1,200 kg car takes a turn of 50 m radius at 72 km/h (20 m·s⁻¹): the centripetal acceleration equals 20²/50 = 8 m·s⁻², and the required force 9,600 N. It is provided entirely by tire friction. If the road is wet and can only provide 6,000 N, the car goes off in a straight line - it does not “fly outward”, it keeps going straight.
- A laboratory centrifuge at 10,000 rpm, at 8 cm from the axis: ω = 1,047 rad·s⁻¹, so a = ω²R ≈ 8.8 × 10⁴ m·s⁻², approximately 9,000 g. This is what separates the components of a sample by density.
- A salad spinner rotating at 3 revolutions per second, radius 10 cm: a = (2π×3)² × 0.10 ≈ 36 m·s⁻², that is 3.6 g. The water is not “expelled”: it keeps going straight and passes through the holes of the basket, which the wall cannot force it to go around.
- A fighter pilot in a tight turn experiences up to 9 g: it is the seat that exerts this centripetal force on them. Beyond that, blood no longer reaches the brain and blackout occurs.
- Roller coasters: at the top of a loop, the weight and the rail’s reaction both point toward the center. This is why you do not fall, provided the speed is high enough that v²/R ≥ g.
- A satellite in orbit is subject only to gravitation, which plays exactly the role of the centripetal force. The weightlessness felt is not due to a centrifugal force that would compensate gravity, but to permanent free fall.
FAQ
Q: Does centrifugal force exist, yes or no? A: No, not in the sense of a physical interaction: no object exerts it, and it appears in none of the four fundamental interactions. It is an inertial force, a mathematical term added in order to apply Newton’s second law in a non-inertial reference frame. Its calculated effects are exact, but its cause does not exist.
Q: Then why am I pressed against the door in a turn? A: Because your body, by inertia, keeps going straight while the car turns. It is the door that comes to you and pushes you - toward the inside of the turn. No force pushes you outward; you simply feel the real force exerted by the door.
Q: What is the difference between centripetal force and centrifugal force? A: The centripetal force is real, directed toward the center, exerted by a string, a rail, friction or gravitation, and it is the cause of circular motion. The centrifugal force is fictitious, directed outward, and exists only if one chooses to reason in the rotating reference frame. They do not cancel each other out: they never appear in the same force balance.
Q: If I cut the string of an object I am spinning, does it fly outward? A: No, it flies off tangentially, in a straight line, in the direction of its velocity at the moment of the break. This is the principle of inertia. If a centrifugal force existed, it would fly off radially, which is never observed.
Q: Does the smartphone accelerometer measure the centrifugal force? A: It measures the phone’s proper acceleration, which, on a turntable, is directed toward the center: this is the centripetal acceleration. The numerical value is the same as the one that would be attributed to the centrifugal force, but the measured vector points inward. Beware of saturation: depending on the model, the sensor caps out between 2 g and 8 g, and a salad spinner quickly exceeds this limit.
Related concepts
Centripetal Acceleration - Uniform Circular Motion - Inertial Reference Frame - Accelerometer - Coriolis Force - Law of Gravitation