An inertial reference frame (also called a Galilean reference frame) is a reference frame in which the principle of inertia is verified: any body subjected to forces with zero resultant remains at rest or in uniform rectilinear motion. No real reference frame is rigorously inertial; it is always an approximation.
Discover FizziQ
How to measure it in class
With FizziQ, the smartphone’s accelerometer gives a direct experimental criterion: a reference frame is all the closer to an inertial frame as the measured acceleration in it is zero, once gravity has been subtracted.
Steps:
- Open FizziQ and select the “linear acceleration” sensor (without g), which measures the acceleration due to motion alone.
- Lay the smartphone on a motionless table and record for one minute: the value oscillates around zero, to within a few hundredths of m/s². The reference frame of the classroom behaves here as an inertial frame.
- Repeat the recording in a train or a car traveling in a straight line at constant speed: the acceleration remains zero, even though the vehicle is moving at several tens of meters per second. This is Galileo’s result: speed cannot be measured from the inside, only acceleration is felt.
- Start again in a curve or during braking: the acceleration becomes clearly nonzero. The vehicle’s reference frame then ceases to be approximately inertial, and inertial forces must be introduced in it to keep using Newton’s laws.
- Swing the smartphone in a circle at arm’s length at constant speed and measure the acceleration: although the speed is constant in magnitude, the acceleration is not zero. Conclude that uniform motion is not enough, it must also be rectilinear.
- Compare the recordings and rank the observed situations from closest to furthest from the inertial case.
Scientific activities on this topic
Several activities make it possible to handle the notion of reference frame concretely and to spot when it ceases to be valid.
- Uniform rectilinear motion - Characterize motion at constant speed and verify the principle of inertia.
- Straight-line course, no compass - Follow a straight trajectory by inertial navigation, like a Mars rover.
- G-forces on a roller coaster - Measure what a passenger feels in a strongly non-inertial reference frame.
- Einstein’s equivalence principle - Observe that acceleration and gravity are locally indistinguishable.
- Gravitational slingshot - Understand why a probe gains speed in the heliocentric reference frame, but not in the planet’s.
- Kinematic study of a goal kick - Analyze a trajectory by video, in the reference frame of the field.
Learn more
The historical origin: the principle of Galilean relativity
The concept goes back to Galileo, who notices at the beginning of the 17th century that inside the closed cabin of a ship moving at constant speed, no mechanics experiment makes it possible to know whether the ship is moving or docked. This is the principle of Galilean relativity: the laws of mechanics are the same in all inertial reference frames, in uniform rectilinear translation with respect to one another. It is also in these frames, and only there, that Newton’s laws apply in their usual form.
The principle of inertia as a criterion, not as a tautological definition
The definition “reference frame where the principle of inertia is verified” seems circular, since one never knows with certainty that a body is isolated. In practice, one proceeds differently: one postulates that a given reference frame is inertial, applies Newton’s laws, and checks whether the predictions are verified. If accelerations appear without any identifiable force, it is the sign that the frame is not inertial at the scale considered. These residual accelerations then translate into inertial forces, which are not interactions between bodies but correction terms linked to the choice of reference frame.
The hierarchy of reference frames
In practice, three nested reference frames are used, less and less approximate:
The terrestrial reference frame is attached to the surface of the Earth. It is approximately inertial for any short and local experiment: a falling marble, a pendulum over a few oscillations, a glider on an air track. It is the frame of all high school lab work.
The geocentric reference frame has its origin at the center of the Earth and its axes pointed toward distant stars: it therefore does not rotate with the Earth. It is suited to the study of satellites and the Moon, over durations of a few days.
The heliocentric reference frame (or Copernican frame) is centered on the Sun, with axes directed toward distant stars. It is the best of the three for celestial mechanics, and the only one suited to the study of planetary motion over years.
None is perfect: the heliocentric frame is itself carried along in the rotation of the Galaxy, with an acceleration of the order of 2×10⁻¹⁰ m/s², totally negligible at the scale of the solar system but not zero.
What betrays the Earth’s rotation
The terrestrial reference frame is not inertial because the Earth rotates on itself in 23 h 56 min. Two phenomena make this visible as soon as the duration or the scale of the experiment is extended.
The Foucault pendulum, installed at the Pantheon in 1851, oscillates in a plane that seems to rotate slowly: in Paris, by about 11° per hour. The plane of oscillation itself does not move: it is the ground that rotates under the pendulum. It is the most direct mechanical demonstration of the Earth’s rotation, and an experiment impossible to interpret if the terrestrial frame is assumed to be inertial.
The Coriolis force deflects to the right, in the northern hemisphere, any body moving in the terrestrial reference frame. It is imperceptible on a football, but it structures air masses: it is what gives depressions their swirl, counterclockwise in the north, clockwise in the south. These two notions are detailed in the entries devoted to the Coriolis force and the centrifugal force.
Frequent errors
Three confusions come up often. First, believing that a “motionless” reference frame is inertial: the classroom is motionless with respect to the ground, but it rotates with the Earth. Next, believing that a moving reference frame cannot be inertial: a train at 300 km/h in a straight line at constant speed is an excellent inertial frame, much better than a merry-go-round spinning in place. Finally, treating inertial forces as real forces: they have no reaction in the sense of Newton’s third law, and they disappear if one changes reference frame.
Orders of magnitude
Centripetal acceleration due to the Earth’s rotation, at the equator: 0.034 m/s², that is 0.3% of g. Acceleration of the Earth in its orbit around the Sun: 5.9×10⁻³ m/s². Acceleration of the solar system in the Galaxy: about 2×10⁻¹⁰ m/s². Angular velocity of the Earth’s rotation: 7.29×10⁻⁵ rad/s. Apparent rotation of the plane of a Foucault pendulum: 15°/h at the pole, about 11°/h in Paris, zero at the equator.
Formula
The principle of inertia, or Newton’s first law, is written in an inertial reference frame:
Σ F⃗ = 0⃗ ⟺ v⃗ = constant
where:
- Σ F⃗: vector sum of the forces applied to the system (N)
- v⃗: velocity vector of the system’s center of mass (m/s)
Newton’s second law, valid only in an inertial reference frame:
Σ F⃗ = m · a⃗
where:
- m: mass of the system (kg)
- a⃗: acceleration vector of the center of mass (m/s²)
Change of inertial reference frame (Galilean transformation), between two frames R and R’ in uniform rectilinear translation:
v⃗(R) = v⃗(R’) + v⃗ₑ and a⃗(R) = a⃗(R’)
where:
- v⃗ₑ: constant relative velocity of R’ with respect to R (m/s)
The acceleration is the same in all inertial reference frames: this is why Newton’s laws take the same form in them.
In a non-inertial reference frame in translation with acceleration a⃗ₑ, the inertial force of transport must be added:
m · a⃗(R’) = Σ F⃗ − m · a⃗ₑ
where:
- a⃗ₑ: acceleration of the reference frame R’ (m/s²)
Application examples
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In a high-speed train traveling at 300 km/h in a straight line, a dropped pen falls vertically for the passenger, exactly as at the station: the train’s reference frame is inertial.
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In an aircraft in stabilized horizontal flight, a glass can be placed on a tray table without sliding; at takeoff, with an acceleration of about 3 m/s², it slides backward.
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The terrestrial reference frame is sufficient to study the fall of a marble over 1 m, which lasts 0.45 s: the Earth has only rotated by 0.002° during that time.
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The geocentric reference frame is necessary to compute the orbit of a geostationary satellite, whose 24 h period makes the Earth’s rotation non-negligible.
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The heliocentric reference frame is indispensable to verify Kepler’s third law: in the terrestrial frame, the trajectory of Mars shows apparent reversals that cannot be explained.
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A marble released on a rotating merry-go-round seems to deviate from its straight trajectory: the merry-go-round’s frame is not inertial, and the centrifugal and Coriolis forces must be introduced in it.
FAQ
Q: Is the terrestrial reference frame inertial, yes or no? A: It is approximately, and this approximation is sufficient for almost all high school experiments. It ceases to be valid for long experiments, like a Foucault pendulum, or extended ones, like atmospheric circulation. The right answer on a test is therefore: “approximately inertial over the duration and at the scale of the experiment”.
Q: How do you know whether a reference frame is inertial? A: A frame in uniform rectilinear translation with respect to a known inertial frame is inertial too. Experimentally, one can place an accelerometer at rest in this frame: if it reads zero once gravity is subtracted, and if no rotation is detected by the gyroscope, the frame behaves as inertial.
Q: Is a reference frame rotating at constant speed inertial? A: No. “Uniform” is not enough, the motion must also be rectilinear. In a rotation, even at constant angular velocity, the direction of the velocity changes constantly: there is a centripetal acceleration, so the frame is not inertial.
Q: Do inertial forces really exist? A: They correspond to no interaction between two bodies, unlike weight or the tension of a string. They are calculation terms that make it possible to keep using Newton’s second law in a non-inertial reference frame. Their effects, however, are quite real: you really are pressed against the car door in a curve.
Q: Why do we not feel the Earth’s rotation, nor its speed around the Sun? A: Because one never feels a speed, only an acceleration. The Earth races at 30 km/s around the Sun, but the corresponding acceleration is small (5.9×10⁻³ m/s²) and, above all, it acts identically on us and on everything around us. This is already Galileo’s argument with his ship.
Related concepts
Reference Frame - Centrifugal Force - Coriolis Force - Linear Acceleration