Gravitation is a fundamental interaction that attracts all massive bodies toward one another. Newton’s law (1687) describes it quantitatively: the force is proportional to the product of the masses and inversely proportional to the square of their distance. It explains both gravity on Earth and orbits.
Discover FizziQ
How to measure it in class
With FizziQ, the smartphone’s accelerometer gives access to the local value of the gravitational field strength g, and the kinematics module makes it possible to recover it from the trajectory of a falling object.
Steps:
- Open FizziQ and select the “absolute acceleration” sensor (acceleration including g). Lay the smartphone flat, motionless, on the table: the displayed value is the local gravitational field strength, close to 9.81 m/s².
- Record this value for about ten seconds and calculate its average, to smooth out the sensor noise.
- Film the fall of a ball against a contrasting background, with an object of known size in the frame for calibration, then import the video into the kinematics module.
- Track the ball frame by frame and plot the curve of vertical position versus time: it should be a parabola with equation z = ½·g·t².
- Compare the g deduced from the video with the g measured by the accelerometer, and discuss the difference (air friction, tracking errors, calibration).
- Repeat the measurement of g at the top and then at the bottom of a building or a hill to test the 1/d² dependence: the variation is on the order of 3×10⁻⁶ m/s² per meter of altitude, and therefore undetectable over a few floors with a smartphone.
Scientific activities on this topic
Several experiments that can be carried out with a smartphone or a tablet allow you to explore gravitation, from free fall to interplanetary trajectories. Seven of them are detailed in an article dedicated to gravity published on our blog.
- Measure g from the duration of a free fall - Determine the gravitational field strength by timing the fall of a smartphone.
- Variation of g with altitude - Test the 1/d² law by comparing g on the ground and in an airplane.
- Variation of g with latitude - Understand why you weigh slightly less at the equator than at the pole.
- Parabolic trajectory: basketball - Analyze by video the motion of a ball subject to gravity alone.
- Orbital period of the Moon - Relate the duration of the lunar revolution to the Earth-Moon distance.
- Solar system and Kepler’s laws - Verify Kepler’s third law on the planets.
- Geostationary orbit: Meteosat - Calculate the radius of the orbit of a satellite that remains above the same point.
- Gravitational slingshot - Understand how a probe gains speed by flying close to a planet.
- Translunar injection - Simulate putting a spacecraft on a trajectory toward the Moon.
- Three-body problem and chaos - Observe that a system of three masses has no simple solution.
- Parabolic flight and microgravity - Recreate the weightlessness of an airplane in free fall.
Learn more
Newton’s breakthrough: a single phenomenon for the apple and the Moon
Before Newton, physics distinguished two worlds: the sublunar world, where “heavy” bodies fall toward the center of the Earth, and the celestial world, where the heavenly bodies describe circles according to laws supposedly of their own. Newton’s decisive contribution, published in the Principia in 1687, is to assert that these two behaviors follow the same law: the apple falls and the Moon stays in orbit for one and the same reason. Newton verified numerically that the acceleration of the Moon, at 60 Earth radii from the center of the Earth, is indeed g/60², or about 0.0027 m/s². It is this unification, more than the formula itself, that constitutes the revolution.
The apple anecdote is authentic in its broad outlines: Newton himself told it to his biographer William Stukeley in 1726. But he speaks of an apple he sees fall from his window, which leads him to wonder why it always falls perpendicular to the ground. The apple never fell on his head; that version is a later embellishment.
Mass and weight: the confusion to avoid
This is the most frequent student error. Mass m is a quantity of matter: it is measured in kilograms, with a beam balance, and it is the same everywhere in the universe. Weight P is a force: it is measured in newtons, with a dynamometer, and it depends on the celestial body you are on, since P = m·g. A 60 kg student has a mass of 60 kg on Earth, on the Moon and in the International Space Station. Their weight is about 589 N on Earth, 97 N on the Moon (g = 1.62 m/s²), and it is still nearly 589 N in the ISS - but they do not feel it, because they are in permanent free fall around the Earth. Saying that an astronaut is “weightless” is therefore inaccurate: they are in a state of apparent weightlessness, which is not the same thing.
Also beware of everyday vocabulary: a “bathroom scale” actually measures a reaction force and displays a mass by assuming g = 9.81 m/s². On the Moon, it would display one sixth of the value, even though the mass would not have changed.
Gravity and gravitation are not exactly identical
The gravitational field strength g measured by an accelerometer in a terrestrial reference frame is not exactly the Earth’s gravitational attraction: it differs from it by the contribution of the Earth’s rotation. This is why g is about 9.780 m/s² at the equator, where the centrifugal effect is maximal and where one is also farther from the center of the Earth because of its flattening, versus about 9.832 m/s² at the pole. This 0.5% difference is measurable and explains why space launch sites are located near the equator.
The limits of Newton’s law
Newton’s law fails when fields become intense or speeds become large. Einstein’s general relativity (1915) gives a deeper description of it, as a curvature of spacetime; Newton’s law remains an excellent approximation in all non-relativistic cases, including for space trajectories. The historical example is the advance of Mercury’s perihelion: Newton’s law predicts part of it, but an unexplained discrepancy of 43 arcseconds per century remains, which general relativity explains exactly. The relativistic correction is not an astronomer’s curiosity: without it, GPS satellites would accumulate a position error of several kilometers per day.
Beyond gravity and orbits, gravitation also manifests itself in the tides and in the quasi-spherical shape of large celestial bodies, which their own attraction rounds out.
Orders of magnitude
Gravitational constant: G = 6.674×10⁻¹¹ N·m²·kg⁻². Gravitational field strength at ground level: g ≈ 9.81 m/s² in metropolitan France; 1.62 m/s² on the Moon; 3.71 m/s² on Mars; 24.8 m/s² on Jupiter. Mass of the Earth: 5.97×10²⁴ kg. Earth’s radius: 6,371 km. Earth-Moon distance: 384,400 km. Gravitational force between two 60 kg students 1 m apart: about 2.4×10⁻⁷ N, the weight of a speck of dust - gravitation is by far the weakest of the four fundamental interactions, but the only one that is always attractive and of infinite range, which makes it dominant on the astronomical scale.
Formula
Law of universal gravitation, for two point-like or spherically symmetric bodies:
F = G · m₁ · m₂ / d²
where:
- F: magnitude of the gravitational force exerted by each body on the other (N)
- G: gravitational constant, G = 6.674×10⁻¹¹ N·m²·kg⁻²
- m₁, m₂: masses of the two bodies (kg)
- d: distance between their centers (m)
The two forces are opposite, of equal magnitude, along the line joining the centers (Newton’s third law).
Strength of the gravitational field created by a body of mass M at distance d from its center:
g = G · M / d²
where:
- g: gravitational field strength (N/kg, or m/s²)
- M: mass of the celestial body (kg)
- d: distance to the center of the body (m)
Weight of a body of mass m:
P = m · g
where:
- P: magnitude of the weight (N)
- m: mass of the body (kg), invariant
- g: gravitational field strength at the location considered (N/kg), variable
Application examples
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A 5 kg bag has a weight of 5 × 9.81 = 49 N on Earth, and 5 × 1.62 = 8.1 N on the Moon; its mass remains 5 kg in both cases.
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The gravitational force between the Earth (5.97×10²⁴ kg) and the Moon (7.35×10²² kg), 3.844×10⁸ m apart, is about 1.98×10²⁰ N: it is what keeps the Moon in orbit.
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Tides result from the difference in lunar attraction between the side of the Earth facing the Moon and the opposite side, a direct consequence of the 1/d² term.
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A geostationary satellite orbits at an altitude of 35,786 km, the only distance for which its orbital period is exactly one sidereal day.
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A space probe uses the gravitational assist of a planet to gain speed relative to the Sun without consuming fuel.
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On a bathroom scale, an elevator starting upward momentarily increases the displayed value: it is the contact force that changes, not the mass.
FAQ
Q: What is the difference between mass and weight? A: Mass is a quantity of matter, in kilograms, identical everywhere in the universe. Weight is a force, in newtons, equal to m·g, which depends on the celestial body you are on. You have the same mass on Earth and on the Moon, but a weight six times smaller on the Moon.
Q: Why do astronauts float in the Space Station? A: Not because gravitation has disappeared: at 400 km altitude, g is still about 8.7 m/s², nearly 90% of its value at ground level. They float because the station and its crew are in permanent free fall around the Earth. No contact force acts on them, so they no longer feel their weight.
Q: Does a heavy object fall faster than a light one? A: No, if air is neglected. The force is twice as large on a mass twice as large, but so is the inertia: the acceleration is the same, g, for all bodies. A feather falls more slowly only because of air resistance; in a tube emptied of air, feather and ball fall together.
Q: Is the value of g really the same everywhere on Earth? A: No. It varies by about 0.5% between the equator (9.780 m/s²) and the poles (9.832 m/s²), because of the Earth’s rotation and the flattening of the globe. It also decreases with altitude, by about 3 millionths of m/s² per meter. A smartphone accelerometer is not sensitive enough to detect these variations.
Q: Is the constant G well known? A: It is paradoxically the least well known of the fundamental constants: its value, 6.674×10⁻¹¹ SI, is established only to about 2×10⁻⁵ in relative value, versus 10⁻¹⁰ or better for other constants. The reason is that gravitation is so weak that the experiment cannot be isolated from surrounding masses. Its first measurement is credited to Cavendish in 1798.
Related concepts
Gravitational Acceleration - Free Fall - Centrifugal Force - Inertial Reference Frame - Absolute Acceleration