Newton (N)
Definition, derivation and practical force measurement in the International System of Units
Force is the quantity that describes how objects act on one another: a force changes an object’s velocity, deforms it or holds it in place against other forces. The newton, symbol N, is the SI unit of force and one of the most widely used derived SI units. Unlike the kilogram, the meter and the second, the newton is not a base unit; it is built from them.
As with the base units, three levels must be kept apart: definition, realization and practical force measurement. The everyday confusion of force with mass belongs to the third level and is treated separately below.
The definition of the newton in the SI system
Within the International System of Units (SI) the newton is defined as:
the force that gives a mass of 1 kilogram an acceleration of 1 meter per second per second.
In base units, 1 N = 1 kg·m·s⁻². The definition is a direct application of Newton’s second law, F = m·a, and therefore rests on three base units:
- the kilogram, defined since 2019 through the Planck constant,
- the meter, defined through the speed of light in vacuum,
- the second, defined through the hyperfine transition frequency of the caesium-133 atom.
The newton therefore has no definition of its own. When the kilogram was redefined in the SI revision of 2019, the newton followed automatically, and no practical force value changed.
Fact box: History of the newton
| Year | Event |
| 1687 | Isaac Newton publishes the Principia with the three laws of motion, including F = m·a. |
| 1901 | The 3rd CGPM fixes the standard acceleration of gravity at 9.806 65 m/s², which defines the kilogram-force (kgf). |
| 1904 | The name newton is proposed for the unit of force in the MKS system. |
| 1948 | The 9th CGPM adopts newton as the name of the MKS unit of force. |
| 1960 | The newton becomes a derived unit of the newly established SI. |
| 2019 | The kilogram is redefined through the Planck constant; the newton follows with its value unchanged. |
The unit is named after the English mathematician and physicist Isaac Newton (1642-1727 by the Julian calendar then used in England, 1643-1727 by the Gregorian calendar).
How much is one newton?
One newton is a small force. The gravitational force on a mass of 102 grams, roughly a small apple, is about 1 N at the Earth’s surface. Some reference values:
- the weight of 1 kg at the Earth’s surface: about 9.8 N,
- the weight of an adult: about 700 N,
- the weight of a passenger car: about 15 kN,
- the thrust of a large rocket engine: several MN.
Before the SI, the kilogram-force (kgf, also called kilopond, kp) was in common use, defined as the weight of 1 kg under standard gravity: 1 kgf = 9.806 65 N exactly. The CGS system used the dyne, where 1 dyn = 10⁻⁵ N.
Newton and kilogram: force is not mass
The question “how many newtons is a kilogram” has no universal answer, because the kilogram measures mass and the newton measures force. The two are connected through the acceleration of gravity g: weight F = m·g. With the standard value g₀ = 9.806 65 m/s², 1 kg corresponds to a weight of 9.806 65 N, and 1 kN corresponds to about 102 kg. On the Moon, where g is roughly one sixth, the same kilogram weighs only about 1.6 N.
An ordinary scale senses a force but displays a mass, because it is calibrated for the Earth’s gravity. Conversions between kilonewtons and kilogram-force therefore always assume a value of g, in practice the standard value.
Definition, realization and use
The three levels of force metrology must be kept apart:
- The definition states what a newton is in principle: a product of mass and acceleration.
- The realization covers the experimental methods by which a known force is produced at national metrology institutes.
- The use covers force measurement with transducers and testing machines in industry, civil engineering and research.
How the newton is realized
The primary realization of force takes place in deadweight force standard machines. A weight of accurately known mass m hangs in the Earth’s gravitational field, and the force is F = m·g·(1 − ρair/ρweight), where the last factor corrects for air buoyancy. The local acceleration of gravity g is measured with absolute gravimeters to a relative uncertainty of the order of 10⁻⁸, and the mass is traceable to the definition of the kilogram.
Deadweight machines are built up to about 1 MN. Larger forces are produced by hydraulic amplification or by build-up systems, in which several calibrated force transducers are loaded in parallel.
Practical force measurement and error sources
In practice, force is almost always measured indirectly with force transducers, most often strain-gauge load cells, in which an elastic body deforms and the deformation is read out electrically. The transducer is calibrated against a deadweight machine or a reference transducer.
The main error sources are creep under sustained load, hysteresis between loading and unloading, temperature dependence of both the material and the electronics, and misalignment of the force relative to the transducer axis. For precision measurement these effects must be characterized and corrected.
The newton and other derived units
The newton appears in several other derived SI units:
| Unit | Quantity | Expression |
| pascal (Pa) | Pressure | N/m² |
| joule (J) | Energy, work | N·m |
| watt (W) | Power | N·m/s |
| kilogram-force (kgf) | Force (older unit) | 9.806 65 N |
Multiples and common conversions
Common multiples are the millinewton (1 mN = 10⁻³ N), the kilonewton (1 kN = 10³ N) and the meganewton (1 MN = 10⁶ N). In civil engineering, loads are almost always given in kN, and strength in N/mm², which is the same as megapascals.
Conversions between units of force are exact, since they only involve multiplication by a constant. Converting between mass and force, on the other hand, is always a calculation of weight at an assumed g, not a unit conversion in the strict sense.
Summary
The newton is the SI unit of force, defined as kg·m·s⁻² directly from Newton’s second law. The unit has no material or physical definition of its own; it inherits its stability from the kilogram, the meter and the second.
It is realized with deadweight machines in which a known mass hangs in an accurately measured gravitational field, and practical force measurement relies on calibrated transducers. This chain of traceability is what makes a newton mean the same thing in every laboratory in the world.