Force

In mechanics, a force is a vector quantity used to characterize an interaction capable of changing the momentum of a body or system. A modern Newtonian statement of the relation between net force and momentum is

$\displaystyle \mathbf F_{\rm net}=\frac{d\mathbf p}{dt},$

where $\mathbf p$ is linear momentum. For a particle of constant mass, $\mathbf p=m\mathbf v$ , so that

$\displaystyle \mathbf F_{\rm net}=m\mathbf a.$

The SI unit of force is the newton,

$\displaystyle 1\,\mathrm{N}=1\,\mathrm{kg\,m\,s^{-2}},$

and the dimensions of force are

$\displaystyle MLT^{-2}.$

Because force has both magnitude and direction, forces combine by vector addition. The vector sum of all external forces acting on a system is its net force. If the net external force vanishes, the total momentum of the system is constant. For a constant-mass particle observed from an inertial frame, zero net force therefore implies zero acceleration.

At the macroscopic level, familiar force models include gravitation, normal contact forces, friction, Tension, elastic forces, drag, and distributed loads. Many everyday contact and elastic forces ultimately arise from electromagnetic interactions among atoms and molecules. The rotational effect of a force is described by torque.

Classical mechanics provides an extremely useful force-based description of motion. Modern physics goes deeper: the known forces of nature are traced to four fundamental interactions, while relativity and quantum theory modify the way the classical force concept is interpreted.

Fundamental interactions

All presently known fundamental interactions are classified into four groups: the gravitational, electromagnetic, strong, and weak interactions. The strong and weak interactions operate over very short distances and govern subatomic processes involving quarks, leptons, nucleons, and nuclei. Electromagnetism acts on Electric Charge, while gravitation couples to mass-energy.

The macroscopic forces used in classical mechanics are not additional fundamental interactions. Friction, normal forces, spring forces, and most other contact forces arise from electromagnetic interactions together with quantum-mechanical constraints such as the Pauli exclusion principle. Likewise, centrifugal force is not a fundamental interaction; it is an inertial or fictitious force that appears when motion is described from an accelerating rotating reference frame.

The history of physics includes several major unifications of apparently different forces. Newton showed that terrestrial falling motion and celestial orbital motion could be described by the same law of gravitation. Faraday and Maxwell unified electric and magnetic phenomena into electromagnetism. In the twentieth century, quantum field theory described electromagnetic, weak, and strong interactions in terms of quantum fields and exchanged gauge bosons. The electromagnetic and weak interactions were further unified in the electroweak theory, whose predictions were subsequently confirmed experimentally.

Comparison of the four fundamental interactions

The following table summarizes the classification used in the source article. The relative strengths are approximate order-of-magnitude comparisons and depend on the characteristic energy or distance scale at which the interactions are compared.

Property / interaction	Gravitation	Weak	Electromagnetic	Strong
		Electroweak		Fundamental	Residual
Acts on	mass-energy	flavor	electric charge	color charge	atomic nuclei
Particles experiencing	all	quarks, leptons	electrically charged	quarks, gluons	hadrons
Particles mediating	graviton (not yet observed)	, ,	photon ( $\gamma$ )	gluons ()	mesons
Relative strength at quark scale	$10^{-41}$	$10^{-4}$			not applicable
Relative strength at proton/neutron scale	$10^{-36}$	$10^{-7}$		not applicable

Gravitational interaction

Newtonian gravitation describes gravity as an attractive force acting at a distance. In that theory, changing the position of one mass changes the gravitational influence associated with it everywhere else without an explicit propagation delay. This instantaneous-action picture is incompatible with special relativity, in which physical influences cannot propagate faster than Light.

General relativity replaces Newton's action-at-a-distance description with a geometrical theory of spacetime. Freely falling bodies follow geodesics of curved spacetime; locally, an ideal freely falling observer can regard the motion as inertial rather than as motion driven by a conventional force. Newtonian gravity remains an excellent approximation when gravitational fields are weak and speeds are small compared with the speed of light.

Historically, the anomalous perihelion precession of Mercury was one of the major tests distinguishing Einstein's theory from Newtonian gravitation. General relativity correctly supplies the additional orbital precession that is absent from the Newtonian prediction.

In ordinary engineering and introductory mechanics, however, it remains useful to describe gravity by a force such as

$\displaystyle \mathbf F_g = -G\frac{m_1m_2}{r^2}\,\hat{\mathbf r},$

or, near Earth's surface and over modest height changes,

$\displaystyle \mathbf F_g\approx m\mathbf g.$

Electromagnetic interaction

Electric and magnetic forces are different manifestations of a single electromagnetic interaction. At the classical level, electromagnetic fields are described by Maxwell's equations together with the Lorentz force law,

$\displaystyle \mathbf F = q\left(\mathbf E+\mathbf v\times\mathbf B\right).$

Classical electromagnetism already has a relativistic structure: electric and magnetic fields mix under changes of inertial frame. At the quantum level, electromagnetic interactions between elementary particles are described by quantum electrodynamics (QED), in which the photon is the gauge boson of the electromagnetic field.

Many macroscopic forces that appear unrelated to electromagnetism are ultimately consequences of electromagnetic interactions among charged particles, combined with quantum-mechanical structure. Examples include most contact forces, chemical bonding, elasticity, and friction.

Strong interaction

The strong interaction is the fundamental interaction described by quantum chromodynamics (QCD). It acts on color charge and directly affects quarks, antiquarks, and gluons. Gluons are the gauge bosons of QCD and, unlike photons, themselves carry the charge associated with the interaction.

The fundamental strong interaction is responsible for binding quarks into hadrons such as protons and neutrons. Because of color confinement, isolated free quarks are not observed under ordinary conditions.

A residual strong interaction remains between color-neutral hadrons. This residual nuclear force binds protons and neutrons inside atomic nuclei and is strong enough at nuclear distances to overcome the electromagnetic repulsion between positively charged protons. At low energies it can be modeled in terms of meson exchange, although the underlying fundamental theory is QCD.

Weak interaction

The weak interaction is responsible for processes in which particle flavor changes and is especially familiar through beta decay and related radioactive processes. It does not form stable bound states in the way the strong and electromagnetic interactions do.

The weak interaction is mediated by the massive

$\displaystyle W^+,\qquad W^-,\qquad Z^0$

bosons. Processes involving $W^\pm$ are called charged-current interactions, while processes involving the electrically neutral $Z^0$

are called neutral-current interactions.

At ordinary energies the weak interaction is much less intense than the strong or electromagnetic interactions, but it is still vastly stronger than gravitation at subatomic distances.

The weak and electromagnetic interactions are two low-energy manifestations of a unified electroweak interaction. At sufficiently high temperatures or collision energies the distinction between electromagnetic and weak interactions disappears. Such conditions existed during the very early universe.

External links

Bibliography

1: Daniel Kleppner and Robert J. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
2: Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Vol. I, New Millennium ed., Basic Books, 2010.
3: Elliott H. Lieb, “The stability of matter: from atoms to stars,” Bulletin of the American Mathematical Society, 22(1), 1–49, 1990.
4: Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, W. H. Freeman, 1973.
5: Yvonne Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, 2009.
6: Wolfgang K. H. Panofsky and Melba Phillips, Classical Electricity and Magnetism, 2nd ed., Dover Publications, 2005.
7: Anthony Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press, 2010.
8: Dave Goldberg, The Standard Model in a Nutshell, Princeton University Press, 2017.
9: Walter Greiner and Berndt Müller, Gauge Theory of Weak Interactions, 4th ed., Springer, 2009.
10: Ruth Durrer, The Cosmic Microwave Background, Cambridge University Press, 2008.
11: Contemporary Physics Education Project, Standard Model of Particles and Interactions, 2000.

Source and license

This PhysicsLibrary entry is an adapted and reorganized derivative of the Wikipedia article “Force”, revision 1369185740 (13 August 2026). The source article is available under the Creative Commons Attribution–ShareAlike 4.0 International License. The present entry has been rewritten and reorganized for PhysicsLibrary rather than copied verbatim.

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