Force

In physics, force is a vector quantity used to describe interactions that can change a body's motion or deform it. In classical mechanics, the net external force on a system equals the rate of change of its momentum. For a body of constant mass, this becomes the familiar relation $\mathbf F_{\mathrm{net}}=m\mathbf a$. Individual forces may balance, so their presence does not necessarily imply acceleration. Force has both magnitude and direction and is measured in newtons. (openstax.org)

Mathematical description and units

Forces combine by vector addition. If several interactions act on a body, their resultant is

Resolving this equation into components allows motion along different coordinate axes to be analyzed separately. Equal forces in opposite directions have a zero resultant; forces at other angles generally do not. A force's point of application also matters for the rotation of an extended body, even when its effect on translational motion is unchanged. (openstax.org)

The International System of Units assigns force the derived unit newton, symbol N:

One newton is therefore the net force that gives a one-kilogram body an acceleration of one metre per second squared. Force has dimensions $MLT^{-2}$, rather than the dimensions of mass or energy. (nist.gov)

Newtonian dynamics

Newton’s laws of motion, formulated by Isaac Newton, establish the central role of force in Newtonian mechanics. The first law identifies motion with constant velocity as the state maintained when the net force is zero. Thus, continued motion does not require a continued net force; a change in motion does. These laws describe ordinary macroscopic motion accurately within their nonrelativistic domain. (openstax.org)

In an inertial reference frame, the second law can be written for a system containing a fixed collection of matter as

With constant mass and $\mathbf p=m\mathbf v$, it reduces to $m\mathbf a$. The momentum form also connects force to impulse: integrating net force over a time interval gives the change in momentum. A short collision can therefore produce a substantial momentum change through a large force acting briefly. (openstax.org)

Force analysis commonly uses a diagram that isolates a selected body and represents the external forces acting on it with arrows. Identifying the body is essential: a support force on an object and the force that object exerts on its support act on different bodies. For an object resting on a horizontal table, its upward support force can balance its downward gravitational force. (openstax.org)

Everyday forces and circular motion

Macroscopic force names usually describe an interaction's circumstances rather than a separate fundamental mechanism. A normal force acts perpendicular to a contact surface; tension transmits a pull through a stretched rope or cable; and friction opposes relative sliding or its tendency at a contact. An elastic restoring force arises when a material is deformed. For an ideal spring within its linear range, the restoring force is proportional to displacement and directed toward equilibrium. (openstax.org)

Weight, in the gravitational sense, is the force exerted on a body by gravity. Near Earth's surface its magnitude is approximately $mg$, where $g$ is the local gravitational acceleration. Weight is distinct from mass, and a support force need not equal weight—for example, when the body accelerates or rests on an inclined surface. (openstax.org)

“Centripetal force” names the inward resultant needed for circular motion, not an additional kind of interaction. In uniform circular motion its magnitude is $mv^2/r$. Gravity, tension, friction, or a combination of forces can supply this resultant. Because the force changes the direction of velocity, acceleration occurs even though speed remains constant. (openstax.org)

Work, potential energy, and torque

A force transfers energy through work when its point of application undergoes displacement with a component along the force. For a constant force,

for a varying force, work is the integral of $\mathbf F\cdot d\mathbf r$ along the path. A force perpendicular to displacement does no work, as illustrated by the inward force in uniform circular motion. (openstax.org)

A conservative force does work that depends only on the endpoints, permitting an associated potential energy. Dissipative forces such as sliding friction generally have path-dependent work. These distinctions connect force-based descriptions of motion with energy-based descriptions. (openstax.org)

The rotational effect of a force is measured by torque:

Here $\mathbf r$ runs from the chosen origin to the point of application. The perpendicular lever arm explains why applying a force farther from a door's hinge produces a greater turning effect. (openstax.org)

Fundamental interactions and modern descriptions

Four fundamental interactions are recognized: gravity, electromagnetism, the strong interaction, and the weak interaction. Electromagnetic interactions underlie electrical and magnetic phenomena; the strong interaction binds quarks and contributes to nuclear binding; and the weak interaction enables processes such as radioactive beta decay. The Standard Model describes the electromagnetic, strong, and weak interactions but does not include gravity. (home.web.cern.ch)

In modern particle physics, interactions are described through quantum field theory, rather than simply through classical pushes and pulls. General relativity gives gravity a different interpretation: freely falling bodies follow the straightest possible paths through curved spacetime. Tidal effects reflect this curvature, while a gravitational force remains a useful approximation in Newtonian calculations. (wwwcompass.cern.ch)