Force

=== Continuum mechanics ===
[[File:Stokes sphere.svg|thumb|upright|When the drag force (<math>F_\text{d}</math>) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (<math>F_\text{g}</math>), the object reaches a state of [[#Dynamic equilibrium|dynamic equilibrium]] at [[terminal velocity]].]]
{{main|Pressure|Drag (physics)|Stress (mechanics)}}
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized [[point particle]]s rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of [[continuum mechanics]] describe the way forces affect the material. For example, in extended [[fluid mechanics|fluids]], differences in [[pressure]] result in forces being directed along the pressure [[gradient]]s as follows:
<math display="block">\frac{\vec{F}}{V} = - \vec{\nabla} P,</math>

where <math>V</math> is the volume of the object in the fluid and <math>P</math> is the [[scalar function]] that describes the pressure at all locations in space. Pressure gradients and differentials result in the [[buoyancy|buoyant force]] for fluids suspended in gravitational fields, winds in [[atmospheric science]], and the [[lift (physics)|lift]] associated with [[aerodynamics]] and [[flight]].<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

A specific instance of such a force that is associated with [[dynamic pressure]] is fluid resistance: a body force that resists the motion of an object through a fluid due to [[viscosity]]. For so-called "[[Drag (physics)#Very low Reynolds numbers – Stokes' drag|Stokes' drag]]" the force is approximately proportional to the velocity, but opposite in direction:
<math display="block" qid=Q824561>\vec{F}_\mathrm{d} = - b \vec{v}, </math>
where:
* <math>b</math> is a constant that depends on the properties of the fluid and the dimensions of the object (usually the [[Cross section (geometry)|cross-sectional area]]), and
* <math> \vec{v}</math> is the velocity of the object.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

More formally, forces in [[continuum mechanics]] are fully described by a [[Stress (mechanics)|stress]] [[tensor]] with terms that are roughly defined as
<math display="block" qid=Q206175>\sigma = \frac{F}{A},</math>
where <math>A</math> is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the [[matrix diagonal]]s of the tensor) as well as [[Shear stress|shear]] terms associated with forces that act [[Parallel (geometry)|parallel]] to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all [[strain (physics)|strains]] (deformations) including also [[tensile stress]]es and [[compression (physical)|compressions]].<ref name=uniphysics_ch2>{{cite book|title=University Physics |last1=Sears |first1=Francis W. |last2=Zemansky |first2=Mark W. |last3=Young |first3=Hugh D. |author-link1=Francis Sears |author-link2=Mark Zemansky |author-link3=Hugh D. Young |title-link=University Physics |pages=18–38 |publisher=Addison-Wesley |edition=6th |year=1982 |isbn=0-201-07199-1}}</ref><ref name=Kleppner>{{cite book |last1=Kleppner |first1=Daniel |last2=Kolenkow |first2=Robert J. |title=An Introduction to Mechanics|year=2014|publisher=Cambridge University Press|location=Cambridge|isbn=978-0521198110|edition=2nd|chapter=Chapter 3: Forces and equations of motion|chapter-url=https://archive.org/details/KleppnerD.KolenkowR.J.IntroductionToMechanics2014/page/n102}}</ref>{{rp|133–134}}<ref name=FeynmanVol2>{{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |title=The Feynman lectures on physics. Vol. II: Mainly electromagnetism and matter |year=2010 |publisher=Basic Books |location=New York |isbn=978-0465024940 |edition=New millennium |title-link=The Feynman Lectures on Physics |author-link1=Richard Feynman |author-link2=Robert B. Leighton |author-link3=Matthew Sands}}</ref>{{rp|((38-1–38-11))}}