Force

== Concepts derived from force ==
=== Rotation and torque ===
[[File:Torque animation.gif|frame|right|Relationship between force (''F''), torque (''τ''), and [[angular momentum|momentum]] vectors (''p'' and ''L'') in a rotating system.]]
{{main|Torque}}
Forces that cause extended objects to rotate are associated with [[torque]]s. Mathematically, the torque of a force <math> \vec{F}</math> is defined relative to an arbitrary reference point as the [[cross product]]:
<math display="block" qid=Q48103>\vec{\tau} = \vec{r} \times \vec{F},</math>
where <math> \vec{r}</math> is the [[position vector]] of the force application point relative to the reference point.<ref name="openstax-university-physics"/>{{rp|497}}

Torque is the rotation equivalent of force in the same way that [[angle]] is the rotational equivalent for [[position (vector)|position]], [[angular velocity]] for [[velocity]], and [[angular momentum]] for [[momentum]]. As a consequence of Newton's first law of motion, there exists [[rotational inertia]] that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous [[angular acceleration]] of the rigid body:
<math display="block">\vec{\tau} = I\vec{\alpha},</math>
where
* <math>I</math> is the [[moment of inertia]] of the body
* <math> \vec{\alpha}</math> is the angular acceleration of the body.<ref name="openstax-university-physics"/>{{rp|502}}

This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the [[Moment of inertia tensor|tensor]] that, when properly analyzed, fully determines the characteristics of rotations including [[precession]] and [[nutation]].<ref name=":0" />{{Rp|pages=96–113}}

Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:<ref>{{cite web |last=Nave |first=Carl Rod |title=Newton's 2nd Law: Rotation |work=HyperPhysics |publisher=University of Guelph |url=http://hyperphysics.phy-astr.gsu.edu/HBASE/n2r.html |access-date=2013-10-28}}</ref>
<math display="block">\vec{\tau} = \frac{\mathrm{d}\vec{L}}{\mathrm{dt}},</math>
where <math> \vec{L}</math> is the angular momentum of the particle.

Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques,<ref>{{cite web |last=Fitzpatrick |first=Richard |title=Newton's third law of motion |date=2007-01-07 |url=http://farside.ph.utexas.edu/teaching/336k/lectures/node26.html |access-date=2008-01-04}}</ref> and therefore also directly implies the [[conservation of angular momentum]] for closed systems that experience rotations and [[revolution (geometry)|revolution]]s through the action of internal torques.

=== Yank ===
The '''yank''' is defined as the rate of change of force<ref name=jazar>{{Cite book |last=Jazar |first=Reza N. |title=Advanced dynamics: rigid body, multibody, and aerospace applications |date=2011 |publisher=Wiley |isbn=978-0-470-39835-7 |location=Hoboken, N.J}}</ref>{{rp|131}}
:<math>\vec Y = \frac{d\vec F}{dt}</math>
The term is used in biomechanical analysis,<ref>{{Cite journal |last=Lin |first=David C. |last2=McGowan |first2=Craig P. |last3=Blum |first3=Kyle P. |last4=Ting |first4=Lena H. |date=2019-09-12 |title=Yank: the time derivative of force is an important biomechanical variable in sensorimotor systems |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6765171/ |journal=The Journal of Experimental Biology |volume=222 |issue=18 |pages=jeb180414 |doi=10.1242/jeb.180414 |issn=0022-0949 |pmc=6765171 |pmid=31515280}}</ref> athletic assessment<ref>{{Cite journal |last=Harry |first=John R. |last2=Barker |first2=Leland A. |last3=Tinsley |first3=Grant M. |last4=Krzyszkowski |first4=John |last5=Chowning |first5=Luke D. |last6=McMahon |first6=John J. |last7=Lake |first7=Jason |date=2021-05-05 |title=Relationships among countermovement vertical jump performance metrics, strategy variables, and inter-limb asymmetry in females |url=https://www.tandfonline.com/doi/full/10.1080/14763141.2021.1908412 |journal=Sports Biomechanics |language=en |pages=1–19 |doi=10.1080/14763141.2021.1908412 |issn=1476-3141}}</ref> and robotic control.<ref>{{Cite journal |last=Rosendo |first=Andre |last2=Tanaka |first2=Takayuki |last3=Kaneko |first3=Shun’ichi |date=2012-04-20 |title=A Yank-Based Variable Coefficient Method for a Low-Powered Semi-Active Power Assist System |url=https://www.fujipress.jp/jrm/rb/robot002400020291/ |journal=Journal of Robotics and Mechatronics |volume=24 |issue=2 |pages=291–297 |doi=10.20965/jrm.2012.p0291|doi-access=free }}</ref> The second (called "tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.<ref name=jazar/>

=== Kinematic integrals ===
{{main|Impulse (physics)|l1=Impulse|Mechanical work|Power (physics)}}
Forces can be used to define a number of physical concepts by [[integration (calculus)|integrating]] with respect to [[kinematics|kinematic variables]]. For example, integrating with respect to time gives the definition of [[Impulse (physics)|impulse]]:<ref>{{Cite book
|title=Engineering Mechanics
|first1=Russell C.
|last1=Hibbeler
|publisher=Pearson Prentice Hall
|year=2010
|edition=12th
|isbn=978-0-13-607791-6
|page=222
}}</ref>
<math display="block">\vec{J}=\int_{t_1}^{t_2}{\vec{F} \, \mathrm{d}t},</math>
which by Newton's Second Law must be equivalent to the change in momentum (yielding the [[Impulse momentum theorem]]).

Similarly, integrating with respect to position gives a definition for the [[work (physics)|work done]] by a force:<ref name=FeynmanVol1/>{{rp|((13-3))}}
<math display="block" qid=Q42213>W= \int_{\vec{x}_1}^{\vec{x}_2} {\vec{F} \cdot {\mathrm{d}\vec{x}}},</math>
which is equivalent to changes in [[kinetic energy]] (yielding the [[work energy theorem]]).<ref name=FeynmanVol1/>{{rp|((13-3))}}

[[Power (physics)|Power]] ''P'' is the rate of change d''W''/d''t'' of the work ''W'', as the [[trajectory]] is extended by a position change <math> d\vec{x}</math> in a time interval d''t'':<ref name=FeynmanVol1/>{{rp|((13-2))}}
<math display="block"> \mathrm{d}W = \frac{\mathrm{d}W}{\mathrm{d}\vec{x}} \cdot \mathrm{d}\vec{x} = \vec{F} \cdot \mathrm{d}\vec{x},</math>
so
<math display="block">P = \frac{\mathrm{d}W}{\mathrm{d}t} = \frac{\mathrm{d}W}{\mathrm{d}\vec{x}} \cdot \frac{\mathrm{d}\vec{x}}{\mathrm{d}t} = \vec{F} \cdot \vec{v},
</math>
with <math qid=Q11465>\vec{v} = \mathrm{d}\vec{x}/\mathrm{d}t</math> the [[velocity]].

=== Potential energy ===
{{main|Potential energy}}
Instead of a force, often the mathematically related concept of a [[potential energy]] field is used. For instance, the gravitational force acting upon an object can be seen as the action of the [[gravitational field]] that is present at the object's location. Restating mathematically the definition of energy (via the definition of [[Mechanical work|work]]), a potential [[scalar field]] <math>U(\vec{r})</math> is defined as that field whose [[gradient]] is equal and opposite to the force produced at every point:
<math display="block">\vec{F}=-\vec{\nabla} U.</math>

Forces can be classified as [[Conservative force|conservative]] or nonconservative. Conservative forces are equivalent to the gradient of a [[potential]] while nonconservative forces are not.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Conservation ===
{{main|Conservative force}}
A conservative force that acts on a [[closed system]] has an associated mechanical work that allows energy to convert only between [[kinetic energy|kinetic]] or [[potential energy|potential]] forms. This means that for a closed system, the net [[mechanical energy]] is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space,<ref>{{cite web
|last=Singh
|first=Sunil Kumar
|title=Conservative force
|work=Connexions
|date=2007-08-25
|url=http://cnx.org/content/m14104/latest/
|access-date=2008-01-04}}</ref> and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the [[contour map]] of the elevation of an area.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

Conservative forces include [[gravity]], the [[Electromagnetism|electromagnetic]] force, and the [[Hooke's law|spring]] force. Each of these forces has models that are dependent on a position often given as a [[radius|radial vector]] <math> \vec{r}</math> emanating from [[spherical symmetry|spherically symmetric]] potentials.<ref>{{cite web
|last=Davis
|first=Doug
|title=Conservation of Energy
|work=General physics
|url=http://www.ux1.eiu.edu/~cfadd/1350/08PotEng/ConsF.html
|access-date=2008-01-04}}</ref> Examples of this follow:

For gravity:
<math display="block">\vec{F}_\text{g} = - \frac{G m_1 m_2}{r^2} \hat{r},</math>
where <math>G</math> is the [[gravitational constant]], and <math>m_n</math> is the mass of object ''n''.

For electrostatic forces:
<math display="block">\vec{F}_\text{e} = \frac{q_1 q_2}{4 \pi \varepsilon_{0} r^2} \hat{r},</math>
where <math>\varepsilon_{0}</math> is [[Permittivity|electric permittivity of free space]], and <math>q_n</math> is the [[electric charge]] of object ''n''.

For spring forces:
<math display="block">\vec{F}_\text{s} = - k r \hat{r},</math>
where <math>k</math> is the [[spring constant]].<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of [[Microstate (statistical mechanics)|microstates]]. For example, static friction is caused by the gradients of numerous electrostatic potentials between the [[atom]]s, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other [[contact force]]s, [[Tension (physics)|tension]], [[Physical compression|compression]], and [[drag (physics)|drag]]. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with [[statistical mechanics]]. In macroscopic closed systems, nonconservative forces act to change the [[internal energy|internal energies]] of the system, and are often associated with the transfer of heat. According to the [[Second law of thermodynamics]], nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as [[entropy]] increases.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />