# Body force

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{{Short description|Force which acts throughout the volume of a body}}
{{More sources needed|date=March 2007}}

In [physics](/source/physics), a '''body force''' is a force that acts throughout the volume of a body.<ref>Springer site - Book 'Solid mechanics'. [https://link.springer.com/chapter/10.1007/978-90-481-3809-8_7 preview paragraph 'Body forces'.]</ref> Forces due to [gravity](/source/gravity), [electric fields](/source/electrostatics) and [magnetic fields](/source/magnetic_fields) are examples of body forces. Body forces contrast with ''[contact force](/source/contact_force)s'' or ''[surface force](/source/surface_force)s'' which are exerted to the surface of an object.
[Fictitious forces](/source/Fictitious_forces) such as the [centrifugal force](/source/centrifugal_force), [Euler force](/source/Euler_force), and the [Coriolis effect](/source/Coriolis_effect) are other examples of body forces.

==Definition==

===Qualitative===
A body force is simply a type of force, and so it has the same [dimensions](/source/dimensional_analysis) as [force](/source/force), [M][L][T]<sup>−2</sup>. However, it is often convenient to talk about a body force in terms of either the [force](/source/force) per unit [volume](/source/volume) or the force per unit [mass](/source/mass). If the [force](/source/force) per unit [volume](/source/volume) is of interest, it is referred to as the [force density](/source/force_density) throughout the system.

A body force is distinct from a contact force in that the force does not require contact for transmission. Thus, common forces associated with [pressure gradient](/source/pressure_gradient)s and [conductive](/source/conductive) and [convective](/source/convective) heat transmission are not body forces as they require contact between systems to exist.

More examples of common body forces include;
*[Gravity](/source/Gravity),
*[Electric force](/source/Electric_force)s acting on an object charged throughout its volume,
*[Magnetic force](/source/Magnetic_force)s acting on currents within an object, such as the braking force that results from [eddy currents](/source/eddy_currents),

[Fictitious force](/source/Fictitious_force)s (or inertial forces) can be viewed as body forces. Common inertial forces are,

*[Centrifugal force](/source/Centrifugal_force),
*[Coriolis force](/source/Coriolis_force),
*[Euler force](/source/Euler_force) (or transverse force), which occurs in a rotating reference frame when the rate of rotation of the frame is changing

However, fictitious forces are not actually forces. Rather they are corrections to [Newton's second law](/source/Newton's_second_law) when it is formulated in an [accelerating reference frame](/source/Non-inertial_reference_frame). (Gravity can also be considered a fictitious force in the context of [General Relativity](/source/General_Relativity).)

===Quantitative===

The body force density is defined so that the [volume integral](/source/volume_integral) (throughout a volume of interest) of it gives the total force acting throughout the body;

:<math>\mathbf{F}_{\mathrm{body}} =  \int\limits_{V}\mathbf{f}(\mathbf{r}) \mathrm{d} V \,,</math>

where d''V'' is an [infinitesimal](/source/Differential_(infinitesimal)) [volume element](/source/volume_element), and '''f''' is the ''external body force density field'' acting on the system.

==Acceleration==
{{main|Newton's second law}}
Like any other force, a body force will cause an object to accelerate. For a non-rigid object, [Newton's second law](/source/Newton's_second_law) applied to a small volume element is

:<math>\mathbf{f} (\mathbf{r})=\rho (\mathbf{r})\mathbf{a} (\mathbf{r})</math>,

where ''ρ''('''r''') is the [mass density](/source/mass_density) of the substance, '''ƒ''' the force density, and '''a'''('''r''') is [acceleration](/source/acceleration), all at point '''r'''.

== The case of gravity ==
In the case of a body in the [gravitational field](/source/gravitational_field) on a planet surface, '''a'''('''r''') is nearly constant ('''g''') and uniform. Near the Earth

:<math>g = 9.81 \text{ }\mathrm{ms}^{-2}</math>.<ref>{{Cite web |title=A force that shapes our planet |url=https://www.esa.int/Applications/Observing_the_Earth/FutureEO/GOCE/A_force_that_shapes_our_planet |access-date=2025-10-16 |website=www.esa.int |language=en}}</ref>

In this case simply
:<math>\mathbf{F}_{\mathrm{body}} =  \int\limits_{V}\rho (\mathbf{r})\mathbf{g}\mathrm{d} V = \int\limits_{V}\rho (\mathbf{r})\mathrm{d} V \cdot \mathbf{g} = m \mathbf{g}</math>

where ''m'' is the mass of the body.

==See also==
*[Action at a distance](/source/Action_at_a_distance)
*[Fictitious force](/source/Fictitious_force)
*[Force density](/source/Force_density)
*[Non-contact force](/source/Non-contact_force)
*[Normal force](/source/Normal_force)
*[Surface force](/source/Surface_force)

==References==
{{Reflist}}

{{DEFAULTSORT:Body Force}}
Category:Force

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Adapted from the Wikipedia article [Body force](https://en.wikipedia.org/wiki/Body_force) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Body_force?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.