# Body force

A body force is a force that acts throughout the volume of a body. Forces due to gravity, electric fields and magnetic fields are examples of body forces. Body forces contrast with contact forces or surface forces which are exerted to the surface of an object.

Normal forces and shear forces between objects are surface forces as they are exerted to the surface of an object. All cohesive surface attraction and contact forces between objects are also considered as surface forces.

Fictitious forces such as the centrifugal force, Euler force, and the Coriolis effect are also examples of body forces.

## Definition

### Qualitative

A body force is simply a type of force, and so it has the same dimensions as force, [M][L][T]−2. However, it is often convenient to talk about a body force in terms of either the force per unit volume or the force per unit mass. If the force per unit volume is of interest, it is referred to as the 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 gradients and conductive and convective heat transmission are not body forces as they require contact between systems to exist. Radiation heat transfer, on the other hand, is a perfect example of a body force.

More examples of common body forces include;

Fictitious forces (or inertial forces) can be viewed as body forces. Common inertial forces are,

However, fictitious forces are not actually forces. Rather they are corrections to Newton's second law when it is formulated in an accelerating reference frame.

### Quantitative

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

${\displaystyle \mathbf {F} _{\mathrm {body} }=\int \limits _{V}\rho \mathbf {g} (\mathbf {r} )\mathrm {d} V\,,}$

where dV is an infinitesimal volume element, ρ is the mass density, and g is the external body force density field acting on the system.

## Acceleration

Like any other force, a body force will cause an object to accelerate. For a non-rigid object, Newton's second law applied to a small volume element is

${\displaystyle \mathbf {f} (\mathbf {r} )=\rho (\mathbf {r} )\mathbf {g} (\mathbf {r} )}$ ,

where ρ(r) is the mass density of the substance, ƒ the force density, and a(r) all at point r.

In the case of gravity on a planet surface, g(r) is simply the approximately constant and uniform gravitational field, like on Earth where:

${\displaystyle g=9.81{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}$