### Body force

A **body force** is a force that acts throughout the volume of a body, in contrast to contact forces. Gravity and electromagnetic forces are examples of body forces. Inertial spin forces such as the Centrifugal force, Euler force, and the Coriolis effect are also examples of body forces.

This can be put into contrast to the classical definition of surface forces which are supposed to be exerted to the surface of an object. Shear forces and normal forces occurring in physical and engineering circumstances are supposed to be **surface forces** and exerted to the surface of an object. All cohesive surface attraction and contact forces between objects are also considered as surface 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;

- Gravity,
- Electric forces acting on an object charged throughout its volume,
- Magnetic forces acting on currents within an object, such as the braking force that results from eddy currents,

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

- Centrifugal force,
- Coriolis force,
- 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 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;

- \mathbf{F}_{\mathrm{body}} = \int\limits_{V} \mathbf{f}(\mathbf{r}) \mathrm{d} V \,,

where d*V* is an infinitesimal volume element and **f** is the force density within 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

- \mathbf{f} (\mathbf{r})=\rho (\mathbf{r})\mathbf{a} (\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, **a**(**r**) is simply the acceleration due to gravity, that is the gravitational field **g**.