# Node GravityNode

Defined in `mime.nodes.body._gravity.GravityNode`.

Algorithm `MIME-NODE-113`.

Weight and buoyancy on a body in a medium at rest: the wrench about the centre of mass, with the buoyancy's couple where the centre of buoyancy is off it.

## Governing equations

$$
\mathbf{F} = m \, \mathbf{g} - \rho_{f} V \, \mathbf{g}, \quad \boldsymbol{\tau} = R(\mathbf{q}) \, \mathbf{c}_{b} \times \left( -\rho_{f} V \, \mathbf{g} \right)
$$

## Discretisation

none declared; a node whose order depends on its construction declares it with the `discretization_order()` hook, which is read from a built node.

## Assumptions

- The medium is at rest and its density uniform: a flow that carries its own hydrostatic load would count the buoyancy twice
- The weight acts at the centre of mass and the buoyancy at the centre of buoyancy, both fixed in the body

## Limitations

none declared

## Declared regimes (not evidenced here)

none declared

## References

[@TatumClassicalMechanics]

## Symbols

| Symbol | Kind | Name | Units | Implemented by |
|---|---|---|---|---|
| $\mathbf{F}$ | state | `wrench` | N; N m | `mime.nodes.body._gravity.GravityNode.wrench` |
| $\boldsymbol{\tau}$ | state | `wrench` | N; N m | `mime.nodes.body._gravity.GravityNode.wrench` |
| $m$ | parameter | `mass` | kg | `mime.nodes.body._gravity.GravityNode.wrench` |
| $\mathbf{g}$ | parameter | `gravity` | m/s^2 | `mime.nodes.body._gravity.GravityNode.wrench` |
| $\rho_{f}$ | parameter | `fluid_density` | kg/m^3 | `mime.nodes.body._gravity.GravityNode.wrench` |
| $V$ | parameter | `buoyant_volume` | m^3 | `mime.nodes.body._gravity.GravityNode.wrench` |
| $\mathbf{c}_{b}$ | parameter | `centre_of_buoyancy` | m | `mime.nodes.body._gravity.GravityNode.wrench` |

## Built by

- [`gravity`](../entries/interaction/gravity.md) 0.1.0