 Line of action

For a nonzero force applied at position ${\mathbf{r}}_0$, its line of action is the straight line through that point parallel to the force:The force direction and line position together determine its torque about a chosen origin. They contain more mechanical information than the magnitude alone.

The moment about the origin is $\mathbf{\tau }={\mathbf{r}}_0\times \mathbf{F}$. Moving the application point by a displacement parallel to $\mathbf{F}$ leaves the moment unchanged because $\Delta \mathbf{r}\times \mathbf{F}=0$. The principle of transmissibility uses this equivalence for the external motion or equilibrium of an ideal rigid body. The lever-arm formulation expresses the same geometry: the torque magnitude is $Fd$, where $d$ is perpendicular distance from the origin to the line.^[ref]

For a constructed two-dimensional example, a force $(0,10)\mathrm{N}$ acts at $(0.30,0.20)\mathrm{m}$. Its line is $x=0.30\mathrm{m}$ and its moment is ${\tau }_z=0.30(10)=3.0\mathrm{N}\mathrm{m}$. Moving the application point to $(0.30,0.80)\mathrm{m}$ changes neither force nor moment. Moving it instead to $x=0.50\mathrm{m}$ gives $5.0\mathrm{N}\mathrm{m}$, so the unchanged force magnitude does not imply the same rigid-body effect.

An equal and opposite pair on different parallel lines forms a couple: the resultant force is zero but a moment remains. Consequently, not every loading system can be replaced by one force with one line of action. A free-body diagram should retain the relevant forces and couples.

Transmissibility is an external rigid-body statement. Applying a load at a different location on a deformable object can change internal stress, deformation and local contact even if the resultant and moment are preserved. A static load-cell comparison and a deformation measurement can therefore answer different questions about apparently equivalent loading arrangements.

The term "line of action" can refer to different concepts depending on the context in which it is used, including physics, biomechanics, and the field of animation or art. Here are a few interpretations: 1. **Physics and Mechanics**: In physics, the line of action refers to the direction along which a force acts on an object. It is an imaginary line that extends infinitely in both directions along the direction of the force vector.