= Line of action
{wiki}
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:
$$
\mathcal L=\left\{\mathbf r_0+s\frac{\mathbf F}{\|\mathbf F\|}:s\in\mathbb R\right\}.
$$
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 $\boldsymbol\tau=\mathbf r_0\times\mathbf F$. 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.<University Physics Volume 1 by William Moebs and others>[Section 10.6, torque vector, line of action and perpendicular lever arm]{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 $\tau_z=0.30(10)=3.0\,\mathrm{N\,m}$. 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 $5.0\,\mathrm{N\,m}$, 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 (mechanics)>: 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.
Back to article page
