Why is force a localized vector and not a free vector?

A force is a vector acting along a line in space. It may act anywhere along the line and the end result will be the same (2nd paragraph).

The reason pure forces act on lines is that if they move away from the line there will be a torque applied also and the resulting motion will not be the same. The line of action of a force is the locus of the point where no net torque exists for this force.

Given a set of forces and torques specified at an arbitrary point, the location (and direction) of the line of action can be found. This line is unique, meaning that only one exists for each set of force and torque components.

In addition, these principles are dual to the idea of rotations about axes. Rotational motion needs to have the location of the rotation axis specified to be fully defined. Any other point away from the rotation axis will exhibit linear motion.

To fully specify the loading on a rigid body you need one of these two pieces of information:

• A force vector $\vec{F}$ and a point which the line of action goes through $\vec{r}$. The equivalent torque at the origin is $\vec{\tau}=\vec{r} \times \vec{F}$.
• A force vector $\vec{F}$ and the equivalent torque at the origin $\vec{\tau}$. The location of the line of action closest to the origin is $\vec{r} = \frac{\vec{F}\times \vec{\tau}}{\|\vec{F}\|^2}$.

To fully specify the motion of a rigid body you need one of these two pieces of information:

• The rotation vector $\vec{\omega}$ and a point which the rotation axis goes through $\vec{r}$. The linear velocity at the origin is $\vec{v}=\vec{r} \times \vec{\omega}$.
• The rotation vector $\vec{\omega}$ and the linear velocity at the origin $\vec{v}$. The location of the rotation axis closest to the origin is $\vec{r} = \frac{\vec{\omega}\times \vec{v}}{\|\vec{\omega}\|^2}$.

Quantities that require one free vector and one line vector to fully specify something are called screws (or screw vectors). All screws transform from one place to another with the same laws. Some quantities that are screws are:

• Small motions (infinitesimal movement)
• Rigid Body Motion (velocity)
• Momenta (linear and angular)
• Impulses and Contacts
• Forces & torques
• Spatial Acceleration (motion of entire rigid body)

Quantities that are not screws are:

• Body Rotations (large angles are not vectors)
• Material Accelerations (motion of each particle on a body)

_{Note $\times$ is the vector cross product, and $\cdot$ the vector dot product}