quality:correction, structural:scholar_edition, structural:reformatting, stylistic:style_transfer, information:extraction
Rigid body

If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis k ^ {\displaystyle \mathbf {\hat {k}} } parallel to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia. The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles.

If a system of n {\displaystyle n} particles, P i, i = 1, …, n {\displaystyle P_{i},i=1,\dots,n}, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point R {\displaystyle \mathbf {R} }, and absolute velocities v i {\displaystyle \mathbf {v} _{i}}: Δ r i = r i − R, v i = ω × (r i − R) + V = ω × Δ r i + V, {\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {R},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {V} ={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V},\end{aligned}}} where ω {\displaystyle {\boldsymbol {\omega }}} is the angular velocity of the system and V {\displaystyle \mathbf {V} } is the velocity of R {\displaystyle \mathbf {R} }.

For planar movement the angular velocity vector is directed along the unit vector k {\displaystyle \mathbf {k} } which is perpendicular to the plane of movement. Introduce the unit vectors e i {\displaystyle \mathbf {e} _{i}} from the reference point R {\displaystyle \mathbf {R} } to a point r i {\displaystyle \mathbf {r} _{i}}, and the unit vector t ^ i = k ^ × e ^ i {\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}, so e ^ i = Δ r i Δ r i, k ^ = ω ω, t ^ i = k ^ × e ^ i, v i = ω × Δ r i + V = ω k ^ × Δ r i e ^ i + V = ω Δ r i t ^ i + V {\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\Delta \mathbf {r} _{i}}{\Delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} =\omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {V} =\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \end{aligned}}}

This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.

Note on the cross product: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.