Physics equations/Equations/Rotational and linear motion analogy

The following table refers to rotation of a rigid body about a fixed axis: $$\mathbf s$$ is arclength, $$\mathbf r$$ is the distance from the axis to any point, and $$\mathbf{a}_\mathbf{t}$$ is the tangential acceleration, which is the component of the acceleration that is parallel to the motion. In contrast, the centripetal acceleration, $$\mathbf{a}_\mathbf{c}=v^2/r=\omega^2 r$$, is perpendicular to the motion. The component of the force parallel to the motion, or equivalently, perpendicular, to the line connecting the point of application to the axis is $$\mathbf{F}_\perp$$. The sum is over $$\mathbf j \ = 1 \ \mathbf{to}\ N$$ particles or points of application.