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.

Analogy between Linear Motion and Rotational motion

Linear motion	Rotational motion	Defining equation
Displacement = $\mathbf {x}$	Angular displacement = $\theta$	$\theta =\mathbf {x} /\mathbf {r}$
Velocity = $\mathbf {v}$	Angular velocity = $\omega$	$\omega =\mathbf {d} \theta /\mathbf {dt} =\mathbf {v} /\mathbf {r}$
Acceleration = $\mathbf {a}$	Angular acceleration = $\alpha$	$\alpha =\mathbf {d} \omega /\mathbf {dt} =\mathbf {a_{\mathbf {t} }} /\mathbf {r}$
Mass = $\mathbf {m}$	Moment of Inertia = $\mathbf {I}$	$\mathbf {I} =\sum \mathbf {m_{j}} \mathbf {r_{j}} ^{2}$
Force = $\mathbf {F} =\mathbf {m} \mathbf {a}$	Torque = $\tau =\mathbf {I} \alpha$	$\tau =\sum \mathbf {r_{j}} \mathbf {F} _{\perp }\mathbf {_{j}} =\sum \mathbf {r_{\perp j}} \mathbf {F} _{\mathbf {_{j}} }$
Momentum= $\mathbf {p} =\mathbf {m} \mathbf {v}$	Angular momentum= $\mathbf {L} =\mathbf {I} \omega$	$\mathbf {L} =\sum \mathbf {r_{j}} \mathbf {p} \mathbf {_{j}}$
Kinetic energy = ${\frac {1}{2}}\mathbf {m} \mathbf {v} ^{2}$	Kinetic energy = ${\frac {1}{2}}\mathbf {I} \omega ^{2}$	${\frac {1}{2}}\sum \mathbf {m_{j}} \mathbf {v_{j}} ^{2}={\frac {1}{2}}\sum \mathbf {m_{j}} \mathbf {r_{j}} ^{2}\omega ^{2}$