# Template:Physeq1/RotationalLinearAnalogyTable

The following table refers to rotation of a rigid body about a fixed axis: ${\displaystyle \mathbf {s} }$ is arclength, ${\displaystyle \mathbf {r} }$ is the distance from the axis to any point, and ${\displaystyle \mathbf {a} _{\mathbf {t} }}$ is the tangential acceleration, which is the component of the acceleration that is parallel to the motion. In contrast, the centripetal acceleration, ${\displaystyle \mathbf {a} _{\mathbf {c} }=v^{2}/r=\omega ^{2}r}$, 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 ${\displaystyle \mathbf {F} _{\perp }}$. The sum is over ${\displaystyle \mathbf {j} \ =1\ \mathbf {to} \ N}$ particles or points of application.
Displacement = ${\displaystyle \mathbf {x} }$ Angular displacement = ${\displaystyle \theta }$ ${\displaystyle \theta =\mathbf {s} /\mathbf {r} }$
Velocity = ${\displaystyle \mathbf {v} }$ Angular velocity = ${\displaystyle \omega }$ ${\displaystyle \omega =\mathbf {d} \theta /\mathbf {dt} =\mathbf {v} /\mathbf {r} }$
Acceleration = ${\displaystyle \mathbf {a} }$ Angular acceleration = ${\displaystyle \alpha }$ ${\displaystyle \alpha =\mathbf {d} \omega /\mathbf {dt} =\mathbf {a_{\mathbf {t} }} /\mathbf {r} }$
Mass = ${\displaystyle \mathbf {m} }$ Moment of Inertia = ${\displaystyle \mathbf {I} }$ ${\displaystyle \mathbf {I} =\sum \mathbf {m_{j}} \mathbf {r_{j}} ^{2}}$
Force = ${\displaystyle \mathbf {F} =\mathbf {m} \mathbf {a} }$ Torque = ${\displaystyle \tau =\mathbf {I} \alpha }$ ${\displaystyle \tau =\sum \mathbf {r_{j}} \mathbf {F} _{\perp }\mathbf {_{j}} }$
Momentum= ${\displaystyle \mathbf {p} =\mathbf {m} \mathbf {v} }$ Angular momentum= ${\displaystyle \mathbf {L} =\mathbf {I} \omega }$ ${\displaystyle \mathbf {L} =\sum \mathbf {r_{j}} \mathbf {p} \mathbf {_{j}} }$
Kinetic energy = ${\displaystyle {\frac {1}{2}}\mathbf {m} \mathbf {v} ^{2}}$ Kinetic energy = ${\displaystyle {\frac {1}{2}}\mathbf {I} \omega ^{2}}$ ${\displaystyle {\frac {1}{2}}\sum \mathbf {m_{j}} \mathbf {v_{j}} ^{2}={\frac {1}{2}}\sum \mathbf {m_{j}} \mathbf {r_{j}} ^{2}\omega ^{2}}$