location description | velocity description | |
---|---|---|

point mass | position vector \( \vec{r}_P \) | velocity vector \( \vec{v}_P \) |

rigid body in 2D |
position vector \( \vec{r}_P \) angle $\theta$ |
velocity vector \( \vec{v}_P \) angular velocity \( \omega \) |

rigid body in 3D |
position vector \( \vec{r}_P \) angles \( \theta,\phi,\psi \) |
velocity vector \( \vec{v}_P \) angular velocity vector \( \vec{\omega} \) |

Neither point masses nor rigid bodies can physically exist, as no body can really be a single point with no extent, and no extended body can be exactly rigid. Despite this, these are very useful models for mechanics and dynamics.

All points on a rigid body have the same angular rotation angles, as we can see on the figure below. Because the angular velocity is the derivative of the rotation angles, this means that every point on a rigid body has the same angular velocity \( \vec{\omega} \), and also the same angular acceleration \( \vec{\alpha} \).

In 2D the angle \( \theta \) of a rigid body the angle of rotation from a fixed reference (typically the \( \hat\imath \) direction), measured positive counter-clockwise. The angular velocity is \( \omega = \dot\theta \) and the angular acceleration is \( \alpha = \dot\omega = \ddot\theta \). The vector versions of these are \( \vec\omega = \omega \, \hat{k} \) and \( \vec\alpha = \alpha\,\hat{k} \), where \( \hat{k} \) is the out-of-plane direction.

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Warning: Labeling angular quantities on rigid bodies. #rkg-ww

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Complete in "Steering geometry".

This refers to "Rigid Bodies".Complete in "Four-Bar Linkages" under the subtitle "Example: Knee joint (constrained motion)".

This refers to "Constrained motion".Complete in "Four-Bar Linkages" under the subtitle "Example: Suspensions with Watt's linkage (constrained motion)".

This refers to "Constrained motion".