Relative and absolute acceleration in rotating frames
Understanding motion in non-inertial reference frames is essential for analyzing systems with rotational dynamics. In this post, I explore how absolute and relative accelerations are connected, breaking down their mathematical representation and physical meaning.
When studying motion in a rotating frame, the acceleration of a point \mathbf{P} can be expressed as:
\mathbf{a}_P = \mathbf{a}_{O_2} + \mathbf{a}_{\text{rel}} + \boldsymbol{\alpha} \times \mathbf{r} - \omega^2 \mathbf{r} + 2\boldsymbol{\omega} \times \mathbf{v}_{\text{rel}}
This equation combines contributions from the frame’s acceleration, relative acceleration within the frame, tangential acceleration due to angular velocity change, radial acceleration from centripetal forces, and Coriolis acceleration.
For planar motion, \boldsymbol{\alpha} and \boldsymbol{\omega} can be expressed as scalars \ddot{\theta} and \dot{\theta}, respectively, perpendicular to the plane. Substituting these into the equation refines the terms and clarifies the interplay between rotational and translational motion:
\mathbf{a}_P = \mathbf{a}_{O_2} + \mathbf{a}_{\text{rel}} + \ddot{\theta} \mathbf{k} \times \mathbf{r} - \omega^2 \mathbf{r} + 2\dot{\theta} \mathbf{k} \times \mathbf{v}_{\text{rel}}
For more insights into this topic, you can find the details here.