Euler’s second law for arbitrary points
In this continuation of my exploration of Euler’s Second Law, I examine how angular momentum and moments are influenced by shifting the reference point from the center of mass to an arbitrary point. The angular momentum of a system about an arbitrary point P can be expressed in terms of its angular momentum about the center of mass C as:
\mathbf{L}_P = \mathbf{L}_C + \mathbf{r}_{PC} \times M \mathbf{v}_C
Here, \mathbf{r}_{PC} represents the vector from the center of mass to the point P, and M \mathbf{v}_C denotes the linear momentum of the system. This formula allows me to relate the motion of the system to the center of mass dynamics.
Taking the time derivative of \mathbf{L}_P, I derive the relationship between the moments about P and C, which includes an additional term accounting for the linear acceleration of the center of mass:
\sum \mathbf{M}_P = \frac{\mathrm d\mathbf{L}_C}{\mathrm dt} + \mathbf{r}_{PC} \times M \mathbf{a}_C
This equation emphasizes how moments about an arbitrary point P depend on the moments about the center of mass C, the angular momentum about C, and the acceleration term.
Through this analysis, I highlight the nuanced interplay between the reference point and the forces and moments acting on a system. Understanding this relationship is essential for applying Euler’s laws effectively in complex mechanical systems.
For more insights into this topic, you can find the details here.