Impulse-momentum relationship for 3d rigid bodies
In my previous posts, I discussed the impulse-momentum relationship for particles and planar rigid bodies. Now, I’ll extend this concept to three-dimensional rigid body motion. The principles remain the same, but the mathematical representation becomes more involved, particularly for angular motion.
The fundamental concept is that the sum of forces acting on a body is related to the time rate of change of its linear momentum:
\sum \mathbf F = \frac{\mathrm d \mathbf P}{\mathrm d t}
Similarly, the angular impulse and the change in angular momentum are related by:
\int_{t_1}^{t_2} \sum \mathbf M_C \, \mathrm d t = \Delta \mathbf L_C = \mathbf L_{C_f} - \mathbf L_{C_i}
For linear motion, integrating the forces over time gives the linear impulse and the change in linear momentum:
\int_{t_1}^{t_2} \sum \mathbf F \, \mathrm d t = \Delta \mathbf P = m \mathbf v_{C_f} - m \mathbf v_{C_i}
Where m is the mass, \mathbf{v}_{C_f} is the final velocity of the center of mass, and \mathbf{v}_{C_i} is the initial velocity of the center of mass.
The key difference in 3D comes when I consider angular momentum. For a 3D rigid body, I express the angular momentum about the center of mass, C, as:
\mathbf L_C = \mathbf I_C \boldsymbol \omega = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & I_{yz} \\ I_{xz} & I_{yz} & I_{zz} \end{bmatrix} \begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix}
Here, \mathbf{I}_C is the inertia tensor about the center of mass, and \boldsymbol{\omega} is the angular velocity vector. The inertia tensor, a 3x3 matrix, describes the distribution of mass within the rigid body and how it resists rotational acceleration.
Therefore, the angular impulse-momentum equation becomes:
\int_{t_1}^{t_2} \sum \mathbf M_C \, \mathrm d t = \mathbf I_C \boldsymbol \omega_f - \mathbf I_C \boldsymbol \omega_i
Where \boldsymbol{\omega}_f is the final angular velocity and \boldsymbol{\omega}_i is the initial angular velocity. This equation allows me to analyze how changes in moments applied to a rigid body affect its rotational motion.
For more insights into this topic, you can find the details here.