Work-energy principle for 3d rigid bodies - rotational energy
In my previous blog post here, I discussed the work-energy principle for translational motion, establishing the relationship between the work done by forces and the change in translational kinetic energy. I now extend this principle to include rotational kinetic energy, deriving the corresponding relationship for rotational motion.
I begin with Euler’s second law for rotational motion:
\sum \mathbf{M}_C = \dot{\mathbf{L}}_C
where \mathbf{M}_C represents the sum of the moments about point C, and \mathbf{L}_C is the angular momentum about point C.
I then take the dot product of both sides with the angular velocity \boldsymbol{\omega} and after few steps I derive:
\Delta W = W_{1 \to 2} = \frac{1}{2} \boldsymbol{\omega} \cdot \mathbf{L}_C \Big|_{t_1}^{t_2} = T_{\omega_2} - T_{\omega_1} = \Delta T_\omega
where \Delta W is the work done by the torques, and \Delta T_\omega is the change in rotational kinetic energy.
Combining this equation with the work-energy principle for translational motion, and generalizing for forces and moments at any point in the body, I get:
\Delta W = W_{1 \to 2} = \int_{t_1}^{t_2} \sum \left(\mathbf{F}_i \cdot \mathbf{v}_i + \mathbf{M}_i \cdot \boldsymbol{\omega} \right) \, dt = \Delta T
This equation represents the generalized work-energy principle, encompassing both translational and rotational kinetic energy.
For more insights into this topic, you can find the details here.