Work-energy principle for planar rigid bodies
The work-energy principle application to rigid bodies reveals the interplay between translational and rotational motion. For rigid bodies, the internal forces cancel each other out, allowing me to focus solely on external forces and moments. This makes the principle straightforward yet highly insightful when analyzing real-world systems.
When I apply a constant force, the work done can be calculated as the dot product of the force with the displacement. For variable forces, I compute the work by integrating the force over the path of motion. These calculations provide a clear way to quantify energy transfer.
For rigid bodies specifically, the kinetic energy comprises two components: translational and rotational. The total kinetic energy change is expressed as:
W = \Delta T = \left( \frac{1}{2}m v_C^2 + \frac{1}{2}I_C \omega^2 \right) \bigg|_{t_1}^{t_2}
In addition, gravity, as a conservative force, performs work depending on the vertical displacement of the rigid body. This is given by:
W = mg \Delta h
where \Delta h is the vertical height change of the body’s center of mass.
Through these principles, I can analyze the motion of rigid bodies in two-dimensional systems, providing clarity in situations where both translational and rotational energies contribute.
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