Kinetic energy of a planar rigid body
In this post, I discuss the kinetic energy of a rigid body undergoing planar motion and how it can be derived by analyzing the motion of its mass elements. For a small mass element, its kinetic energy depends on the square of its velocity. By integrating over the entire body, I obtain an expression for the total kinetic energy.
The velocity of any point in the rigid body can be expressed as the sum of the velocity of the center of mass and the rotational velocity due to angular motion. This allows me to separate the kinetic energy into two distinct components: translational kinetic energy, proportional to the square of the velocity of the center of mass, and rotational kinetic energy, dependent on the angular velocity and the moment of inertia about the center of mass.
The final result is elegant and widely used:
T = \frac{1}{2} m v_C^2 + \frac{1}{2} I_C \omega^2
where m is the mass of the rigid body, v_C is the velocity of the center of mass, I_C is the moment of inertia about the center of mass, and \omega is the angular velocity.
This concise expression highlights the dual nature of kinetic energy in rigid body dynamics and is fundamental in physics and mechanical engineering analyses.
For more insights into this topic, you can find the details here.