Kinetic Energy for a Rigid Body

Consider a rigid body undergoing a general planar motion. Its center of mass is located at point G.

The kinetic energy of the i-th particle of the rigid body, relative to an arbitrary point A, can be expressed using the relative velocity definition. The position vector r_A extends from point A to mass element i.

Using the scalar product, expressing the equation in an integral form, and using a vector identity, the kinetic energy for the entire body is expressed in terms of its angular momentum.

Furthermore, if point A is the center of mass of the rigid body, then the integral of the position vector and mass element becomes zero.

Here, the kinetic energy is expressed as the sum of the kinetic energy of the center of mass of the body and the rotational kinetic energy of the body.

If point A is a fixed point on the rigid body, the kinetic energy equation gets simplified. Using the definition of angular momentum of the rigid body, the kinetic energy equation can be expressed in component form.