16.7 : Kinetic Energy for a Rigid Body

Imagine a solid object involved in a general planar movement, with its center of mass pinpointed at a spot labeled G. The object's kinetic energy relative to an arbitrary point A can be quantified for each of its particles - the ith particle in this case. This measurement is achieved through the employment of the relative velocity definition. The position vector, known as r_A, extends from point A to the mass element i.

The calculation of the kinetic energy for the entire object involves several steps. First, the scalar product comes into use. Following that, the equation is expressed in its integral form. Lastly, a vector identity is utilized to complete the calculation. The complexity of the kinetic energy equation can be reduced if point A is deemed as a fixed point on the solid object. By applying the definition of the object's angular momentum, the equation can then be represented as follows

An interesting scenario unfolds when point A coincides with the center of mass of the solid object. In this case, the integral of the position vector and the mass element equates to zero. This leads to a simplified expression of the kinetic energy. It is represented as the sum of two components: the kinetic energy of the center of mass of the object and the rotational kinetic energy of the object.