Spatial acceleration

In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity. In this article the notion of spatial acceleration is explored, which entails looking at a fixed (unmoving) point in space and observing the changes of velocity of whatever particle/point happens to coincide with the observation point. This is similar to the acceleration definition fluid dynamics where typically one can measure velocity and/or accelerations on a fixed locate inside a testing apparatus. In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity. In this article the notion of spatial acceleration is explored, which entails looking at a fixed (unmoving) point in space and observing the changes of velocity of whatever particle/point happens to coincide with the observation point. This is similar to the acceleration definition fluid dynamics where typically one can measure velocity and/or accelerations on a fixed locate inside a testing apparatus. Consider a moving rigid body and the velocity of a particle/point P along the body being a function of the position and velocity of a center particle/point C and the angular velocity ω → {displaystyle {vec {omega }}} . The linear velocity vector v → P {displaystyle {vec {v}}_{P}} at P is expressed in terms of the velocity vector v → C {displaystyle {vec {v}}_{C}} at C as: v → P = v → C + ω → × ( r → P − r → C ) {displaystyle {vec {v}}_{P}={vec {v}}_{C}+{vec {omega }} imes ({vec {r}}_{P}-{vec {r}}_{C})} where ω → {displaystyle {vec {omega }}} is the angular velocity vector. The material acceleration at P is: a → P = d v → P d t {displaystyle {vec {a}}_{P}={frac {{ m {d}}{vec {v}}_{P}}{{ m {d}}t}}} a → P = a → C + α → × ( r → P − r → C ) + ω → × ( v → P − v → C ) {displaystyle {vec {a}}_{P}={vec {a}}_{C}+{vec {alpha }} imes ({vec {r}}_{P}-{vec {r}}_{C})+{vec {omega }} imes ({vec {v}}_{P}-{vec {v}}_{C})} where α → {displaystyle {vec {alpha }}} is the angular acceleration vector.