Rahul Sharma (Editor)

Spatial acceleration

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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.

Definition

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 → .

The linear velocity vector v → P at P is expressed in terms of the velocity vector v → C at C as:

v → P = v → C + ω → × ( r → P − r → C )

where ω → is the angular velocity vector.

The material acceleration at P is:

a → P = d v → P d t

a → P = a → C + α → × ( r → P − r → C ) + ω → × ( v → P − v → C )

where α → is the angular acceleration vector.

The spatial acceleration ψ → P at P is expressed in terms of the spatial acceleration ψ → C at C as:

ψ → P = ∂ v → P ∂ t

ψ → P = ψ → C + α → × ( r → P − r → C )

which is similar to the velocity transformation above.

In general the spatial acceleration ψ → P of a particle point P that is moving with linear velocity v → P is derived from the material acceleration a → P at P as:

ψ → P = a → P − ω → × v → P

References

Spatial acceleration Wikipedia
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