Linear-rotational Analogs - Galilean Frame Transforms

Galilean Frame Transforms

For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Motion of entities Inertial frames Accelerating frames
Translation

V = Constant relative velocity between two inertial frames F and F'.
A = (Variable) relative acceleration between two accelerating frames F and F'.

 Relative position

Relative velocity

Equivalent accelerations

 Relative accelerations

Apparent/fictitious forces
Rotation

Ω = Constant relative angular velocity between two frames F and F'.
Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.

 Relative angular position

Relative velocity

Equivalent accelerations

 Relative accelerations

Apparent/fictitious torques
Transformation of any vector T to a rotating frame

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