Rotation of Coordinate Systems A x z y x* z* y* Rotation Matrix.

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Presentation transcript:

Rotation of Coordinate Systems A x z y x* z* y* Rotation Matrix

Properties of a Rotation Matrix Rotation around z axis: q z x y x* y*

Euler Angles x z y x* z*y* x’ q f y Steps: 1.Rotate around z such that x’ is perpendicular to z* 2.Rotate around x’ by q 3.Rotate around z* by y

Euler Angles (continued)

Rotating System x z y x* z*y* x’ q f (t) y Now the rotation around z* is a function of time

Special Matrix Totally antisymmetric tensor

Cross Product Matrix x y z w wzwz f

Second Derivative Centripetal Acceleration Appears to throw object outward For position vector: Coriolis Acceleration Appears to push object perpendicular to velocity

Stationary Orbit Satellite For stationary orbit: at equator!

Falling Body Observed on Earth Time to fall: Apparent velocity as a function of height: Distance:

Falling Body Observed in Space Ellipitical Orbit Point of Impact – east and south Distance: Motion of building during fall See Mathematica notebook

Coriolis Force Derivation Time of fall: Vertical velocity: Horizontal acceleration: Horizontal velocity: Horizontal position: