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Question Number 31566 by ajfour last updated on 10/Mar/18

Commented by ajfour last updated on 10/Mar/18

Frame XYZ is fixed. Frame xyz  rotates with angular velocity Ω.  Relate acceleration of particle   in fixed frame to that in rotating  frame.

$${Frame}\:{XYZ}\:{is}\:{fixed}.\:{Frame}\:{xyz} \\ $$$${rotates}\:{with}\:{angular}\:{velocity}\:\Omega. \\ $$$${Relate}\:{acceleration}\:{of}\:{particle}\: \\ $$$${in}\:{fixed}\:{frame}\:{to}\:{that}\:{in}\:{rotating} \\ $$$${frame}. \\ $$

Answered by ajfour last updated on 11/Mar/18

v_P ^� = (v_P ^� )_(xyz) +Ω^� ×r^�   a_P ^�  = (a_P ^� )_(xyz) +Ω^� ×(v_P ^� )_(xyz) +(dΩ^� /dt)×r^� +              Ω^� ×(v_P ^� )_(xyz) +Ω^� ×(Ω^� ×r^� )

$$\bar {{v}}_{{P}} =\:\left(\bar {{v}}_{{P}} \right)_{{xyz}} +\bar {\Omega}×\bar {{r}} \\ $$$$\bar {{a}}_{{P}} \:=\:\left(\bar {{a}}_{{P}} \right)_{{xyz}} +\bar {\Omega}×\left(\bar {{v}}_{{P}} \right)_{{xyz}} +\frac{{d}\bar {\Omega}}{{dt}}×\bar {{r}}+ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\bar {\Omega}×\left(\bar {{v}}_{{P}} \right)_{{xyz}} +\bar {\Omega}×\left(\bar {\Omega}×\bar {{r}}\right) \\ $$

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