IV Rotating coordinate system

            We now consider a system S, and a system S’ rotating with angular velocity w relative to S (fig 1); Then if we have basis, íçe_n> ý in S, and í çe_i> ý in S’. The projections of the basis vectors of  S’ into the basis in S.  i. e: <e_nçe_j>, must satisfy

                                         d/dt<e_nçe_j> = w ´ çe_j>                        ...(6)                                                     

Similarly to seccion III, we define êr>, êR> and êr’>. And:

                                          êr> = êR> + êr’>                                                                

            The derivative of this equation  using the basis íçe_n> ý is written:

                                êe_n> d/dt <e_nçr> = êe_n> d/dt <e_nêR> + êe_n> d/dt <e_nêr’>

or alternatively by means of the unity operator êe_i> <e_i ê:

               êe_n> <e_nçv> = êe_n> <e_nêU> + êe_n> d/dt <e_nêe_i><e_i êr’>

In this case d/dt<e_nêe_i> is not zero but is given by (6), then:

               êe_n> <e_nçv> = êe_n> <e_nêU> + êe_n><e_nçw ´êe_i><e_i êr’> + êe_n> <e_nêe_i>d/dt<e_i êr’>

d/dt<e_i êr’> = <e_i êv’> are the coordinates of the velocity of the particule relative to O’ expressed in the í çe_i> ý basis, forgetting unity operators, we get the equation in abstract form.

                                 êv> =  êU> + w´ êr’> + çv’>                                                   ...(9)

These abstract vectors have the following physical meaning: êU> is the velocity of the origin O’ relative to O, w´ êr’> Is the velocity of the point P fixed in S’ relative to the point P fixed in S; and  çv’> is the velocity of the particule relative to the point P fixed in S’.

            Upon differentation of eq. (9) and using the identities: d/dt <e_nçv> = <e_nça>;          d/dt <e_nçU> =  <e_nçA>; d/dt <e_nêw´çe_m> =  <e_nêw¢´çe_m> , w¢ = d/dt w. We obtain:

êe_n> d/dt <e_nçv> = êe_n> d/dt <e_nçU> + êe_n> d/dt <e_n êw´ êr’> + êe_n> d/dt <e_nêv’>

In order to express the temporal derivatives in term of the defined identities we conveniently multiply by the unity operator, çe_n><e_nç= I  = çe_j><e_jç.

êe_n> <e_nça> = êe_n> <e_nçA> + êe_n> d/dt <e_nêw´çe_m><e_mçe_j><e_j êr’>  + çe_n> d/dt <e_nêe_j><e_jêv’>

ça> = çA> +  êe_n><e_n êw¢´çe_m><e_mçe_j><e_j êr’> + êe_n><e_n êw´çe_m><e_mçw ´ çe_j><e_j êr’>  +                                                                                                                       + çe_n><e_nêw´çe_j><e_jêv’> + çe_n> <e_nçw´êe_j><e_jêv’> +çe_n> <e_nêe_j><e_jêa’>

Forgetting unity operators

                               êa> = êA> +  w¢ ´ êr’> + w ´w ´ êr’>  + 2 w ´ êv’> +  ça’>

êa> is the acceleration of the particule relative to the origin O, êA>  is the acceleration of the origin O’ relative to O; w¢ ´ êr’>  is the acceleration of the point P relative to O’ due to the acceleration of the angular velocity of the coordinates axes. w ´w ´êr’> is the centripetal acceleration of the point P relative to O’. 2 w ´ êv’> is the acceleration of the particule relative to the point P fixed in S’ and is called Coriolis acceleration.

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