User talk:Egm6321.f12.team5.eakin

R*1.2 Derive (7) and (1), and show the similarity with the derivation of the Coriolis force.

$$ \displaystyle \frac{df(Y^1(t),t))}{dt} = \frac{\partial f(Y^1(t),t) \dot Y^1}{\partial S} + \frac{\partial f(Y^1(t),t)}{\partial t} $$        (7)

$$ \displaystyle \frac{d^2f}{dt^2}=f_{,S}(Y^1,t) \ddot Y^1+f_{,SS}(Y^1,t)(\dot Y^1)^2+ 2f_{,St}(Y^1,t) \dot Y^1 + f_{,tt}(Y^1,t) $$    (1)

The procedures of deriving (7) and (1) are shown in the (5) and (6) of R*1.2

$$ \displaystyle \frac{df}{dt}=\frac{df(Y^1(t),t)}{dt}=\frac{\partial f(Y^1(t),t)}{\partial S} \frac{\partial S}{\partial t}+\frac{\partial f(Y^1(t),t)}{\partial t} \frac{\partial t}{\partial t}= f_{,S}(Y^1(t),t)\dot Y^1+f_{,t}(Y^1(t),t) $$ (5)

$$ \displaystyle \frac{d^2f}{dt^2}=\frac{\partial f_{,s}(Y^1(t),t)\dot Y^1}{\partial t}+\frac{\partial f_{,t}(Y^1(t),t)}{\partial t}= \frac{\partial f_(,S)(Y^1(t),t)}{\partial S} \frac{\partial S}{\partial t} \dot Y^1 + \frac{\partial f_{,S}(Y^1(t),t)}{\partial t} \frac{\partial t}{\partial t} \dot Y^1 + f_{,S}(Y^1(t),t)\ddot Y^1 + \frac{\partial f_{,t}(Y^1(t),t)}{\partial S} \frac{\partial S}{\partial t} + \frac{\partial f_{,t}(Y^1(t),t)}{\partial t} \frac{\partial t}{\partial t} = f_{,S}(Y^1,t)\ddot Y^1 + f_{,SS}(Y^1,t)(\dot Y^1)^2 + 2f_{,St}(Y^1,t)\dot Y^1 + f_{,tt}(Y^1,t) $$ (6)

We know that a point motions constantly relative to the O’ coordinate system has the coriolis acceleration.

$$ \displaystyle r(t)=r_{0}(t)+r'(t) $$  (8)

For example, P motions constantly relative to system O’, as the picture below shown.

$$ \displaystyle v'(t)= \frac{dr'}{dt}=constant $$  (9)

$$ \displaystyle a'= \frac{dv'}{dt}=0 $$  (10)

$$ \displaystyle v=\frac{Dr}{Dt}=\frac{D(r_{0}+r')}{Dt}=\frac{Dr_{0}}{Dt}+\frac{dr'}{dt}+w * r' $$  (11) $$ \displaystyle v=v'+v_{0}+w * r'=v'+v_{f} $$ (12)

$$ \displaystyle a=\frac{D(v'+v_{0}+w * r')}{Dt}=\frac{Dv'}{Dt}+w * v'+a_{0}+\frac{Dv_{0}}{Dt}+\frac{Dw}{Dt}*r'+w*\frac{Dr'}{Dt} $$        (13)

$$ \displaystyle a=a'+a_{0}+w*v'+w*(\frac{dr'}{dt}+w*r') $$     (14)

$$ \displaystyle a=a_{0}+2w*v'+w*(w*r') $$       (15) $$ \displaystyle a_{cor}=2w*v' $$          (16)

$$ \displaystyle a_{f}=a_{0}+w*(w*r') $$             (17)

$$ \displaystyle a=a_{f}+a_{cor} $$ (18)