Complex substitution into the equation of motion.

[ultimateguy]

Homework Statement

The equation of motion of a mass m relative to a rotating coordinate system is
m\frac{d^{2}r}{dt^2} = \vec{F} - m\vec{\omega} \times (\vec{\omega} \times \vec{r}) - 2m(\vec{\omega} \times \frac{d\vec{r}}{dt}) - m(\frac{d\vec{\omega}}{dt} \times \vec{r})

Consider the case F = 0, \vec{r} = \hat{x} x + \hat{y} y, and \vec{\omega} = \omega \hat{z}, with \omega a constant.

Show that the replacement of \vec{r} = \hat{x} x + \hat{y} y by z = x + iy leads to

\frac{d^{2}z}{dt^2} + i2\omega\frac{dz}{dt} - \omega^2z=0.

Note, This ODE may be solved by the substitution z=fe^{-i\omega t}

Homework Equations

None.

The Attempt at a Solution

I've calculated that -\vec{\omega} \times (\vec{\omega} \times z) = \omega^2 z.

As far as figuring out how -2(\vec{\omega} \times \frac{d\vec{r}}{dt}) -(\frac{d\omega}{dt} \times \vec{r}) gives i2\omega\frac{dz}{dt} I'm lost.

 

[ultimateguy]

I solved it. Turns out that -2(\vec{\omega} \times \frac{d\vec{r}}{dt}) = -2i\omega\frac{dz}{dt} and since F = 0 then \frac{d\omega}{dt} = 0.