In a rotating frame the Coriolis force (2 \(m \vec{v} \times \vec{\Omega}\)) and centrifugal force \((m \vec{\Omega} \times \vec{r} \times \vec{\Omega})\) come into play, where \(\vec{v}\) and \(\vec{r}\) respectively are the velocity and position in the rotating frame.
a) Write down the equation of motion in vector form for a particle of mass m in a rotating coordinate system, in the absence of any external forces.
b) Take \(\vec{\Omega}\) along the z axis with \(\vec{r} = (x, y, 0)\), and write the equations of motion for x, y etc as a function of time. Don't solve the equations of motion yet.
c) Multiply the x equation by x, the y equation by y, and take an appropriate linear combination to show that
\(\frac{1}{2}mv^2 - \frac{1}{2}m\Omega^2r^2 = \text{constant}\), i.e. the dynamics are equivalent to a harmonic oscillator (isotropic in the plane) with a negative spring constant. Express the effective spring constant in terms of m, \(\Omega\) etc.
d) Multiply the x equation by y, and the y equation by x, and take an appropriate linear combination to show that the z component of the angular momentum is \(l_z = -mr^2\Omega + \text{constant}\) with respect to the rotating frame.
e) In polar coordinates, use the result from part (c) to determine r(t), given r(0) = r_0, and \(\dot{r}(0) = 0\).
f) Use this result, and the result from part (d) to determine \(\phi(t)\) given \(\phi(0) = \phi_0\).
