Topic 1: (Non-Inertial Frame) A space station is traveling around the Earth in a circular
orbit of radius R. Observers in the station see a piece of space junk moving nearby, also
presumably gravitationally bound to the Earth. To describe its motion, they use a reference
frame moving with the space station, such that the z axis always points away from Earth
and the y axis points in the direction of the space station's motion.
(a) Show that the equations of motion for the piece of junk, in the given reference frame,
are approximately (for $x \ll R$ and $y \ll R$)
$$ \ddot{x} - 2\omega\dot{y} - 3\omega^2x = 0 $$
$$ \ddot{y} + 2\omega\dot{x} = 0, $$
where $\omega$ is the space station's orbital angular velocity. [Ignore any gravitational at-
traction between the space station and the space junk, and treat the Earth itself as an
inertial frame, i.e., neglect the Earth's motion around the Sun.]
(b) Find the decoupled equation of motion for $x$ by integrating the second equation once
and substituting into the first. [You can assume that $v_{0y} = 0$ for the space junk.]
(c) Solve the equation of motion in $x$, expressing your answer in terms of the initial con-
dition $x_0$ and assuming $v_{0x} = 0$. Explain qualitatively the result that you obtain; it
might help to consider what the space junk's orbit would look like in an inertial frame.
