Energy question, what are the quantities conserved?

[Clara Chung]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't know how to do the last part. What are the additional conservation laws? I don't think momentum is conserved because there is force acting on the particle. I don't think angular momentum is conserved too because there is a torque acting on the particle.
Here is the numerical answer:

 

[TSny]

Note that the equation for $\ddot x$ involves only $x$ (not $y$). Similar comment for $\ddot y$.

Imagine the x equation described a particle moving only along the x-axis. Anything conserved in this x-motion?

 

[Clara Chung]

Note that the equation for $\ddot x$ involves only $x$ (not $y$). Similar comment for $\ddot y$.

Imagine the x equation described a particle moving only along the x-axis. Anything conserved in this x-motion?

Thank you, I think the energy along x and y-axis are conserved too. Then I can find the answer.