Circular Motion Due to Coriolis Force

Homework Statement

A particle has velocity v on a smooth horizontal plane. Show that the particle will move in a circle due to the rotation of the earth and find the radius of the circle. (Ignore all fictitious forces except the Coriolis force.)

Homework Equations

Coriolis force, given by

FC = -2m(ωxv)

The Attempt at a Solution

I started off by writing out the vector ω:

ωxyz

But the x component of the earth's angular velocity is zero.

The velocity of the particle is horizontal, so has no z component.

So I tried calculating the cross product of ω and v and ended up with:

FC=-2m[(-ωzvy)i + (ωzvx)j + (-ωyvx)k]

This is where I got stuck..I'm not sure how I'm supposed to show that this causes circular motion. I'd assume I'm looking for a force perpendicular to v...

If anyone can give me any advice or let me know what I'm doing wrong I'd appreciate it!

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