Help understanding centrefugal force

[Liquidxlax]

Homework Statement

Basically i have to prove a lagrange equation using a non-inertial reference frame. A bead on a hoop where the hoop rotates as in the image below.

F = F_g + F_cf

Homework Equations

mR(d²θ/dt²) = mΩ²Rcosθsinθ -mgsinθ

what I'm supposed to get

The Attempt at a Solution

F_g = -mgsinθ

F_cf = m(ΩxR)xΩ = mΩ²Rcosθsinθ

I know that

Ω = Ω(0 , sinθ, cosθ)

R = R( 0 , k I'm just lost and frustrated :(

 

[Spinnor]

This is probably late but does this work?