Help understanding centrefugal force

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The discussion revolves around proving a Lagrange equation using a non-inertial reference frame, specifically involving a bead on a rotating hoop. The user outlines the forces acting on the bead, including gravitational force (Fg) and centrifugal force (Fcf), and provides the relevant equations. They express confusion regarding the application of these forces and the overall solution process. The user seeks clarification on whether their approach is correct, indicating a sense of frustration with the problem. Understanding the dynamics of non-inertial frames and the forces involved is crucial for solving this type of physics problem.
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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.
mN8wY.png


F = Fg + Fcf

Homework Equations



mR(d2θ/dt2) = mΩ2Rcosθsinθ -mgsinθ

what I'm supposed to get

The Attempt at a Solution



Fg = -mgsinθ

Fcf = m(ΩxR)xΩ = mΩ2Rcosθsinθ

I know that

Ω = Ω(0 , sinθ, cosθ)

R = R( 0 , k I'm just lost and frustrated :(
 
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This is probably late but does this work?
 

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Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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