Classical mech non-inertial frame bead on a rotating ring

[Liquidxlax]

Homework Statement

Consider the bead threaded on a cicular hoop of example 7.6 (pg 260), working in a frame that rotates with the hoop. find the equation of motion of the bead, and check that your result agrees with eq 7. 69. Using a free body diagram explain the result 7.71 for equilibrium positions

Homework Equations

d²θ/dt² = (ω²cosθ - g/R)sinθ 7.69

θ_o = ±arccos(g/ω²R) 7.72

The Attempt at a Solution

In the inertial frame there is going to be a centrifugal force coriolus force and force of gravity

F_eff = F_g + F_cf + F_cor

by the diagram

F_cf = mRω²sinθ Not sure about the direction

( I'm thinking it wouldn't really be in the r hat direction. more like Rcosθ. sounds redundant but I'm not sure how to explain it.)

F_cor = -2mvΩcosθ because it is in the southern hemisphere so it would deflect left which would oppose the gravitational force

I assume F_eff = mR²[d²θ/dt²]



I can solve this problem by lagrange method, but I'm not fully understanding this non-inertial ref frame.

 

[Liquidxlax]

Suprisingly enough i actually did it right

F_eff is mR(d²θ/dt²) = mRΩ²sinθ(rho_hat) -mgRsinθ

where rho_hat is cosθ

Deduced this by saying r' = r therefore R has to equal zero, but capital R is not the same as the radius in the equation above