# Classical mech non-inertial frame bead on a rotating ring

## 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

θo = ±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

Feff = Fg + Fcf + Fcor

by the diagram

Fcf = 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.)

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

I assume Feff = mR2[d2θ/dt2]

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

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Suprisingly enough i actually did it right

Feff 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