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Classical mech non-inertial frame bead on a rotating ring

  1. Oct 22, 2011 #1
    1. The problem statement, all variables and given/known data

    haEse.png

    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

    2. Relevant equations

    d2θ/dt2 = (ω2cosθ - g/R)sinθ 7.69

    θo = ±arccos(g/ω2R) 7.72

    3. 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.
     
  2. jcsd
  3. Nov 1, 2011 #2
    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
     
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