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Polar Coordinates Conversion

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data

    For a Foucalt Pendulum:
    Relative to horizontal Cartesian x and y axes fixed to the earth (with x as East) the equations of motion for horizontal motion are:

    x′′ + ω02x -2ωy′ = 0 and y′′ + ω02y + 2ωx′ = 0

    [where x′, x′′, y′, y′′ are first and second time derivatives of x and y]

    Convert into standard polar coordinates (ρ,φ) where x=ρcosφ and y=ρsinφ and show that:

    ρ′′ + ρ(ω02-φ′2-2ωφ′) = 0

    and

    ρφ′′ + 2ρ′(φ′+ω) = 0


    2. Relevant equations



    3. The attempt at a solution

    I'm just not sure how to convert the derivatives of x and y into polar coordinate form, eg., how to express x′ in terms of ρ′ and φ′ etc. There is no cos or sin term in the resulting equations and I'm not sure where they go...I'd appreciate some help here!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 30, 2012 #2

    rude man

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    Homework Helper
    Gold Member

    I guess it's just going thru the motions:

    x' = ρ'cosψ - ρψ'sinψ
    x'' = ρ''cosψ - ρ'ψ'sinψ - {ρ'ψ'sinψ + ρψ''sinψ + ρψ'2cosψ}

    etc. for y' and y''

    then equating your given equations to each other and to 0 and substituting the x', x'', y' and y'' expressions now in terms of ρ, ψ and their 1st and 2nd derivatives.
     
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