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Direct substitution in the analysis of periodic motion

  1. Dec 9, 2008 #1
    1. The problem statement, all variables and given/known data

    By direct substitution, show that equation (3) is a solution of the differential equation (2).

    2. Relevant equations

    (2) (d^2 θ)/(dt^2 )=-g/l θ (Second derivative of θ(t)=-g/l θ.)


    (3) θ(t)=θ_0 cos⁡(√(g/l) t)


    3. The attempt at a solution

    I tried to integrate equation (2) and derive equation (3) but it didn't come out correctly.
     
  2. jcsd
  3. Dec 9, 2008 #2

    tiny-tim

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    Welcome to PF!

    Hi gsmith12! Welcome to PF! :smile:

    i] The question doesn't want you to integrate …

    it says use direct substitution … which means simply put θ(t) = θ0cos⁡(√(g/l) t) into (d^2 θ)/(dt^2 ), and show that it comes out as -g/l θ :wink:

    ii] but if you still want to integrate, multiply both sides by dθ/dt first :smile:
     
  4. Dec 9, 2008 #3
    Thanks for the help. I think I am on the right track but am still running into a bit a difficulty :confused:.

    To plug equation (3) into equation (2) do I first need to solve for theta? I am sorry but I am not completely clear on the set up for the direct substitution.

    Thanks
     
  5. Dec 10, 2008 #4

    tiny-tim

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    welcome to reality!

    Hi gsmith12! :smile:

    I think you're slightly in denial about reality …
    is a solution …

    equation (3) has solved for θ. :smile:

    So just plug-and-play! :wink:
     
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