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Two connected springs and potential energy as a function of x and y

  1. Dec 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Two springs each of natural length a and spring constant C are connected at one end
    (see figure). Consider a two dimensional displacement given by [itex](x, y)[/itex]
    (a) Write the potential energy as a function of x and y.
    (b) Find the force vector for a given [itex](x, y)[/itex] pair.
    springs.jpg


    2. Relevant equations
    Hooke's Law. Potential Energy.


    3. The attempt at a solution

    a) The stretch lengths of A and B springs Da and Db are
    [tex] D_A = \sqrt{(a+x)^2 + y^2} - a [/tex]
    [tex] D_B = \sqrt{(a-x)^2 + y^2} - a [/tex]
    Since potential energy of a spring is
    [tex] U_{spring} = 1/2kx^2 [/tex]
    The total potential energy U can be written by
    [tex] U(x,y) = U_A + U_B = C/2(D_A^2+D_B^2) = c/2((\sqrt{(a+x)^2 + y^2} - a)^2 + (\sqrt{(a-x)^2 + y^2} - a )^2) [/tex]
    b) [tex] \vec{F} = -\vec{\nabla} U[/tex] and etc

    Would you check my solution ? Is my answer correct ? Thanks for help in advance.
     
  2. jcsd
  3. Dec 11, 2011 #2
    looks right to me
     
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