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Potential energy equilibrium positions

  1. Jun 6, 2009 #1
    1. The problem statement, all variables and given/known data
    A particle has potential energy

    U(x) = x + sin ((2 rad/m) x)

    over the range of x greater or equal to 0 meters and less than or equal to pi meters
    Where are the equilibrium positions in this range and for each is it a point of stable or unstable equilibrium?

    2. Relevant equations
    derv of u(x) set = to 0 and then the second derv of u(x)

    3. The attempt at a solution

    I solved derv of u(x) = 1 + 2 cos(2x) then set = to 0

    so 2 cos(2x) = -1 divide by 2 and cos(2x) = -1/2
    2x = arch cos of (-1/2) = 2pi/3 rad

    x = 1 pi/3 rad

    I get this part but the answer says that x can = 2pi/3 rad as well and I do not understand why.

    To determine if equilibrium is unstable or stable I took the second derv and at x = pi/3 and second derv of - 4 sin(2x) I got less than 0 so a maximum and therefore unstable equilibrium

    Can someone please help me understand why I should have known to consider 2pi /3??? Thanks for your efforts, Frostking
  2. jcsd
  3. Jun 6, 2009 #2


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    Homework Helper

    There are an infinite number of solutions to [itex]\cos(2x) = -1/2[/itex]. You just found one - there's another solution which will give you [itex]2\pi/3[/itex] as the final answer.
  4. Jun 6, 2009 #3
    Thanks, yes I should have realized I needed to check for other values less than or equal to pi!!!!
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