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Force & Potential Energy

  1. Nov 18, 2014 #1
    1. The problem statement, all variables and given/known data
    A potential energy function for a two-dimensional force is of the form U = 3x3 * y - 7x.
    Find the force that acts at the point (x, y).


    2. Relevant equations
    In a 1-dimensional case:
    ΔU = -∫Fx dx
    dU = -Fx dx
    Fx = -dU/dx

    3. The attempt at a solution
    I know how to find the force in a 1-dimensional case; it's the gradient at the given x.

    But I can't wrap my head around it when there are two variables.

    Could it perhaps be the sum of the derivatives; Fx = -(dU/dx + dU/dy)?
     
  2. jcsd
  3. Nov 18, 2014 #2

    LCKurtz

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    The force will be a vector. If ##\phi(x,y)## is a potential for ##\vec F(x,y)## then$$
    \vec F(x,y) = \langle \phi_x,\phi_y\rangle$$
     
  4. Nov 18, 2014 #3
    So it's the combined vector of Vy and Vx? Correct me if I'm wrong, but does that means that the force at (x, y) would be the net vector of the Epot up and downwards against Ekin?
    Finding those two would involve finding the integral of both Fx and Fy
     
  5. Nov 18, 2014 #4

    LCKurtz

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    I don't know what you mean by "combined vector" and "net vector" and "upwards and downwards". It is a force vector field having two components or a magnitude and direction. And I don't know what Fx and Fy you are talking about. You get the vector field from the potential by taking the partials of the potential, not integrating, as I gave in the formula above.
     
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