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I A double in this example problem

  1. Aug 28, 2016 #1
    I am currently self-studying Mathematical Methods for Studying by Weber, Arfken and Harris. In the chapter Matrix Eigenvalue problems, I'm stuck in a particular step in a particular problem. Please look at the Image attached. The expression which is boxed, $$\frac{F_x}{F_y}\neq\frac{x}{y}$$ in the textbook as we can see, it is said that the force will not be directed toward the minimum at $x=0$ & $y=0$. Actually this is the step I dont understand, how the expression came up here. I am trying to visualize this problem physically, but I am unable to come up with this expression. Please give me some hint or information if possible.
     

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  2. jcsd
  3. Aug 28, 2016 #2
    If the ratios are equal it means that the angle of the force can be the same as the angle from the origin to the position (x,y).
    $$cot^{-1}(\frac{x}{y}) = \theta_{X} \;\;\;\;cot^{-1}(\frac{F_x}{F_y}) = \theta_{F} $$
    $$\frac{x}{y} =\frac{F_x}{F_y}\; \text{implies}\;\theta_{X}=\theta_{F}$$
    That is not the case for the given problem, where most of the places the angles are different. Another way of writing this could be
    $$\vec{F} \neq f(\vec{r}) \vec{r},\, \text{most } \vec{r} $$
    So the force vector is not proportional to the position vector, thus the problem is not a "central force" problem with the center at the origin. So it's different from problems like the Kepler problem of two objects with gravity.
     
  4. Aug 29, 2016 #3
    Or otherwise can I say, External torque along $${\hat{k}}$$ direction for a central force is zero. i.e, $$F_x{y}-F_y{x}=0$$, is my reasoning correct?
     
    Last edited: Aug 29, 2016
  5. Aug 29, 2016 #4

    Mark44

    Staff: Mentor

    On this site, use a pair of # symbols at each end (inline LaTeX) or a pair of $ symbols at each end (standalone LaTeX). A single $ at each end doesn't do anything, nor does a single # at each end.
     
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