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Homework Help: What direction of r vector would decrease the potential energy most rapidly?

  1. Nov 13, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider a point mass, M, at the origin and a mass, m, at the point r(vector)=(x,y,z). The gravitational force on m is F(r)= (-GmM/||r||^3)(r). For simplicity, lets set GmM=1. Note that this force is directed towards the origin. the gravitational potential is a real valued function of ||r||=r given by V(r)=-1/r

    a)what direction from r=(x,y,z) would decrease the potential energy most rapidly?
    b)show that F(r)=-delta V(r). what does this say about the force?
    c) if the force and the potential are related as in part b, what type of a force field would we have if V(r)=r if V(r)=ln r?

    2. Relevant equations



    3. The attempt at a solution

    we first originally thought the ||r||^3 was a typo of ||r||^2. So we just simply made the equation F(r)=-1/||sqrt(x^2+y^2+z^2)||^2 * (x,y,z). This becomes really ugly very quickly. We would also take the inverse since that would be the rate of decrease.
     
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  3. Nov 13, 2011 #2

    SammyS

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    The magnitude of the force vector, [itex]\vec{F}(\vec{r})[/itex] is given by [itex]\displaystyle \left\|\vec{F}(\vec{r})\right\|=\frac{GmM}{r^2}[/itex], with [itex]\vec{F}(\vec{r})[/itex] directed toward the origin.

    Thus [itex]\vec{F}(\vec{r})[/itex] can be written as

    [itex]\displaystyle \vec{F}(\vec{r})=-\frac{GmM}{r^2}\hat{r}[/itex]
    [itex]\displaystyle =-\frac{GmM}{\left\|\vec{r}\right\|^3}\vec{r}[/itex]​
     
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