1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
  2. jcsd
  3. Nov 13, 2011 #2

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    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]​
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: What direction of r vector would decrease the potential energy most rapidly?
  1. Direction Vector (Replies: 5)

  2. Direction vector (Replies: 3)

Loading...