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

[Minihoudini]

Homework Statement

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, let's 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?

Homework Equations

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.

 

[SammyS]

Homework Statement

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, let's 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?

Homework Equations

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.

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

Thus $\vec{F}(\vec{r})$ can be written as

$\displaystyle \vec{F}(\vec{r})=-\frac{GmM}{r^2}\hat{r}$

$\displaystyle =-\frac{GmM}{\left\|\vec{r}\right\|^3}\vec{r}$​