Thin Rod Gravitational Potential and Field Vector

[erb12c]

Homework Statement

I am having a hard time understanding where to begin with this problem. Here it is:

Consider a thin rod of length L and constant density n that lies on the x-axis with endpoints at x=0 and x=L.
(i) Find a formula for the gravitational potential Φ = Φ(x) at the (variable) point P located on the positive x-axis at coordinate x>L.
(ii) Find the gravitational field vector E=E(x) = -E(x)i at the point P and show that the gravitational intensity is E(x)=m/x(x-L), where m is the mass of the rod.

Homework Equations

Total mass due to mass density μ over Volume V m=∫μ dv

Potential Due to Mass Density η over Curve C Φ=∫η/r dl

The Attempt at a Solution

I do not know how to start a solution because first I am confused as to which equation I would use and then second I get confused as to how I find the variables. I have done this problem with a circular disk instead of a thin rod and it was easier because I could find r but now I am lost.

 

[LCKurtz]

Homework Statement

I am having a hard time understanding where to begin with this problem. Here it is:

Consider a thin rod of length L and constant density n that lies on the x-axis with endpoints at x=0 and x=L.
(i) Find a formula for the gravitational potential Φ = Φ(x) at the (variable) point P located on the positive x-axis at coordinate x>L.
(ii) Find the gravitational field vector E=E(x) = -E(x)i at the point P and show that the gravitational intensity is E(x)=m/x(x-L), where m is the mass of the rod.

Homework Equations

Total mass due to mass density μ over Volume V m=∫μ dv

Potential Due to Mass Density η over Curve C Φ=∫η/r dl

The Attempt at a Solution

I do not know how to start a solution because first I am confused as to which equation I would use and then second I get confused as to how I find the variables. I have done this problem with a circular disk instead of a thin rod and it was easier because I could find r but now I am lost.

Consider a segment of the rod of length $ds$ located at $(s,0)$ on the rod, where $s$ is between $0$ and $L$. What is its mass? What force is exerted at the point $(x,0)$ by that segment? Then add up (integrate) that over the length of the rod and see what you get.

 

[erb12c]

deleted because of formatting

 

[erb12c]

Consider a segment of the rod of length $ds$ located at $(s,0)$ on the rod, where $s$ is between $0$ and $L$. What is its mass? What force is exerted at the point $(x,0)$ by that segment? Then add up (integrate) that over the length of the rod and see what you get.

Okay so if I am looking at s the mass would be ∫(ρ)∫(θ) η/√(x^2+s^2) dΘdρ ?
And then just split it up and integrate? Also would the bounds be (0,L) for ρ and (0, 2π) for Θ?

Am I on the right track?

 

[LCKurtz]

Consider a segment of the rod of length $ds$ located at $(s,0)$ on the rod, where $s$ is between $0$ and $L$. What is its mass? What force is exerted at the point $(x,0)$ by that segment? Then add up (integrate) that over the length of the rod and see what you get.

Okay so if I am looking at s the mass would be ∫(ρ)∫(θ) η/√(x^2+s^2) dΘdρ ?
And then just split it up and integrate? Also would the bounds be (0,L) for ρ and (0, 2π) for Θ?

Am I on the right track?

No. You have introduced a new variable (what is $\theta$?) and you didn't answer either of my questions. Try again:
1. What is the mass of that $ds$ segment?
2. What force is exerted at the point $(x,0)$ by it?
Then we can talk about the integral.

 

[erb12c]

Consider a segment of the rod of length $ds$ located at $(s,0)$ on the rod, where $s$ is between $0$ and $L$. What is its mass? What force is exerted at the point $(x,0)$ by that segment? Then add up (integrate) that over the length of the rod and see what you get.

No. You have introduced a new variable (what is $\theta$?) and you didn't answer either of my questions. Try again:
1. What is the mass of that $ds$ segment?
2. What force is exerted at the point $(x,0)$ by it?
Then we can talk about the integral.

The reason why I added θ because I thought I had to put it into spherical coordinates. I am honestly so confused I do not know where to go with this.

 

[Ray Vickson]

No. You have introduced a new variable (what is $\theta$?) and you didn't answer either of my questions. Try again:
1. What is the mass of that $ds$ segment?
2. What force is exerted at the point $(x,0)$ by it?
Then we can talk about the integral.

The question said to find the force at a point on the x-axis. So, you have a simple 1-dimensional problem.

If you had been asked for the gravitational force (or potential) at a general point (x,y,z), then it might be the case that spherical or polar coordinates would be helpful---it might also not be the case. First set up the required integration, then decide what type of coordinates would make the problem easiest.