(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

GRAVITATIONAL POTENTIAL AND FIELD DUE TO A “THIN” ROD

A thin rod of length L lies along the +y-axis, with one end at the

origin (see diagram).

Assume:

• The rod has length only- no thickness in other directions.

• The density of the rod increases proportionally to the

y-coordinate: λ = ky, where k is a known constant and λ is in

kg/m

• Gravitational potential is zero at infinity: φ (∞) = 0

a) Find the gravitational potential φ ( x) at a point (x,0) by direct integration.

b) Find the gravitational field g at a point (x,0) by direct integration.

2. Relevant equations

dφ = -(G dm)/r

3. The attempt at a solution

Still stuck on part a, so that's really the brunt of my question for now (though assistance with part b is more than welcome!).

Using the given density function to solve for dm and substituting √(x^2+y^2) for r, I have an expression for dφ:

dφ = -(Gk y dy)/√(x^2+y^2)

...but I have no idea how to manipulate this to get a soluble integral :( I've been messing around with partial derivatives and polar coordinates for hours, but nothing seems to work.

PLEASE HELP!

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# Gravitational Potential Due to a Thin Rod of Varying Density.

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