What Is the Gravitational Field Strength Within a Uniform Thin Disk?

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The discussion centers on calculating the gravitational field strength (g) within a uniform thin disk, specifically in the context of dark matter and galaxy rotation curves. The user seeks to understand how g varies with distance (r) from the disk's center, noting that existing resources primarily address the z-axis rather than the x-axis, which is crucial for simulating galaxies. An integral approach was attempted to determine g, but it resulted in an infinite value, prompting the user to seek assistance in identifying the error in their calculations. The user is willing to share their step-by-step process for further analysis. This inquiry highlights the complexities of gravitational modeling in astrophysics.
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I'm researching dark matter and how it affects galaxy rotation curves, I came up with the problem below.

Imagine a very thin, flat disk which has uniform mass per unit area.

What is the gravitational field strength (g) within the disk itself? How does g vary with respect to r, the distance from the center of the disk.

The area density of the disk is δ and the radius of the disk is R.
 
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Most of the threads focus on the z-axis. But in order for me to simulate a galaxy, I have to focus on the x-axis.


I came up with an integral to find the value of g, but its value ends up being infinite.

I can write it down step-by-step to let you guys find out where I went wrong.
 
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