Integration of gravitational force on a test particle in a uniform density disk

Alan Silverman

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So imagine a disk of radius "R" with a point "p" at arbitrary distance "r" from the center. I'm going to describe three regimes within this disk as follows:

1. The area within the disk described by the circle at radius "r". Call this area "A"

Now draw a diameter of the disk at "R" through the center and point "p"

and now draw a tangent at point "p" perpendicular to the diameter just drawn.

2. The area to the right of the tangent. Call this area "B"


3. The area to the left of the tangent excluding area "A" which we'll call area "C"


I'd like to find the total gravitational force on point "p" from the three regimes of the disk. F = GM/(d^2)


Since the disk is of uniform density, the gravitational force from area "A" is just G x rho x pi x (d^2)/(d^2) = G x rho x pi


Now let's look at area "C"


I'd like to integrate the force produced by a ring of mass at radius "r" and then integrate out in radius


pick an arbitrary point "e" on the ring at radius "r" above the diameter, and an equidistant point "f" below, where the distance to "p" is d



Now the transverse components of the force from "e" and "f" will cancel, while the radial components will reinforce - 2 x G x rho x cos(theta)/ d

I have to now integrate around the semi-circle from point "p" to the opposite side - how do I do the integration?
 
So imagine a disk of radius "R" with a point "p" at arbitrary distance "r" from the center. I'm going to describe three regimes within this disk as follows:

1. The area within the disk described by the circle at radius "r". Call this area "A"

Now draw a diameter of the disk at "R" through the center and point "p"

and now draw a tangent at point "p" perpendicular to the diameter just drawn.

2. The area to the right of the tangent. Call this area "B"


3. The area to the left of the tangent excluding area "A" which we'll call area "C"


I'd like to find the total gravitational force on point "p" from the three regimes of the disk. F = GM/(d^2)


Since the disk is of uniform density, the gravitational force from area "A" is just G x rho x pi x (d^2)/(d^2) = G x rho x pi


Now let's look at area "C"


I'd like to integrate the force produced by a ring of mass at radius "r" and then integrate out in radius


pick an arbitrary point "e" on the ring at radius "r" above the diameter, and an equidistant point "f" below, where the distance to "p" is d



Now the transverse components of the force from "e" and "f" will cancel, while the radial components will reinforce - 2 x G x rho x cos(theta)/ d

I have to now integrate around the semi-circle from point "p" to the opposite side - how do I do the integration?

can you write the expression you wish to integrate?
 
can you write the expression you wish to integrate?
Well, the constant outside the integral is 2 x G x rho, while the integrand is cos(theta)/d where theta is the angle ope. But I'm not sure if I should integrate over angle or over distance, and I probably shouldn't have called the distance "d" since if I integrate over distance the differential would be "dd".
 
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