I ordinarily would put this up in a homework/ coursework forum, but this isn't either one, its just something I was curious about given that I am in a Mechanics class.
So, I have done calculations for finding the gravitational force from a sphere and from a ring, and a flat circular plate. All these are symmetrical, tho. I was thinking about what happens with a square plate.
Now, here's the thing. Let's put our particle at a distance z from the surface of the plate. We wil put it in the center, to make the forces as symmetrical as possible.
Kind of intuitively, I would say that the way to do that calculation is to pretend you are dealing with a wire, (kind of like what you do calculating the electrical fields with a square wire). But this is a plate, so it's different (I think).
We have the mass of the plate. At any "radius" b from the center, if the plate is density ρ, with a side of "a" units, the mass is going to be [itex]2b^2ρ[/itex]. So I thought we could have a dM = 4bρ.
That wold say to me that th eintegral would look like this: [tex] \int dF_z = \int_0^a \frac{mG4d\rho dM}{s^2}[/tex] where s^2 is the distance to that dM, and that will be [itex]\sqrt{b^2+z^2}[/itex] so the integral would end up as: [tex] \int dF_z = mG\rho \int_0^a \frac{4b db}{\sqrt{b^2+z^2}}[/tex]
but anyway, I'd be curios to know if I am approaching this right. I was thinking of the old science fiction tropes with these huge space stations, and wondering what wold happen if one was constructed as a plane square -- also Stephen Baxter -- a scientist himself -- wrote a novel called Raft which takes place in a universe where gravitational forces are a billion times stronger, and a rat of metal plates actually has enough "pull" to live on. (there's more to it than that, but y'all get the idea).
So, I have done calculations for finding the gravitational force from a sphere and from a ring, and a flat circular plate. All these are symmetrical, tho. I was thinking about what happens with a square plate.
Now, here's the thing. Let's put our particle at a distance z from the surface of the plate. We wil put it in the center, to make the forces as symmetrical as possible.
Kind of intuitively, I would say that the way to do that calculation is to pretend you are dealing with a wire, (kind of like what you do calculating the electrical fields with a square wire). But this is a plate, so it's different (I think).
We have the mass of the plate. At any "radius" b from the center, if the plate is density ρ, with a side of "a" units, the mass is going to be [itex]2b^2ρ[/itex]. So I thought we could have a dM = 4bρ.
That wold say to me that th eintegral would look like this: [tex] \int dF_z = \int_0^a \frac{mG4d\rho dM}{s^2}[/tex] where s^2 is the distance to that dM, and that will be [itex]\sqrt{b^2+z^2}[/itex] so the integral would end up as: [tex] \int dF_z = mG\rho \int_0^a \frac{4b db}{\sqrt{b^2+z^2}}[/tex]
but anyway, I'd be curios to know if I am approaching this right. I was thinking of the old science fiction tropes with these huge space stations, and wondering what wold happen if one was constructed as a plane square -- also Stephen Baxter -- a scientist himself -- wrote a novel called Raft which takes place in a universe where gravitational forces are a billion times stronger, and a rat of metal plates actually has enough "pull" to live on. (there's more to it than that, but y'all get the idea).