- #1
Tazerfish
- 119
- 23
irst of all an apology : I was very uncertain where to put this. .I am doing this for fun. It isn't really homework so I don't care about any specifics or numbers.Additionally, I couldn't really follow the template with my question.
I also wasn't sure how difficult this problem really is.
Sorry, if this is the wrong place or phrased the wrong way
.
I am confused about multiple integrals.Specifically, when integrating over angles.
Calculate the gravitational force on an object sitting in the middle of a hemispherical "planet".
(By middle, I mean middle of the flat surface)
The thing is I "know" the solution. I just don't understand it.
You integrate the downward force from "shells"
Solution:## dF=\int_0^{2\pi} d\phi \int_0^\frac{\pi}{2} d\theta r^2 sin(\theta) dr \rho \frac{Gm}{r^2} cos(\theta) ##
I don't know why this equation looks like it does...
Does ## \int_0^{2\pi} d\phi \int_0^\frac{\pi}{2} d\theta r^2 dr ## give us some infinitesimal volumes ?
Or rather, how to interpret the above formula?(Or the part about the volume in the first equation if I did mess something up)
Why is there a sin and a cos in the formula?
I suppose it isn't really that hard and I just don't "see" the thought behind it.
I would be glad if someone answered
PS: How to make the Integrals big in tex?
EDIT: I think I get some part of it now.
The integrals of the angles together with the r^2 produce the shells.
The theta integral makes a curve on the surface of the hemisphere.And the phi integral rotates it around to make it an area.The last integral with dr integrates the shells into volumes ... right ?
The cos is there because the component of the force downward is ##cos (\theta) F##
And the sin is there because the rings the phi integral would produce without the theta integral would have the radius ##r_0 * sin(\theta)=r_{of the ring}##
I think I understood it now.
But i dont think anyone else will by reading my rambling ....
Is there a way o delete my post ?
I also wasn't sure how difficult this problem really is.
Sorry, if this is the wrong place or phrased the wrong way

I am confused about multiple integrals.Specifically, when integrating over angles.
Calculate the gravitational force on an object sitting in the middle of a hemispherical "planet".
(By middle, I mean middle of the flat surface)
The thing is I "know" the solution. I just don't understand it.
You integrate the downward force from "shells"
Solution:## dF=\int_0^{2\pi} d\phi \int_0^\frac{\pi}{2} d\theta r^2 sin(\theta) dr \rho \frac{Gm}{r^2} cos(\theta) ##
I don't know why this equation looks like it does...
Does ## \int_0^{2\pi} d\phi \int_0^\frac{\pi}{2} d\theta r^2 dr ## give us some infinitesimal volumes ?
Or rather, how to interpret the above formula?(Or the part about the volume in the first equation if I did mess something up)
Why is there a sin and a cos in the formula?
I suppose it isn't really that hard and I just don't "see" the thought behind it.
I would be glad if someone answered
PS: How to make the Integrals big in tex?
EDIT: I think I get some part of it now.
The integrals of the angles together with the r^2 produce the shells.
The theta integral makes a curve on the surface of the hemisphere.And the phi integral rotates it around to make it an area.The last integral with dr integrates the shells into volumes ... right ?
The cos is there because the component of the force downward is ##cos (\theta) F##
And the sin is there because the rings the phi integral would produce without the theta integral would have the radius ##r_0 * sin(\theta)=r_{of the ring}##
I think I understood it now.
But i dont think anyone else will by reading my rambling ....
Is there a way o delete my post ?
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