- #1
Tazerfish
- 119
- 24
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 don't 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 don't think anyone else will by reading my rambling ...
Is there a way o delete my post ?
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