Force of Gravity on the tip of a giant snowcone

1. The problem statement, all variables and given/known data
I'm studying for a physics exam and made up the following problem to practice. Since I made it up there is no way to check my answer and I would greatly appreciate any and all feedback.

Find the force of gravity exerted on a point mass m at the tip of a snowcone (portion of a ball cut out by a cone where the radius of the sphere = the length of the side of the cone) with constant density ϱ and radius R. The internal angle of the cone is pi/4 in both directions.
snowcone.png


2. Relevant equations
F = -GmM(r)/r^2


3. The attempt at a solution
I used spherical coordinates
M(r)=ϱV(r)

M(r)=[tex]\frac{-\rho}{3}r^{3}\theta\cos\phi[/tex]

[tex]dM(r)=r^{2}\sin\phi dr d\phi d\theta[/tex]

[tex]F=\int_0^{\pi/4}\int_0^{\pi/4}\int_0^R \frac{-Gm\rho r^{2}}{r^{2}} \sin\phi dr d\phi d\theta[/tex]

[tex]F=-\frac{\pi}{4} Gm\rho\int_0^{\pi/4} \sin\phi d\phi \int_0^R dr [/tex]

[tex]F=-\frac{\pi}{4} Gm\rho R(\cos0-\cos\frac{\pi}{4})[/tex]

[tex]F=-\frac{\pi}{8} (2-\sqrt{2})Gm\rho R[/tex]

Also, I can't think of any way to find F(d) where d is the distance from the point of the cone. It might just be wishful thinking though.
Thanks in advance!
 
Last edited:

LowlyPion

Homework Helper
3,055
4
I might consider solving by superposition.

The sum of the conical solid and the spherical one.

For the cone you can sum the individual disks of dm = ϱ*a(x)dx and then add the spherical section calculated from the distance of the cone height.

I guess I don't see the need to use the spherical coordinates.
 

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