91Ó°ÊÓ

The equation of cone is $z^2=x^2+y^2,0<z<h$

Gravitational force is an attractive force that exists between all objects with mass; an object with mass attracts another object with mass; the magnitude of the force is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between the two objects.

The vertical force on the unit mass exerted by the bit of mass is given by the following,

$$\begin{gathered} d{F}_z=dF\cos \mathrm{θ} \\ =G\frac{dM}{r^2}\cos \mathrm{θ} \end{gathered}$$

Where $$\begin{gathered} dM=sdsd\mathrm{θ}dz\mathrm{ÒÏ},\cos \mathrm{θ}=\frac{z}{\sqrt{s^2+z^2}},r^2=s^2+z^2 \\ {F}_z={∫}_v^{}G\frac{dM}{r^2}\cos \mathrm{θ} \\ =G\mathrm{ÒÏ}{∫}_v^{}\frac{sdsd\mathrm{θ}zdz}{{\left(s^2+z^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\cos \mathrm{θ} \\ =G\mathrm{ÒÏ}{∫}_0^{h}zdz{∫}_z^h\frac{sds}{{\left(s^2+z^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{∫}_0^{2\mathrm{Ï€}}d\mathrm{θ} \end{gathered}$$

$$\begin{gathered} F_z=2\mathrm{Ï€GÒÏ}{∫}_0^{\mathrm{h}}\mathrm{zdz}{∫}_{\mathrm{z}}^{\mathrm{h}}\frac{\mathrm{d}\left({\mathrm{s}}^2+{\mathrm{z}}^2\right)}{{\left({\mathrm{s}}^2+{\mathrm{z}}^2\right)}^{3/2}}\frac{1}{2} \\ =\mathrm{Ï€GÒÏ}{∫}_0^{\mathrm{h}}\mathrm{zdz}\frac{-2}{\sqrt{{\mathrm{s}}^2+{\mathrm{z}}^2}}\overline{){}_{\mathrm{z}}^{\mathrm{h}}} \\ =-2\mathrm{Ï€GÒÏ}{∫}_0^{\mathrm{h}}\mathrm{zdz}\left[\frac{1}{\sqrt{{\mathrm{h}}^2+{\mathrm{z}}^2}}-\frac{1}{\mathrm{z}\sqrt{2}}\right] \\ =-2\mathrm{Ï€GÒÏ}{∫}_0^{\mathrm{h}}\mathrm{zdz}\left[\frac{1}{\mathrm{z}\sqrt{2}}-\frac{1}{\sqrt{{\mathrm{h}}^2+{\mathrm{z}}^2}}\right] \\ =2\mathrm{Ï€GÒÏ}\left[\frac{\mathrm{h}}{\sqrt{2}}-\frac{1}{2}{∫}_0^{\mathrm{h}}\frac{\mathrm{d}\left({\mathrm{h}}^2+{\mathrm{z}}^2\right)}{\sqrt{{\mathrm{z}}^2+{\mathrm{h}}^2}}\right] \\ =2\mathrm{Ï€GÒÏ}\left[\frac{\mathrm{h}}{\sqrt{2}}-\sqrt{{\mathrm{z}}^2+{\mathrm{h}}^2}\overline{){}_0^{\mathrm{h}}}\right] \\ =\mathrm{Ï€³ÒÒϳó}\left(2-\sqrt{2}\right) \end{gathered}$$

Hence, the gravitational attraction on a unit mass at the origin due to a mass (of constant density) occupying the volume inside the cone $z^2=x^2+y^2,0<z<h$ is $F_z=\mathrm{Ï€³ÒÒϳó}\left(2-\sqrt{2}\right)$