Distance of the centre of mass of a solid uniform cone from its vertex is `z_0`. It the radius of its base is R and its height is h then `z_0` is equal to:

Explanation

Consider a cone of height 'h' base radius 'R' and density `rho`. To find centre of mass of the cone imagine a small element of radius 'r' and thickness 'dx' at a distance 'x' form 'O' . Mass of small element,`dm=(pir^(2)) dx rho`.
from figure.

`r/R=x/hrArrr =(Rx)/h`
`x_(CM)=(intdmx)/(intdm)=(int_(0)^(h)(pir^(2)rhodx)x)/(int_(0)^(h)pir^(2)dx rho)`
`=(int_(0)^(h)((R^(2)x^(2))/(h^(2)))xdx)/(int_(0)^(h)(R^(2)x^(2))/(h^(2))dx)=(int_(0)^(h)x^(3)dx)/(int_(0)^(h)x^(2)dx)=((x^(4))/4)^(h))/(((x^(3))/3)_(0)^(h))=((h^(4))/4)/((h^(3))/3=(3h)/4`
Therefore, centre of mass of cone is at a distance `(3h)/4` from vertex on its line of symmetry.