Determine the coordinates of the centre of mass of a right circular soild cone of base radius `R` and height `h`.

Explanation

Fig. shown a unifrom soild cone of base radius `R` and height `h`. It is supposed to be made up of a large number of circular discs of different radii and masses.
Consider one such disc of radius `BC = x` and mass `dm` at a height `y` from the base.
`Delta^(s) ABC` and `ADO` are similar,
`:. (AO)/(AC) = (DO)/(BC) or (h)/(h - y) = (R )/(x)`
or `x = (R(h - y))/(h)` ..(i)
Mass of elementary disc, `dm =` Volume xx density
`dm = pi x^(2) .dy xx rho = pi rho (R^(2)(h - y)^(2))/(h^(2)) dy` ..(ii)
`:.` Co-ordinates of centre of mass of one are
`y_(cm) = (1)/(M) int y dm = (1)/(M) int y. (pi rho R^(2))/(h^(2)) (h - y)^(2) dy`
`= (pi rho R^(2))/(M h^(2)) underset(0) overset(h)int (h^(2) y + y^(3) - 2hy^(2)) dy`
`y_(cm) = (pi rhoR^(2))/((1)/(3) pi R^(2)h rho xx h^(2)) xx(h^(4))/(12) = (h)/(4)`
`( :' M = (1)/(3) pi r^(2) h rho)`
Similarly `x_(cm) = 0, z_(cm) = 0`
Hence co-ordinates of the centre of mass of soild
cone are `(0, (h)/(4), 0)`.