Find the centre of mass of a uniform solid cone.

Explanation

Let us consider a uniform solid cone of mass M, radius R and height h.
`X_(cm)=0` (by symmetry)
Let us consider a small element (disc) of mass .dm., radius r and thickness dy at a distance y the from base as showm.
Then, `rho = (3M)/(pi R^(2)h)=(dm)/(pi r^(2)dy)`
`dm=(3Mr^(2))/(R^(2)h)dy`
`Y_(CM)=(1)/(M)int y dm =(1)/(M)int y (3Mr^(2))/(R^(2)h)dy = (3)/(R^(2)h)int yr^(2)dy`
`=(3)/(h)int_(0)^(h)y(1-(y)/(h))^(2)dy " " [because (h-y)/(r )=(h)/(R )rArr r = (1-(y)/(h))R]`
On solving, we get `Y_(CM)=(h)/(4)`
`therefore` C.M. of a uniform solid cone is `(0, (h)/(4))` from the centre of base.