Find the centre of mass of a uniform solid cone.
Let us consider a uniform solid cone of mass M, radius R and heightt h
Xcm=0 (by symmetry)
Let us consider a small element (disc) of dm, radius r and thickness dy at a distance y the from base as shown
Then, ρ=3MπR2h=dmπr2dy⇒dm=3Mr2R2hdy
YCM=1M∫ydm=1M∫y3Mr2R2hdy=3R2h∫yr2dy=3h∫h0y(1−yh)2dy[∵h−yr=hR⇒r=(1−yh)R]
On solving we get YCM=h4
∴ CM of a uniform solid cone is $(0,\cfrac{h}{4}) from the centre of base.