Find out the centre of gravity of a right circular cone:

Find out the centre of gravity of a right circular cone of height H & the base radius a.

Solution

Refer to Figure, where radius of the circle is illustrated as a and semi vertical angle of the cone is α

∴ a = H tan α

Figure

At any level z from apex O, radius of thin slice of volume dV is specified by

x = z tan α.

As dz is very small thickness of the slice,

dV = π x ²  dz

= π z ²  tan ²  α dz

{Moment of dV about O }   = dV × z

By applying the theorem of moments around O, if G is the centre of gravity of cone where, OG =  z¯ ,

= π a² × ( H/3) ( z¯ ) = π tan ²  α [H⁴/4]  

= π (tan ²  α H ² ) ×  H²/4

= π a ² ×  H²/4

z¯ =  (¾) H from apex O

∴          Location of G from base of Cone =  H/4