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 a^{2} × ( H/3) ( z¯ ) = π tan^{ 2} α [H^{4}/4]
= π (tan ^{2} α H ^{2} ) × H^{2}/4
= π a ^{2} × H^{2}/4
z¯ = (¾) H from apex O
∴ Location of G from base of Cone = H/4