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Centres of Mass, Integral Method

We apply the formulae,

 

 

To find the distancesof the centre of gravity about certain axes. Suppose we want to find the distance of the centre of gravity of a uniform circular cone from it's base.

 

By symmetry,andwill be on a vertical line through the centre of the cone. We must apply the third equation above to find

The mass of the cone isThe mass contained in the cross section shown above, remembering that the cone is circular, is. The slope of the cone isand by taking the origin at the centre of the base we can find the equation of the side of the cone. The coordinates of the top of the cone are

The moment of the slice of cone shown above is, but from our equation

betweenandso

which simplifies toNow integrate.