A Level Maths Notes:M3 – Centres of Mass, Integral Method.
We apply the formulae,
of
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,
and
will
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 is
The
mass contained in the cross section shown above, remembering that the
cone is circular, is
.
The slope of the cone is
and
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
between
and
so
which simplifies to
Now
integrate.
