Question

how to find Venter of mass of a a solid and Hallows cone

Answer 1

I am not sure about this formula. Lets start by taking the vertex of the solid cone to be O(0,0,0)O000 in cylindrical coordinates (rr, θθ, zz). Then take the height of the cone to be hh and the base of the cone to have radius aa. In this case the we know that

r=ahz.
rahz
The formula for the center of mass of this cone can be written as

Mzm=∫h0zdm,
Mzm0hzdm
where MM is the total mass of the (solid) cone and zmzm is the location of the center of mass. We can write dmdm as

dm=πρa2h2z2dz,
dmπρa2h2z2dz
where we have considered dmdm to be the mass of a thin disk at height zz and of radius rr, with thickenss dzdz. Now we can write the full equation for the center of mass as

Mzm=πρ∫h0a2h2z3dz,
Mzmπρ0ha2h2z3dz
this becomes

Mzm=ρVzm=14πρa2h2.
MzmρVzm14πρa2h2
We know that the volume of a cone V=13πa2hV13πa2h, so we find

zmρ13πa2h=14πρa2h2,
zmρ13πa2h14πρa2h2
so

zm=34h.
zm34h
Which is the distance from the vertex of the cone.

I hope this helps.