please derive the centre of mass of a solid cone and a hollow cone.

Dear Student,

**For a solid cone**,

Let (X_{c},Y_{c}) are the coordinates of center of mass. Then,

For symmetry consider the center of mass at X_{c}=0.

Divide the cone in horizontal disks of mass then,

dm=ρdV where, ρ is the density of material. (constant)

In this case,

where r is the radius of the cone at an arbitrary height *dy*.

radius of cone depends upon the height of the cone. So,

y=0 ; r=R (R is the base radius of the cone)

y=h ; r=0.

Using point-slope form:

Now, the volume

So,

So, the center of mass of solid cone is at the quarter height on the line joining from center of the base to the vertex.

**For hollow cone**,

Let (X_{c},Y_{c}) are the coordinates of center of mass. Then,

For symmetry consider the center of mass at X_{c}=0.

For hollow cone,

dm=ρ dA

So, the center of mass of a hollow cone is at one-third of the height on the line joining from center of the base to the vertex.

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