Question

find center of mass of right circular cone with hight h​

Answer 1

Answer:

Explanation:

Consider a right circular cone of uniform density,  ρ  and mass  M,  with radius  R  and height H as shown in the figure.

By symmetry, it is clear that the centre of mass lies on the vertical line from the vertex to the base. What is required to be determined is the height of the centre of mass from the base.

Consider a thin slab of the cone of thickness  dh  at a height  h  from the vertex.

The mass of this slab is,  dm=ρπr2dh.  

⇒  The distance of the centre of mass from the vertex is

hcm=1M∫0Hhdm=1H∫0Hhρπr2dh.  

From the properties of similar triangles, we get  rh=RH.  

⇒r=RhH.  

⇒hcm=1M∫0Hhρπr2dh=1M∫0Hhρπ(RhH)2dh  

=ρπR2MH2∫0Hh3dh=ρπR2MH2[h44]H0=ρπR2H24M.  

M=ρπR2H3.  

⇒hcm=ρπR2H24×3ρπR2H=3H4.  

⇒  The height of the centre of mass from the base is  H4.  

⇒  The centre of mass of a cone is on the vertical line from the vertex to the base and its distance from the base is one fourth the height of the cone.