how to determine the centre of mass of a solid hemisphere

							
 
 consider the  hemisphere  shown below the solution , let the mass of it be M & radius be R
then , mass density , ρ = M / {2/3 ¶ R ^3}
 from the symmetry of object we can say the C.M. will lie on the vertical axis shown in the figure
take a disk of dl length at the l distance from O, its C.M. will be at its centre means at the l distance from O
its mass  dm =  ρ  dv
                    = ρ  ¶ r^2 dl
                    =   ρ  ¶  ( R^2 - l^2 ) dl
 
C.M. of mass of hemisphere =
                                          = 1/M  ∫ dm l
                                          = 1/M ∫ ρ  ¶  ( R^2 - l^2 ) dl  *l
                                           =  ρ  ¶ / M ∫ l ( R^2 - l^2 ) dl 
                                           =  ρ  ¶ / M [R^2 * l^2 /2   -  l^4 /4  ]0 R
                                           = ρ  ¶ / M ( R^4) /4
                                            = ρ  ¶ (R^4) / { ρ *2/3 *4 *  ¶ R ^3}
                                            = 3 R/ 8       from O at the axis of symmetry 
 
 
 
 
 

						

							
the solid hemisphere is a symmtrical body about the coordinate axes, i mean is symmtrical about X , Y axes. therefore its centre of mass lies on its centre.