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how to determine the centre of mass of a solid hemisphere

how to determine the centre of mass of a solid hemisphere

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2 Answers

Pratham Ashish
17 Points
12 years ago

 

 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

 

 

 

 

 

4567-704_4788_7.bmp

satyapreet singh
18 Points
12 years ago

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.

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