From the figure it is clear that mass is distributed over the surface area of hollow hemisphere. One thing is clear that hollow hemisphere is symmetric about x axis means positive x axis (right side) and negative x axis (left side) mass distribution is same.
Hence Xcm will no need to calculate it will be at origin for Xcm.
Along y axis its mass is different for different value of y.
Let total mass of hollow hemisphere is M and its radius is R, Now taking a elemental ring of mass dm, now the thickness of elemental ring will be Rdϴ as shown in figure.
Now the radius of elemental ring is Rcosϴ as shown above figure. Now the Ring above the origin that is y = Rsinϴ as shown in figure.
Ycm = ∫ ydm/∫dm now put the value of y = Rsinϴ
Now dm can’t be integrated, Because dm is not written in terms of y, Hence we need to find out the mass of elemental dm first.
So to find out the mass of elemental ring, we have to consider surface mass density. which is define as mass per unit surface area, It is denoted by ? .
Hence ? = M/A = M/2?R² Since mass is only distribute on surface area of hollow sphere
So we can write ? =dm/dA or dm = ?dA.
Now we can calculate the area of elemental ring and it will be area .
of elemental circle of radius Rcosϴ and thickness Rdϴ.
Hence Area = 2?Rcosϴ*Rdϴ So for elemental area dA = 2?Rcosϴ*Rdϴ hence from here dm = ?*2?Rcosϴ*Rdϴ.
Now use the equation for Ycm = ∫ ydm/∫dm = ⎰Rsinϴ *?*2?Rcosϴ*Rdϴ/M
Here ∫dm = M since total mass is M.
Ycm = R³*?*?*1/M⎰2sinϴ*cosϴdϴ now putting value of ? and 2sinϴ*cosϴ = sin2ϴ then
Ycm = R³xM/2?R²x?/M⎰sin2ϴdϴ now take limit from 0 to ?/2 total mass will be cover hence after simplification we will get .
Ycm = R/2⎰sin2ϴdϴ
Ycm = R/2⎡-cos2ϴ/2⎤ now put the value we will get
Ycm = R/2 (2/2) = R/2 , Ycm = R/2
Center of mass for solid hemisphere
Solid hemisphere can be made by many hollow hemisphere so we will use this technique to find the center of mass for solid hemisphere.
This is 3-D because it is solid lets suppose an elemental strip is taken from origin at a r distance then just we have seen in case of hollow hemisphere its center of mass is at r/2 distance from origin.
Hence this elemental strip center of mass will be at r/2 distance from origin as shown above figure
Now apply center of mass formula Ycm = ∫ ydm/∫dm .
Here it is clear y = r/2
now this is solid so its mass density will be mass per unit volume and it is denoted by ?
? = M/V = M/(2x?r³)/3 = 3M/2x?r³ since half volume of solid sphere
Ycm = ⎰r/2dm/⎰dm = ⎰r/2dm/M ⎰dm = M.
Since total mass is M
now for dm mass dm/dv =? or dm = ?dv
now for dv = Area x thickness = 2?r²xdr
dm = ?2?r²dr now put the value in above equation we will get
Ycm = ⎰r/2x?2?r²drx1/M = put ? = 3M/2?r³
Ycm = ⎰r/2*3M/2?r³*2?r²dr/M after simplification and integration we will get
Ycm = 3R/8
Now we will continue in next post. I hope you have enjoyed learning Center of mass Physics .If you like comment and share thanks for learning and share.
Dated 3rd Nov 2018