Please show all intermediate steps neatly, thanks! Find the position of the cent

Question

Please show all intermediate steps neatly, thanks!

Find the position of the centre of mass of a homogeneous solid hemisphere of radius R and mass M. dV = r2 sin theta d theta dPhi dr (Hints: Let the base of the hemisphere be in the x,y plane. Hence you only need find the z co-ordinate (why?). You will need to change the variables in the (Cartesian) expression for the position of the centre of mass to polar co-ordinates. Which variables do you need to integrate over? What are the limits of each integration?)

Explanation / Answer

by symmetry xcm and ycm are zero Consider the hemisphere as made up of a bunch of disks each of thickness dz if the disk is centered at z, then its radius will be given by r^2 +z^2 = R^2 r = sqrt(R^2-z^2) the hemisphere has density, d = M/V = M/ (2/3 pi R^3) = 3 M/(2 pi R^3) so z cm = 1/M integral of z dM z cm = 1/M integral of z d dV for the disk, dV = pi r^2 dz z cm = pi d/M integral of z r^2 dz z cm = 3/2 R^3 integral of z (R^2-z^2) dz from 0 to R z cm = 3/2 R^3 ( R^2 z^2/2 - z^4/4) from 0 to R z cm = 3/2 R^3 ( R^4/2- R^4/4) = 3/8 R

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