Please show all intermediatesteps! Show that the infinitesimal volume element fo

Question

Please show all intermediatesteps!

Show that the infinitesimal volume element for a sphere in these co-ordinates is given by dV = r2 sin theta d theta d dr Find the position of the centre of mass of a homogeneous solid hemisphere of radius R and mass M. (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

Let us pretend that the hemisphere is described by 0 0, and let us pretend that the density function is p(x; y; z) = z. (The general form of the density function will be p(x; y; z) = kz for some constant k, but this will not alter the center of mass and will only multiply the mass we calculate by k.) Let E be the region that the hemisphere occupies. The mass is given by m =triple Integral[p(x,y,z)]dV = triple integral(zdV)............ The shape of the region strongly suggests we use spherical coordinates, so in spherical coordinates this integral becomes.. Integral(0 to pi/2)..itegral(0 to 2pi)...itegral(0 to a)..[pcosphi)(p^2sinphi)dp*d(theta)*d(phi)......... This equals...on solving: pia^4 / 4.............................. : For the center of mass, notice that symmetry immediately shows the x; y coordinates of the center of mass are 0. (Indeed, the region E as well as the density function are invariant under the various functions (x; y; z) 7! (x;y; z)) We only need to focus on computing the z-coordinate. To do so, we want to compute the integral triple integral(z^2dV)......... In spherical coordinates, this equals::::::::::; Integral(0 to pi/2)..itegral(0 to 2pi)...itegral(0 to a)..[pcos(phi)]^2(p^2sinphi)dp*d(theta)*d(phi)......... This equals on solving: 2pia^5 / 15.................... : To obtain the z-coordinate of the center of mass, we need to divide the above integral by the mass, which gives a z-coordinate of 8a/15.

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