A uniform solid hemisphere H of mass M and radius R is fixed to a coordinate system, where the center of the hemisphere is located at the origin and the x-axis is normal to the plane surface of the hemisphere as shown in the figure below. The hemisphere is divided into numerous mass elements, each of them has the form of a disk with infintesimal thickness, volume, and mass given by dx, dV , and dm respectively. A mass element of the hemisphere is highlighted in the figure.

• The x coordinate of the center of mass of the hemisphere is defined as xcm = 1/M /hdm where x is the coordinate of the center of the mass element and dm is the infinitesimal mass of the element. The integration includes all masses in H. Denote the density of the hemisphere as . (a) Express M in terms of and R. (b) Show that dm = R3 cos3 d. [Hint: dm = dV , where dV is the infinitesimal volume of the mass element.] 3R (c) Show that xcm = 3R/8 .