Question

Let s be the part of the sphere x2 + y2 + z2 = 100 that lies above the plane z = 6. let s have constant density k. (a) find the center of mass.

Answer 1

center of mass = (0, 0, 96/13) exactly (0, 0, 7.3846) approximately. The object described is a spherical cap for a sphere with a radius of 10. Since the sphere is centered at the origin, the center of mass will have X and Y coordinates of 0 and we only need to find the Z coordinate. The formula for the geometric centroid of a spherical cap is: z = 3(2R - h)^2 / 4(3R - h) where z = distance from the center of the sphere R = radius of sphere h = distance from base of spherical cap to top of spherical cap And for a spherical cap of uniform density, the geometric centroid is also known as the center of mass. Since the sphere has a radius of 10 and is cut by the plane z=6, the value h will be 10-6 = 4. So substitute the known values into the formula: z = 3(2R - h)^2 / 4(3R - h) z = 3(2*10 - 4)^2 / 4(3*10 - 4) z = 3(20 - 4)^2 / 4(30 - 4) z = 3(16)^2 / 4(26) z = 3(256) / 104 z = 768/104 z = 96/13 z ~= 7.384615385