Question

Find the mass center of a cylinder with height L and a hemisphere glued to the top with radius R. L=2R. The cylinder is standing up on its bottom end with hemisphere on top. Im so lost on this and cant use the formula to work.

Answer 1

I assume radius of circular cylinder is r
Volume of cylinder = (2 r* pi r^2) = 2 pi r^3
moment of this cylinder about base = r(2 pi r^3) = 2 pi r^4

Volume of hemisphere = (1/2) (4/3) pi r^2 = (2/3) pi r^3

now to find cg of hemi
moment of hemi about base of hemi:
hemi goes from d = 0 to d = r where d is height of slice above top of cyllinder
the radius at d = sqrt(r^2-d^2)
the area at d = pi(r^2-d^2)
the moment at d = pi(d r^2 - d^3)
integrate over d from d = 0 to d = r
pi (r^4/2 -r^4/4) = pi r^4/4
so hemi cg above base of hemi = (r^4/4)/(2 r^3/3) = r/6
so the cg of the hemi is r/6 above base of hemi
whioh is
2 r+ r/6 = 13 r/6 above the ground
so moment of hemi above ground = (13 r/6)(2/3 pi r^3) = (13/9) pi r^4
now
total volume = 2 pi r^3 + (2/3) pi r^3 = (8/3) pi r^3
total moment = 2 pi r^4 + (13/9 ) pi r^4
= (31/9) pi r^4
so
cg above ground = (31/9)(3/8)r
=(31/24) r
check my arithmetic !