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January 26, 2015

January 26, 2015

Posted by **spencer** on Sunday, February 8, 2009 at 6:53pm.

- dynamics -
**Damon**, Sunday, February 8, 2009 at 8:24pmI 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 !

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