dynamics
posted by spencer .
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.

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^2d^2)
the area at d = pi(r^2d^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 !
Respond to this Question
Similar Questions

math
Find the ratio of the volume of the cylinder to the volume of the hemisphere, given that the height, h, of the cylinder is equal to the diameter, d, of the hemisphere 
math
Find the ratio of the volume of the cylinder to the volume of the hemisphere, given that the height, h, of the cylinder is equal to the diameter, d, of the hemisphere. 
Calc
A hemisphere of radius 5 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the … 
algebra
a storage bin is shaped like a cylinder with a hemisphere shaped top. the cylinder is 45 inches tall. the volume of the bin is 4131 pi cubic inches. find the radius of the bin. i think [4(pi)r(cubed)]/3=volume of hemisphere and h (pi) … 
Math  Calculus 2
An underground tank full of water has the following shape: Hemisphere  5 m radius. at the bottom Cylinder  radius 5 m and height 10m in the middle Circular cone radius 5 m and height 4 m at the top The top of the tank is 2 m below … 
Calculus 2
An underground tank full of water has the following shape: Hemisphere  5 m radius. at the bottom Cylinder  radius 5 m and height 10m in the middle Circular cone radius 5 m and height 4 m at the top The top of the tank is 2 m below … 
Calculus 2 / Physics
An underground tank full of water has the following shape 1) hemisphere of radius 5 m at the bottom 2) a cylinder of radius 5 m and height 10 m in the middle 3) a circular cone with base radius 5 and height 4 m at the top The top of … 
Math
A hemisphere of radius 7 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the … 
Mathematics
Given solid object is made up of a hemisphere snd a cyinder.the radius of a hemisphere is equal to the radius of a cylinder.The height of a cylinder is 80 cm and the height of the solid object is 94cm.find the total surface area of … 
Physics
A hemisphere of radius r is placed on a horizontal plane and a small mass of "m" is placed on the top of the hemisphere. Find the height from the ground to the mass m,at the moment it loses it's contact with the hemisphere.And the …