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^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 !
To find the mass center of the combined cylinder and hemisphere object, you need to determine the centroid of each individual shape and then calculate their weighted average based on their masses.
Let's break down the problem into two parts: the cylinder and the hemisphere.
1. Finding the mass center of the cylinder:
Since the cylinder is standing up on its bottom end, its mass center will be in the middle of its height. Since the height (L) of the cylinder is given as 2R, its mass center will be located at L/2 or (2R)/2 = R above the bottom end.
2. Finding the mass center of the hemisphere:
The mass center of a hemisphere lies along the axis of symmetry, which is also the height of the cylinder. Since the hemisphere is glued to the top of the cylinder, its mass center will coincide with the mass center of the cylinder at a height of R.
Now, to find the mass center of the combined object, we need to consider the mass distribution of each component. Since the hemisphere and cylinder are glued together, their masses are combined.
Let's assume the mass of the cylinder is M1 and the mass of the hemisphere is M2.
As the cylinder occupies the lower part of the object, its mass center at R is more influential. Whereas, the hemisphere's mass center coinciding with the cylinder's mass center doesn't shift the overall mass center vertically.
So, the mass center of the combined object will be at a height of R from the bottom end of the cylinder, which is also the overall height of the object.
Therefore, the mass center of the cylinder with a height L=2R and a hemisphere glued to the top with radius R is located at an overall height of R from the bottom end of the cylinder.