Masses and Moments in Three Dimensions(3-D)

University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
21
Masses and Moments in Three Dimensions(3-D)
This section shows how to calculate the masses and moments of three dimensional objects in Cartesian coordinates.
Mass and moment formulas for solid objects in space(3D)
If ?(x, y, z) is the density of material then
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
22
EXAMPLE 13: Find the center of mass of a solid of constant density ? bounded below by the disk R: x2 + y2 ? 4 in the plane z = 0 and above by the paraboloid z = 4-x2 - y2 .
Solution:
Polar coordinates
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
23
EXAMPLE 14: Find Ix, Iy, Iz for the rectangular solid of constant density
shown in Figure .
Solution:
- ? x ? , - ? y ? , - ? z ?