MATH 220 SUPPLEMENTAL NOTES 25

MASSES AND MOMENTS IN THREE DIMENSIONS

Here are the formulas for mass and moments in dimensions with density d (x, y, z).

MASS

FIRST MOMENTS ABOUT THE COORDINATE PLANES

CENTER OF MASS

MOMENTS OF INERTIA

MOMENT OF INERTIA ABOUT A LINE L

RADIUS OF GYRATION ABOUT A LINE L

EXAMPLE 1:

Find the center of mass of a solid of constant density bounded below by the paraboloid z = x 2 + y 2 and above by the plane z = 4.

SOLUTION:

Now to convert to polar coordinates.

Now convert to polar coordinates.

 

EXAMPLE 2:

Given the tetrahedron whose vertices are the points (0, 0, 0), (1, 0, 0), (0,1, 0), and (0, 0, 1) and has constant density d , find the mass, the moment of inertia I x, and the radius of gyration about the x-axis.

SOLUTION:

First I must find the equation of the plane that goes through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1).

 Now using the point (1, 0, 0), I will write the equation of the plane that forms this tetrahedron.

When working these types of problems, you will realize that once you get the bounds of integration set up for the mass, then the other required parts just require the use of the correct formula. Also notice that the problems get quite lengthy. I know I skipped several algebra steps in the last example, but with the TI-92, it is easily done. Work through these two examples, and if you have any questions, please feel free to contact me.

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