Moments and Center of Mass

    The moment of an object about an axis measures the tendency of the object to rotate about that axis. It is defined as the product of the object's mass and the signed distance from the axis. The center of mass of the lamina covering a region is the point where the mass can be concentrated without affecting the moments and ; that is,



The center of mass may also be thought of as the point from which the lamina may be suspended without movement. If is a continuous density function on a lamina corresponding to a plane region then the moments of mass with respect to the -axis and -axis, respectively, are



Furthermore, if is the mass of the lamina, the center of mass is where



If the density is constant, the point is called the centroid of the region.

Example (Moments and Center of Mass of a Planar Lamina)

(a) Find the mass and the center of mass of a triangular lamina with vertices and if the density function is
    
    Solution. The mass of the lamina is  









Then the center of mass is given by









and









(b) A lamina with over the region bounded by the curve and the line in the first quadrant.
    
    Solution. The mass of the lamina is  









Then the centroid is given by where









and









and so the centroid is

Example (Finding the Centroid of a Planar Lamina)  A lamina with over the region between the line and the parabola
    
    Solution.The mass of the lamina is  









Then the centroid is given by where









and









and so the centroid is

Example (Finding the Center of Mass of a Planar Lamina)

(a) Find the center of mass for the planar lamina with density function over
    
    Solution. The mass of the lamina is  











Then the center of mass is given by where



and









and so the center of mass is

(b) Find the center of mass for the planar lamina with density function over the region bounded by and
    
    Solution. The mass of the lamina is  









Then the center of mass is given by where










and









and so the center of mass is  

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Cite this as:
Moments And Center Of Mass
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/moments-and-center-of-mass.html