Masses Distributed Along a Line
Masses Distributed over a Plane Region
Centroids

Moments and Centers of Mass
Many structures behave as if their masses were concentrated at one point,
called the center of mass. Often, it is important to know where this point is,
in order to understand and visualize the motion of a structure

Moment, Mass, and Center of Mass of a Thin Rod or Strip
Along the x-axis with Density Function
Moment about the origin:


Mass:


Center of mass:



See Examples 1 - 2, pages 442 - 443.


Moments, Mass, and Center of Mass of a Thin Plate
Covering a Region in the xy-plane
Moment about the x-axis:


Moment about the y-axis:


Mass:


Center of Mass:
,

See Examples 3 - 6, pages 445 - 448.


The Centroid of a Shape
When the density function is constant, it cancels out of the numerator and
denominator of the formulas for the coordinates    and  .
When this happens, the location of the center of mass depends only on the
geometry - the shape - of the object and not on the material out of which the
object is made.
In this case, the center of mass is also called the centroid of the shape.

See Example 6, page 448.


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