Subtractive Objects: In many situations we need to locate the center of mass of an object when from a given body a part is removed. Such systems are called subtractive objects. In such cases we first locate the center of mass of whole body which is at its geometric center if the body is symmetric in shape and the part which is removed also has its center of mass at its geometric center if it is also symmetric in shape.
Now by using mass moment property of center
of mass we can locate the center of mass of the subtractive object. See the video
below for localization of center of mass of subtractive objects –
Continuous Objects: When
we wish to calculate the center of mass of an extended body(Continuous Object
System), we consider an elemental mass in the body at a position vector defined
with respect to the coordinate system associated with the body. Now just like
the case of a multiple body system we consider this elemental mass as one
particle of the body system and use the expression of position vector of center
of mass of the multi body system.
In case of an extended body when we use the
expression of position vector for center of mass of multi body system we need
to replace the summation of all mass moments with the integration of mass
moments with respect to the origin of the coordinate system. See the video
below on localization of center of mass of a continuous extended body –