In Section 8.5 we calculated the center of mass by considering objects
composed of a number of point masses or objects that, by symmetry,
could be represented by a finite number of point masses. For a solid object
whose mass distribution does not allow for a simple determination of the
center of mass by symmetry, the sums of Eqs. (8.28) must be generalized to
integrals
where and are the coordinates of the small piece of the object that
has mass . The integration is over the whole of the object. Consider a
thin rod of length , mass , and cross-sectional area . Let the origin
of the coordinates be at the left end of the rod and the positive -axis lie
along the rod. (a) If the density of the object is uniform,
perform the integration described above to show that the -coordinate of the
center of mass of the rod is at its geometrical center. (b) If the density of
the object varies linearly with that is, , where a is a
positive constantcalculate the -coordinate of the rod's center of mass.