In Section 8.5 we calculated the center of mass by considering objects
composed of a \(finite\) 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 $$x_{cm} = {1\over M}\int x \space dm \space y_{cm} = {1 \over
M}\int y \space dm$$
where \(x\) and \(y\) are the coordinates of the small piece of the object that
has mass \(dm\). The integration is over the whole of the object. Consider a
thin rod of length \(L\), mass \(M\), and cross-sectional area \(A\). Let the origin
of the coordinates be at the left end of the rod and the positive \(x\)-axis lie
along the rod. (a) If the density \(\rho = M/V\) of the object is uniform,
perform the integration described above to show that the \(x\)-coordinate of the
center of mass of the rod is at its geometrical center. (b) If the density of
the object varies linearly with \(x-\)that is, \(\rho = ax\), where a is a
positive constant\(-\)calculate the \(x\)-coordinate of the rod's center of mass.
