CALC? 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 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 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, where ? is a positive constant—calculate the x-coordinate of the rod’s center of mass.

Solution 115P Introduction Here the density as a function of position is given, we have to calculate the position of the center of mass using the given formula. Step 1 The position of the center of mass is given by In the first case the density of the object is constant, and is given by . Now the volume element will be So the mass element can be written as Hence the above integration will become Solving the above integration we will have Now we know that Hence it is proved that the position of the center of mass will be at the geometrical center.