For solid ojbject whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums must be generalized to integrals where x and y are the coordinates of a small piece of an 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 p = 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, p = ax, where a is a positive constant - calculate the x-coordinate of the rod's center of mass.
For solid ojbject whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums must be generalized to integrals where x and y are the coordinates of a small piece of an 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 p = 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, p = ax, where a is a positive constant - calculate the x-coordinate of the rod's center of mass.
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