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# CALC In Section 8.5 we calculated the center of mass by

## Problem 115CP Chapter 8

University Physics | 13th Edition

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Problem 115CP

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.

Step-by-Step Solution:

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.

Step 2 of 2

##### ISBN: 9780321675460

This textbook survival guide was created for the textbook: University Physics, edition: 13. This full solution covers the following key subjects: mass, center, rod, Object, Positive. This expansive textbook survival guide covers 26 chapters, and 2929 solutions. The answer to “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.” is broken down into a number of easy to follow steps, and 187 words. Since the solution to 115CP from 8 chapter was answered, more than 579 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 115CP from chapter: 8 was answered by Patricia, our top Physics solution expert on 05/06/17, 06:07PM. University Physics was written by Patricia and is associated to the ISBN: 9780321675460.

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