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Center of mass of a system of masses in the x-y plane

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Masses are placed on the x-y plane as shown. Find the center of mass of this system. See attached file for full problem description.

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Solution Summary

I have shown how to find the x and y coordinates of the center of mass of a given system of masses place along the sides of a square.

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Center of Mass of Cone, Cylindrical Coordinate System

We are given a cone of height H and angle alpha with constant density. We want to calculate the center of mass using triple integrals in cylindrical coordinates. This requires a description of the solid in such coordinates and the use of the element of volume in the same system of coordinates:

a. Write the element of volume dV in a cylindrical coordinate system.

b. Calculate the Center of Mass of a cone with mass M and height H, and the angle at its head is alpha (or 2 alphas to cover the entire angle of the top). Assume constant density.

c. Now we remove the top from the cone, and its height is now h. What is the center of the mass now?

Please see the attached figures.

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