Center of mass and moments

 The expressions of mass, center of mass, moments of inertia expressed in double integrals can be modified by replacing the double integrals with triple integrals.

Example

Solution

The region Q is a tetrahedron meeting the axes at the point (6,0,0), (0,3,0) and (0,0,2) (see figure below). To find the limits of integration, let z = 0 in the slanted plane z = 1/3(6-x-2y). Then for x and y find the projection of Q on the the xy plane which is bounded by the axes and the line x_+ 2y = 6. The mass is calculated as follow:

 

Finding the moments of inertia of a solid in two dimensions.

 Goal: Find the moment of inertia of a solid in two dimensions

Let's go back to the lamina considered in earlier post where we calculated its mass. In order to do that we considered the region R occupied by the lamina. We divided the lamina into tiny subrectangles. Our goal now is to find the different moments of inertia of the solid.

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Let's find the moment of inertia about the x-axis.

 The moment of inertia of a solid about an axis is equal to its mass by the square of its distance from this axis.

The moment of inertia of the lamina about the x-axis is equal to the sum of the moments of the tiny subrectangles of the lamina.

The moment of inertia of a subrectangle about the x-axis is equal to the product of its mass by the square of its distance from the x-axis. This moment is calculated as: 

Let's add all the moments of the subrectangles. This comes to take the Rieman sum of the product and to determine its limit.                                                                                                                                         

           

The moment of inertia of a subrectangle about the y-axis is equal to the product of its mass by the square of its distance from the t-axis.  Hence, the moment is given by:                                                                      

                                                                                                                     

Adding all the moments together allows to find the moment of the lamina about the y-axis. This leads to use the Rieman sum of the product and take its limit.

Example

Solution

Practice

Center of mass in two dimensions

 The center of mass of an object is called the center of gravity if the object is located in a gravitational field. If the object has a uniform density, the center of mass is the geometric center of the object, which is called the centroid. The following figure shows the point P as the center of mass of a lamina (flat plate).

 

Restating the center of mass in terms of integrals, we have:

If the object has uniform density, the density function ⍴(x,y) is constant and the formulas become:

Example

Solution

Practice