﻿ Adaptive Map - Centroids via the First Moment Integral

The Method of Composite Parts

The method of composite parts can be used to find the centroid or the center of mass of any body with clearly defined dimensions and masses. This method is an alternative to using the first moment integral as discussed on previous pages, and is often easier and faster. The method works by breaking the shape or volume down into a number of more basic shapes, identifying the centroids or centers of masses of each part, and then combing the results to find the overall centroid or center of mass.

A key aspect of the method is the use of centroid tables. This is a set of tables that lists the centroids (and also moments of inertia) for a number of common areas or volumes. Links to some centroid tables can be found in the sidebar to the right. When breaking the shape down into parts, it is usually beneficial to break it down into shapes or volumes that can be found in these tables. The centroid or center of mass of any shape that is not listed in the table will need to calculated via the first moment integral.

Finding the Centroid via the Method of Composite Parts

To use the method of composite parts, we must first break a complex shape down into several simpler shapes. This may include areas or volumes (which we will count as positive areas or volumes) or holes (which we will count as negative areas or volumes). Each of these shapes should have a centroid we can look up in the table if possible.

Once we have identified the different parts, we will create a table indicating the area or volume, and the x and y coordinates (or x, y, and z coordinates in 3D) for each shape. It is important to remember that each coordinate you list should be relative to the same base origin point. The centroid positions listed in the table will be axes shown in the table. You will need to adjust this to take into account where the shape is positioned.

Once you have the area and centroid coordinates for each shape, you can find the x and y coordinate of the centroid for the overall shape with the following formulas. Remember that areas for any shape that is a hole in the design will be a negative area in your formula.

Finding the Center of Mass via the Method of Composite Parts

To use the method of composite parts to find the center of mass, we must first break a complex volume down into several simpler volumes. This may include volumes (which we will count as positive areas) or cavities (which we will count as negative volumes). Each of these shapes should have a center of mass that we can look up in our table (remember that the centroid and center of mass are the same point for any part with a uniform density).

Once we have identified the different parts, we will create a table indicating the mass of each part, and the x, y, and z coordinate of the center of mass for each part. It is important to remember that each coordinate you list should be relative to the same base origin point.

Finally, once you have the mass the and center of mass coordinates for each shape, you can find the coordinates of the center of mass for the overall volume with the following formulas. Remember that any volumes that represent a cavity in the design should be represented as negative mass (The mass the cavity would have if it was made of the same material as whatever it was cut out of).

All of the calculations above use mass, rather than volumes. If the composite shape is made of a uniform material with a constant density however, then volume can be used interchangeably with the mass in the formulas. This is because the mass will be the density times the volume, and if all the densities are the same then the densities on the top end up canceling out with the densities on the bottom of our formulas.

Worked Problems:

Question 1:

Find the x and y coordinates of the centroid of the shape shown below.

Question 2:

Find the x and y coordinates of the centroid of the shape shown below.

Question 3:

Find the x and y coordinates of the centroid of the shape shown below.

Question 4:

The shape shown below consists of a solid semicircular hemisphere on top of a hollow cylinder. Based on the dimensions below, determine the location of the centroid.

Question 5:

A spherical steel tank (density = 8050 kg/m3) is half filled with water (density = 1000 kg/m3) as shown below. Find the overall mass of the tank and the current location of the center of mass of the tank (measured from the base of the tank).