Centroids & Centers of Mass

Centroids of Areas Consider an arbitrary area A in the x-y plane (Figure 1): Divide the area into parts A1, A2,…, AN & denote the positions of the parts by (x1, y1), (x2, y2),…, (xN, yN)

Centroids of Areas The centroid or average position of the area can be found using the following equations: (1) To reduce the uncertainty in the positions of areas A1, A2,…, AN, divide A into smaller parts: But we would still obtain only approximate values of

Centroids of Areas To determine the exact location of the centroid, we must take the limit as the sizes of the parts approach zero: We obtain this limit by replacing Eqs. (1) by the integrals: (2) (3)

Centroids of Areas Where x & y are the coordinates of the differential element of area dA (Figure2) The subscript A on the integral sign means the integration is carried out over the entire area The centroid of the area is:

Centroids of Areas Keeping in mind that the centroid of an area is its average position will often help you locate it: E.g. the centroid of a circular area or a rectangular area obviously lies at the center of the area If an area has “mirror image” symmetry about an axis, the centroid lies on the axis If an area is symmetric about 2 axes, the centroid lies at their intersection (Figure 3)

Example 1 Centroid of an Area by Integration Determine the centroid of the triangular area shown in the Figure. Strategy Determine the coordinates of the centroid by using an element of area dA in the form of a “strip” of width dx. Fig. 4

Example 1 Centroid of an Area by Integration Solution Let dA be the vertical strip. The height of the strip is (h/b)x, so dA = (h/b)x dx. To integrate over the entire area, we must integrate with respect to x from x = 0 to x = b. The x coordinate of the centroid is:

Example 1 Centroid of an Area by Integration Solution To determine , we let y in Eq. (3) be the y coordinate of the midpoint of the strip:

Example 1 Centroid of an Area by Integration Solution The centroid is shown:

Example 1 Centroid of an Area by Integration Critical Thinking Always be alert for opportunities to check your results: In this example, we should make sure that our integration procedure gives the correct result for the area of the triangle:

Example 2 Area Defined by 2 Equations Critical Thinking Notice the generality of the approach we use in this example: It can be used to determine the x & y coordinates of the centroid of any area whose upper & lower boundaries are defined by 2 functions

2 Centroids of Composite Areas Composite area: an area consisting of a combination of simple areas The centroid of a composite area can be determined without integration if the centroids of its parts are known The area in the figure 5 consists of a triangle, a rectangle & a semicircle, which we call parts 1, 2 & 3

2 Centroids of Composite Areas The x coordinate of the centroid of the composite area is: (4) From the equation for the x coordinate of the centroid of part 1: We obtain:

2 Centroids of Composite Areas Using this equation & equivalent equations for parts 2 & 3, we can write Eq. (4) as: The coordinates of the centroid of a composite area with an arbitrary number of parts are: (5)

2 Centroids of Composite Areas The area in the figure 6 consists of triangular area with a circular hole or cutout: Designate the triangular area (without the cutout) as part 1 of the composite area & the area of the cutout as part 2

2 Centroids of Composite Areas The x coordinate of the centroid of the composite area is: Therefore, we can use Eqs. (5) to determine the centroids of composite areas containing cutouts by treating the cutouts as negative areas

2 Centroids of Composite Areas Determining the centroid of a composite area requires 3 steps: 1.Choose the parts — try to divide the composite area into parts whose centroids you know or can easily determine. 2.Determine the values for the parts — determine the centroid & the area of each part. Watch for instances of symmetry that can simplify your task. 3.Calculate the centroid — use Eqs. (5) to determine the centroid of the composite area.

Example 3 Centroid of a Composite Area Determine the centroid of the area shown in the Figure 7. Strategy Divide the area into parts (the parts are obvious in this example), determine the centroids of the parts & apply Eqs. (5). Fig. 7

Example 3 Centroid of a Composite Area Solution Choose the Parts: Divide the area into a triangle, a rectangle & a semicircle, which we call parts 1, 2 & 3 respectively. Determine the Values for the Parts: The x coordinates of the centroids of the parts:

Example 3 Centroid of a Composite Area Solution The x coordinates, the areas of the parts & their products are summarized in Table 1. Table 1 Information for determining the x coordinate of the centroid Ai Part 1 (triangle) Part 2 (rectangle) Part 3 (semicircle)

Example 3 Centroid of a Composite Area Solution Calculate the Centroid: The x coordinate of the centroid of the composite area is:

Example 3 Centroid of a Composite Area Solution Repeat the last 2 steps to determine the y coordinates of the centroid: Table 2 Information for determining the y coordinate of the centroid Ai Part 1 (triangle) Part 2 (rectangle) Part 3 (semicircle)

Example 3 Centroid of a Composite Area Solution Using the information summarized in Table 2:

Example 3 Centroid of a Composite Area Critical Thinking Although the area in this example may appear very artificial, many of the areas dealt with in engineering applications consist of combinations of simple areas such as these Even when that is not the case, an area can be approximated by combining these kinds of simple areas

Example 4 Centroid of an Area with a Cutout Fig. 8 Determine the centroid of the area in Fig. 8. Strategy Instead of attempting to divide the area into parts, a simpler approach is to treat it as a composite of a rectangular area with a semicircular cutout. Then we can apply Eq. (5) by treating the cutout as a negative area.

Example 4 Centroid of an Area with a Cutout Solution Choose the Parts We call the rectangle without the semicircular cutout & the area of the cutout parts 1 & 2, respectively:

Example 4 Centroid of an Area with a Cutout Solution Determine the Values for the Parts: The x coordinate of the centroid of the cutout is: The information for determining the x coordinate of the centroid is summarized in Table 3. Notice that we treat the cutout as a negative area.

Example 4 Centroid of an Area with a Cutout Solution Calculate the Centroid: Because of the symmetry of the area, Table 3 Information for determining Ai (mm2) Part 1 (rectangle) Part 2 (cutout)

Example 4 Centroid of an Area with a Cutout Critical Thinking If you try to divide the area in Fig. 8 into simple parts, you will gain appreciation for the approach we used: We were able to determine the centroid by dealing with 2 simple areas, the rectangular area without the cutout & the semicircular cutout Determining centroids of areas can often be simplified in this way