Chap9. Center of Gravity & Centroid

Objectives

  • To discuss the concept of center of gravity, center of mass, and the centroid
  • To show how to determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitray shape.
  • To use the theorems of Pappus and Guldinus for finding the surface area and volume for a body having axial symmetry.
  • To present a method for finding the resultant of a general distributed loading and show how it applies to finding the resultant force of a pressure loading caused by a fluid

9.1 Center of Gravity, Center of Mass, and the Centroid of a Body

(1) Center of Gravity

  • A body is composed of infinite particles, and is located within a gravitational field. Each particle's weight forms an approximately parallel force => total weight of the body wich passes through the center of gravity G.

\[W = \int dW, \quad \overline{x} = \frac{\int \widetilde{x}dW}{\int dW}, \quad \overline{y} = \frac{\int \widetilde{y}dW}{\int dW}, \quad \overline{z} = \frac{\int \widetilde{z} dW}{\int dW} \\ \widetilde{x} : the\ coordinates\ of\ each\ particle\ in\ the\ body\]

(2) Center of Mass

\[\overline{x} = \frac{\int \widetilde{x}dm}{\int dm}, \quad \overline{y} = \frac{\int \widetilde{y}dm}{\int dm}, \quad \overline{z} = \frac{\int \widetilde{z} dm}{\int dm}\]

(3) Centroid of a Volume

\[\overline{x} = \frac{\int \widetilde{x}dV}{\int dV}, \quad \overline{y} = \frac{\int \widetilde{y}dV}{\int dV}, \quad \overline{z} = \frac{\int \widetilde{z} dV}{\int dV}\]

(4) Centroid of an Area

\[\overline{x} = \frac{\int \widetilde{x}dA}{\int dA}, \quad \overline{y} = \frac{\int \widetilde{y}dA}{\int dA}\]

(5) Centroid of a Line

\[\overline{x} = \frac{\int \widetilde{x}dL}{\int dL}, \quad \overline{y} = \frac{\int \widetilde{y}dL}{\int dL}, \quad dL = \sqrt{(dx)^2 + (dy)^2}\]

9.3 Theorems of Pappus and Guldinus

(1) Surface Area

\[dA = 2\pi rdL,\ A = 2\pi \int rdL=w\pi \overline{r}L=\theta \overline{r}L\]

  • The area of a surface of revolution equals the product of the length of the generating curve and the distance traveled by the centroid of the curve in generating the surface area

(2) Volume

\[dV = 2\pi rdA,\ V=2\pi \int rdA = 2\pi \overline{r}A=\theta \overline{r}A\]

  • The volume of a body of revolution equals the product of the generating area and the distance traveled by the centroid of the area in generating the colume

(3) Composite Shapes

\[A=\theta \sum(\overline{r}L), \quad V=\theta \sum(\overline{r}A)\]

9.4 Resultant of a General Distributed Loading

(1) Magnitude of Resultant Force

\[F_R=\int_A p(x,y)dA = \int_V dV = V\]

(2) Location of Resultant Force

\[\overline{x} = \frac{\int_A xp(x,y)dA}{\int_A p(x,y)dA} = \frac{\int_V xdV}{\int_V dV}, \quad \overline{y} = \frac{\int_A yp(x,y)dA}{\int_A p(x,y)dA} = \frac{\int_V ydV}{\int_V dV}\]

9.5 Fluid Pressure

  • \(p=\gamma z = \rho gz​\) : valid only for incompressible fluid, Pascal's law
  • If the surface is horizontal, the loading will be uniform
  • The resultant for tilted loading can be determined by finding the volume under the loading curve or \(F_R = \gamma \overline{z} A​\), where $\(\overline{z}​\) is the depth to the centroid of the plate area. The line of action passes through the centroid of the volume.