Centroids and Centers of Gravity

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Centroids of Composite Figures

Center of gravity of a homogeneous flat plate
$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

 

Centroids of areas
$A \, \bar{x} = \Sigma ax$

$A \, \bar{y} = \Sigma ay$

 

Centroids of lines
$L \, \bar{x} = \Sigma lx$

$L \, \bar{y} = \Sigma ly$

 

Center of Gravity of Bodies and Centroids of Volumes

Center of gravity of bodies
$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

$W \, \bar{z} = \Sigma wz$

 

Centroids of volumes
$V \, \bar{x} = \Sigma vx$

$V \, \bar{y} = \Sigma vy$

$V \, \bar{z} = \Sigma vz$

 

Centroids Determined by Integration

Centroid of area
$\displaystyle A \, \bar{x} = \int_a^b x_c \, dA$

$\displaystyle A \, \bar{y} = \int_a^b y_c \, dA$

 

Centroid of lines
$\displaystyle L \, \bar{x} = \int_a^b x_c \, dL$

$\displaystyle L \, \bar{y} = \int_a^b y_c \, dL$

 

Center of gravity of bodies
$\displaystyle W \, \bar{x} = \int_a^b x_c \, dW$

$\displaystyle W \, \bar{y} = \int_a^b y_c \, dW$

$\displaystyle W \, \bar{z} = \int_a^b z_c \, dW$

 

Centroids of volumes
$\displaystyle V \, \bar{x} = \int_a^b x_c \, dV$

$\displaystyle V \, \bar{y} = \int_a^b y_c \, dV$

$\displaystyle V \, \bar{z} = \int_a^b z_c \, dV$

 

Centroids of Common Geometric Shapes

Rectangle Area and Centroid
 
centroid and area of rectangle
 
$A = bd$

$\bar{x} = \frac{1}{2}b$

$\bar{y} = \frac{1}{2}d$

 

Triangle Area and Centroid
 
centroid and area of triangle
 
$A = \frac{1}{2}bh$

$\bar{y} = \frac{1}{3}h$

 

Circle Area and Centroid
 
000-circle.gif
 
$A = \pi r^2$

$\bar{x} = 0$

$\bar{y} = 0$

 

Semicircle Area and Centroid
 
centroid and area of semicircle
 
$A = \frac{1}{2}\pi r^2$

$\bar{x} = 0$

$\bar{y} = \dfrac{4r}{3\pi}$

 

Semicircular Arc Length and Centroid
 
centroid and length of semicircular arc
 
$L = \frac{1}{2}\pi r^2$

$\bar{x} = \dfrac{2r}{\pi}$

$\bar{y} = 0$

 

Quarter Circle Area and Centroid
 
centroid and area of quarter circle.gif
 
$A = \frac{1}{4}\pi r^2$

$\bar{x} = \dfrac{4r}{3\pi}$

$\bar{y} = \dfrac{4r}{3\pi}$

 

Sector of a Circle Area and Centroid
 
centroid and area of circular sector
 
$A = r^2 \theta_{rad}$

$\bar{x} = \dfrac{2r \sin \theta}{3\theta_{rad}}$

$\bar{y} = 0$

 

Circular Arc Length and Centroid
 
centroid and length of circular arc
 
$L = 2r \theta_{rad}$

$\bar{x} = \dfrac{r \sin \theta}{\theta_{rad}}$

$\bar{y} = 0$

 

Ellipse Area and Centroid
 
centroid and area of ellipse
 
$A = \pi ab$

$\bar{x} = 0$

$\bar{y} = 0$

 

Half Ellipse Area and Centroid
 
centroid and area of half ellipse
 
$A = \frac{1}{2}\pi ab$

$\bar{x} = 0$

$\bar{y} = \dfrac{4b}{3\pi}$

 

Quarter Ellipse Area and Centroid
 
centroid and area of quarter ellipse
 
$A = \frac{1}{4}\pi ab$

$\bar{x} = \dfrac{4a}{3\pi}$

$\bar{y} = \dfrac{4b}{3\pi}$

 

Parabolic Segment Area and Centroid
 
centroid and area of parabolic segment
 
$A = \frac{2}{3} bh$

$\bar{x} = \frac{3}{8}b$

$\bar{y} = \frac{2}{5}h$

 

Spandrel Area and Centroid
 
000-spandrel.gif
 
$A = \dfrac{1}{n + 1} bh$

$\bar{x} = \dfrac{1}{n + 2}b$

$\bar{y} = \dfrac{n + 1}{4n + 2}h$

 

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