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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