E-Liberary Pakistan: Centroids and Centers of Gravity

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

$W \, \bar{x} = \Sigma wx$

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

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

Centroids of volumes

$V \, \bar{x} = \Sigma vx$

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

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

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
	$A = bd$ $\bar{x} = \frac{1}{2}b$ $\bar{y} = \frac{1}{2}d$

Triangle	Area and Centroid
	$A = \frac{1}{2}bh$ $\bar{y} = \frac{1}{3}h$

Circle	Area and Centroid
	$A = \pi r^2$ $\bar{x} = 0$ $\bar{y} = 0$