centroid

715 Semicircle and Triangle | Centroid of Composite Figure

Problem 715
Determine the coordinates of the centroid of the area shown in Fig. P-715 with respect to the given axes.
 

Semicircle surmounted on top of a right triangle

 

714 Inverted T-section | Centroid of Composite Figure

Problem 714
The dimensions of the T-section of a cast-iron beam are shown in Fig. P-714. How far is the centroid of the area above the base?
 

Inverted T-section for centroid problem

 

709 Centroid of the area bounded by one arc of sine curve and the x-axis

Problem 709
Locate the centroid of the area bounded by the x-axis and the sine curve $y = a \sin \dfrac{\pi x}{L}$ from x = 0 to x = L.
 

708 Centroid and area of spandrel by integration

Problem 708
Compute the area of the spandrel in Fig. P-708 bounded by the x-axis, the line x = b, and the curve y = kxn where n ≥ 0. What is the location of its centroid from the line x = b? Determine also the y coordinate of the centroid.
 

Centroid and area of spandrel under the curve y = kx^n

 

707 Centroid of quarter ellipse by integration

Problem 707
Determine the centroid of the quadrant of the ellipse shown in Fig. P-707. The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.
 

Centroid of quarter ellipse in the first quadrant

 

706 Centroid of quarter circle by integration

Problem 706
Determine the centroid of the quarter circle shown in Fig. P-706 whose radius is r.
 

Centroid of quarter circle

 

705 Centroid of parabolic segment by integration

Problem 705
Determine the centroid of the shaded area shown in Fig. P-705, which is bounded by the x-axis, the line x = a and the parabola y2 = kx.
 

Open to the right parabola in the first quadrant

 

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

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

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

 

240 How to locate the centroid of metal plate with circular hole

Problem 240
The shaded area in Fig P-240 represents a steel plate of uniform thickness. A hole of 4-in. diameter has been cut in the plate. Locate the center of gravity the plate. Hint: The weight of the plate is equivalent to the weight of the original plate minus the weight of material cut away. Represent the original plate weight of plate by a downward force acting at the center of the 10 × 14 in. rectangle. Represent the weight of the material cut away by an upward force acting at the center of the circle. Locate the position of the resultant of these two forces with respect to the left edge and bottom of the plate.
 

Rectangular plate with circular hole

 

Centers of a Triangle

This page will define the following: incenter, circumcenter, orthocenter, centroid, and Euler line.
 

Incenter
Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle.
 

incenter-incircle.jpg

 

The radius of incircle is given by the formula

$r = \dfrac{A_t}{s}$

where At = area of the triangle and s = ½ (a + b + c). See the derivation of formula for radius of incircle.
 

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