# 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.

## 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?

## 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 = kx^{n} where n ≥ 0. What is the location of its centroid from the line x = b? Determine also the y coordinate of the centroid.

## 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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.

## 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.

## 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 y^{2} = kx.

## Centroids and Centers of Gravity

## Centroids of Composite Figures

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

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

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

## Center of Gravity of Bodies and Centroids of Volumes

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

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

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

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

## 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.

## 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.

The radius of incircle is given by the formula

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

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