Centroid of a Composite Area

  • Calculate centroid of composite shapes, with or without cavities, using formula for centroid of primitive shapes.

Centroid of a composite area is found out by breaking that shape into series of smaller shapes for which we can locate centroid locations.

\[\bar{x} = \dfrac{\Sigma A_i x_i}{\Sigma A_i}, \ \bar{y} = \dfrac{\Sigma A_i y_i}{\Sigma A_i}\] In the above formula, if a shape is added (let’s say by welding) then the terms related to it will be positive. If a shape is removed (let’s say, a hole, cut or cavity) then the terms related to it will be negative.

Solved Examples

Solved Example:

26-2-01

The centre of gravity of a triangle is at the point where three:

Solution:
A centroid of a triangle is the point where the three medians of the triangle meet. A median of a triangle is a line segment from one vertex to the mid point on the opposite side of the triangle.

Correct Answer: C

Solved Example:

26-2-02

If an area has one line of symmetry the centroid will:

Solution:
The centroid of a 2D surface is a point that corresponds to the center of gravity of a very thin homogeneous plate of the same area and shape. If the area (or section or body) has one line of symmetry, the centroid will lie somewhere along the line of symmetry.

Correct Answer: A

Solved Example:

26-2-03

Calculate centroid of the figure shown.

26.2-03



Solution:

  • For quarter circle Area = 314.16 \(cm^2\)

    x = \(\dfrac{4R}{3\pi}\) = 8.4883 cm
    y = \(\dfrac{4R}{3\pi}\) = 8.4883 cm

  • For semi circle Area = 157.08 \(cm^2\)

    x = \(\dfrac{20}{2}\) = 10 cm
    y = \(\dfrac{4R}{3\pi}\) = 4.2441 cm

\[\bar{x} = \dfrac{\Sigma A_x}{\Sigma A} = \dfrac{1095.87}{157.08} = 6.98\ cm\] \[\bar{y} = \dfrac{\Sigma A_y}{\Sigma A} = \dfrac{2000}{157.08} = 12.73\ cm\]

Correct Answer: A

Solved Example:

26-2-04

Considering bottom left corner point as origin, what is the y coordinate of the centroid of the object shown?

26.2-04



Solution:
Let part 1 be the rectangle and part 2 be the semicircle. \[A_1 = 6 \times 3 = 18\ cm^2\] \[y_1 = 1.5\] \[A_2 = \dfrac{(\pi)(4)}{2} = 6.28\ cm^2\] \begin{equation*} \begin{split} y_2 & = 3 - \dfrac{4R}{3\pi} \\ & = 3 - \dfrac{8}{3 \pi} \\ & = 2.1511\ cm \end{split} \end{equation*} \begin{equation*} \begin{split} \bar{y} & = \dfrac{A_1 y_1 - A_2 y_2}{A_1 -A_2} \\ & = \dfrac{18 \times 1.5 - 6.28 \times 2.1511}{18 - 6.28} \\ & = \dfrac{27 - 13.509}{11.72} \\ & = 1.15\ cm \end{split} \end{equation*}

Correct Answer: D

Solved Example:

26-2-05

For the composite lamina shown in red in the figure, determine the coordinates of its centroid. (CBCGS Engineering Mechanics- Dec 2018)

26.2-05



Solution:

Part I = Large Rectangle
Area = 90 $\times$ 60 = 5400 mm$^2$
$\bar{x}$ = -30, $\bar{y}$ = 5

Part II = Small Rectangle
Area = 50 $\times$ 40 = 2000 mm$^2$
$\bar{x}$ = 25, $\bar{y}$ = -20

Part III = Quarter Circle
Area = $\dfrac{\pi}{4}$ $\times$ 50$^2$ = 1963.5 mm$^2$
$\bar{x}$ = $\dfrac{4R}{3\pi}$=21.22
$\bar{y}$ = $\dfrac{4R}{3\pi}$=21.22

Part IV = Triangle (Removed part)
Area = $\dfrac{1}{2}$ $\times$ 75 $\times$ 90 = 3375 mm$^2$
$\bar{x}$ =-35, $\bar{y}$ =-10

\begin{align*} \bar{x} &= \dfrac{\Sigma A_i x_i}{\Sigma A_i}\\ &= \dfrac{5400(-30) + 2000 (25) +1963.5 (21.22) - 3375(-35)}{5400 + 2000 + 1963.5 - 3375}\\ &= \dfrac{47790.5}{5988.5} = 7.98\ mm \end{align*} \begin{align*} \bar{y}&= \dfrac{\Sigma A_i y_i}{\Sigma A_i}\\ &= \dfrac{5400(5) + 2000(-20) + 1963.5 (21.22) -3375(-10)}{5400 + 2000 + 1963.5 - 3375}\\ &= \dfrac{62415.5}{5988.5} = 10.42\ mm \end{align*}

Correct Answer: A