Wize University Statics Textbook (Master) > Center of Gravity and Centriod

Centroids of Area

In transitioning from particles to rigid bodies, you may have noticed that the size of objects was on longer negligible. In all rigid body questions thus far, you have been provided with the location of the mass center – where the weight of the body acts. In this section, we will review methods to determine the centroids of an area. Objects in this course will typically have uniform and homogenous material, and therefore the mass center will act at the geometric center, or the centroid. Symmetric objects will have centroids along their lines of symmetry. Centroids of common geometric shapes are provided in the table below:

Given some geometric object, there are two ways to determine the location of its centroid:

1) Integration

Like any averaged property, integrating the first area moment and dividing by the area provides us with the location of the centroid. This can be computed as follows:

Hibbler: $\bar{x}=\frac{\int_{A} \tilde{x} d A}{\int_{A} d A}$ $\bar{y}=\frac{\Sigma \widetilde{y} A}{\Sigma A}$

Beer: $\overline{x}=\frac{\int_{ }^{ }xdA}{A}$ $\bar{y}=\frac{\int y d A}{A}$

2) Composite sections

A more common method to calculate centroids is based on composite shapes. We decompose the shape into multiple smaller sections, each with a centroid easy to determine (ie. square, triangle, circle, etc.), and average its properties to determine the centroid of the overall object. It is best to determine this to compute this is the format of a table. The integral formula simplifies to:

$\overline{x}=\frac{\Sigma^n_{i_=1}\tilde{x}_iA_i }{\Sigma^n_{i_=1}A_i }$

$\overline{y}=\frac{\bar{y}=\Sigma^n_{i_=1}\tilde{y}_iA_i }{\bar{y}=\Sigma^n_{i_=1}A_i }$

Note: bodies with cavities (holes) can be treated as negative area for purposes of this analysis.

In transitioning from particles to rigid bodies, you may have noticed that the size of objects was on longer negligible. In all rigid body questions thus far, you have been provided with the location of the mass center – where the weight of the body acts. In this section, we will review methods to determine centroids of an area. Objects in this course will typically have uniform and homogenous material, and therefore the mass center will act at the geometric center, or the centroid. Symmetric objects will have centroids along their lines of symmetry. Centroids of common geometric shapes are provided in the table below:

Given some geometric object, there are two ways to determine the location of its centroid:

Integration

Like any averaged property, integrating the first area moment and dividing by the area provides us with the location of the centroid. This can be computed as follows:

$\bar{x}A= \int{x}dA$ $\bar{y}A= \int{y}dA$

2) Composite sections

$\overline{x}=\frac{\Sigma^n_{i_=1}\tilde{x}_iA_i }{\Sigma^n_{i_=1}A_i }$ $\overline{y}=\frac{\bar{y}=\Sigma^n_{i_=1}\tilde{y}_iA_i }{\bar{y}=\Sigma^n_{i_=1}A_i }$

Note: bodies with cavities (holes) can be treated as negative area for purposes of this analysis.

Determine the location of the centroid.

Image illustrating Example: Centroids of Area - Geometric (DO THIS!)

Determine the location of the centroid.

Image illustrating Example: Centroids of Area - Integration (OPTIONAL)

Cut out one Strip,

1) For X-Bar,

$\bar{x}=\frac{\int_{A} \tilde{x} d A}{\int_{A} d A}$

$\text { Substitute } x / 2 \text { for } \tilde{x}, y^{1 / 3} d y^{\text {for }} d A, \text { and } y^{1 / 3} \text { for } x$

$\begin{aligned} \bar{x} &=\frac{\int_{0}^{1} \frac{1}{2}\left(y^{1 / 3}\right) \cdot\left(y^{1 / 3} d y\right)}{\int_{0}^{1} y^{1 / 3} d y} \\ &=\frac{\frac{1}{2} \int_{0}^{1} y^{2 / 3} d y}{\int_{0}^{1} y^{1 / 3} d y} \end{aligned}$

Apply Limits

$\bar{x}=\frac{\frac{1}{2}\left|y^{\left(\frac{2}{3}+1\right)} /\left(\frac{2}{3}+1\right)\right|_{0}^{1}}{\left|y^{\left(\frac{1}{3}+1\right)} /\left(\frac{1}{3}+1\right)\right|_{0}^{1}}$

$\begin{aligned}=& \frac{\frac{3}{10}\left|y^{(5 / 3)}\right|_{0}^{1}}{\frac{3}{4}\left|y^{(4 / 3)}\right|_{0}^{1}} \\=& \frac{\frac{3}{10}\left[(1)^{5 / 3}-(0)^{5 / 3}\right]}{\frac{3}{4}\left[(1)^{4 / 3}-(0)^{4 / 3}\right]} \end{aligned}$

$=\left(\frac{3}{10} \times \frac{4}{3}\right) \mathrm{m}$

$\overline{x}=0.4\ m$

2) For Y-bar use the same strip, dA does not change.

$\bar{y}=\frac{\int_{A} \tilde{y} d A}{\int_{A} d A}$

$\bar{y}=\frac{\int_{0}^{1}(y) \cdot\left(y^{1 / 3} d y\right)}{\int_{0}^{1} y^{1 / 3} d y}$

$\bar{y}=\frac{\int_{0}^{1} y^{4 / 3} d y}{\int_{0}^{1} y^{1 / 3} d y}$

$\bar{y}=\frac{\left|y^{\left(\frac{4}{3}+1\right)} /\left(\frac{4}{3}+1\right)\right|_{0}^{1}}{\left|y^{\left(\frac{1}{3}+1\right)} /\left(\frac{1}{3}+1\right)\right|_{0}^{1}}$

$=\frac{\frac{3}{7}\left|y^{(7 / 3)}\right|_{0}^{1}}{\frac{3}{4}\left|y^{(4 / 3)}\right|_{0}^{1}}$

$=\frac{\frac{3}{7}\left[(1)^{7 / 3}-(0)^{7 / 3}\right]}{\frac{3}{4}\left[(1)^{4 / 3}-(0)^{4 / 3}\right]}$

$\begin{aligned} &=\left(\frac{3}{7} \times \frac{4}{3}\right) \mathrm{m} \\ \bar{y} &=0.5714 \mathrm{m} \end{aligned}$

Practice

Find the centroid of the following shape.

$\bar{x}$ Enter Answer to 3 Sig Figs, Do Not Include Units. $\bar{y}$ Enter Answer to 3 Sig Figs, Do Not Include Units.

I don't know

Practice

Find the centroid $\overline{y}\$ of the following shape

$\bar{y}$ Enter Answer to 3 Sig Figs, Do Not Include Units.

I don't know

Determine the location $\bar{y}$ of the centroid C of the beam having the cross-sectional area shown working from the top-down.

Illustration for practice question: Determine the location of the centroid C of the beam having the cross-sectional area shown working from the top-down.

Y-BAR (mm.) - Enter Answer to 3 Sig Figs.

I don't know

Quiz: Centroids Practice - Geometric Quiz (New)

Determine the location of the centroid.

x-centroidy-centroid

I don't know

Determine the location of the centroid, as measured by $\bar{y}$

Illustration for practice question: Determine the location of the centroid, as measured by

Your answer

I don't know

Practice

Find the location of the centroid of the following body.

$\bar{x}$ Enter Answer to 3 Sig Figs, Do Not Include Units. $\bar{y}$ Enter Answer to 3 Sig Figs, Do Not Include Units. $\bar{z}$ Enter Answer to 3 Sig Figs, Do Not Include Units.

I don't know

Practice

Find the centroid of the following shape

$\bar{x}$ Enter Answer to 3 Sig Figs, Do Not Include Units. $\bar{y}$ Enter Answer to 3 Sig Figs, Do Not Include Units.

I don't know

Find the centroid of the following shape:

Illustration for practice question: Centroids (2D) Integration

$\bar{x}$ Enter Answer to 3 Sig Figs, Do Not Include Units. $\bar{y}$ Enter Answer to 3 Sig Figs, Do Not Include Units.

I don't know

Quiz: Centroids Practice - Integration Quiz (New)

Determine the location of the centroid.

Your answer

I don't know

Extra Practice

Centroid / Center of Mass & Moment of Inertia (2D)

Determine the distance (y bar) to the object's centroid AND the moment of inertia about the x' axis.

Centroids of Area

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Quiz: Centroids Practice - Geometric Quiz (New)

Practice

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Quiz: Centroids Practice - Integration Quiz (New)

Extra Practice