5. Centroid of an Area

by M. Bourne


Typical (straight sided) Problem:

In tilt-slab construction, we have a concrete wall (with doors and windows cut out of it) which we need to raise into position. We have only one rope and we must attach it to the centre of mass of the wall.

In this section we'll see how to find the centroid of an area with straight sides, then we'll extend this to areas with curved sides (where we'll use integration).

Moment

The moment of a mass is a measure of its tendency to rotate about a point. Clearly, the greater the mass (and the greater the distance from the point), the greater will be the tendency to rotate.

The moment is defined as:

Moment = mass × distance from a point

Example

math

In this case, there will be a total moment about O of:

(Clockwise is regarded as positive in this work.)

M = 2 × 1 − 10 × 3 = -28 kgm

Centre of Mass

We now aim to find the centre of mass of the system and this will lead to a more general result.

Example

We have 3 masses of 10 kg, 5 kg and 7 kg at 2 m, 2 m and 1 m distance from O as shown.

math

We wish to replace these masses with one single mass to give an equivalent moment. Where should we place this single mass?

Centre of Mass (Centroid) for a Thin Plate

1) Rectangle:

math

The centroid is (obviously) going to be exactly in the centre of the plate, at (2, 1).

2) More Complex Shapes:

We divide the complex shape into rectangles and find math (the x-coordinate of the centroid) and math (the y-coordinate) by taking moments about the y and x coordinates respectively.

Because they are thin plates with a uniform density, we can just calculate moments using the area.

Example:

Find the centroid of the shape:

math

In general, we can say:

math

math

This idea is used more extensively in the next section.

Centroid for Curved Areas

Taking the simple case first, we aim to find the centroid for the area defined by a function f(x), and the 2 vertical lines x = a and x = b as indicated in the following figure.

math

To find the centroid, we use the same basic idea that we were using for the straight-sided case above. The "typical" rectangle indicated has width Δx and height y = f(x).

Generalizing from the above rectangular areas case, we can find the coordinates \large{(\bar{x},\bar{y})} of the centroid using the total moments in the x-direction, given by:

\large{\bar{x}=\frac{\text{total\ moments}}{\text{total\ area}}=\frac{1}{A}\int_a^bx\ f(x)\ dx}

And, considering the moments in the y-direction about the x-axis and re-expressing the function in terms of y,

\large{\bar{y}=\frac{\text{total\ moments}}{\text{total\ area}}=\frac{1}{A}\int_c^dy\ f(y)\ dy}

Of course, there may be a rectangular portion to consider separately, as in the given diagram above.

Alternate method: Depending on the function, it may be easier to use the following alternative formula for the y-coordinate, which is derived from considering moments in the x-direction (Note the "dx" and the upper and lower limits are along the x-axis):

\large{\bar{y}=\frac{\text{total\ moments}}{\text{total\ area}}=\frac{1}{A}\int_a^b\frac{[f(x)]^2}{2}dx}

Another advantage of this second formula is there is no need to re-express the function in terms of y.

Centroids for Areas Bounded by 2 Curves

math

We extend the simple case given above. The "typical" rectangle indicated has width Δx and height y2 − y1, so the total moments in the x-direction over the total area is given by:

\large{\bar{x}=\frac{\text{total\ moments}}{\text{total\ area}}=\frac{1}{A}\int_a^bx(y_2-y_1)dx}

For the y coordinate, we have 2 different ways we can go about it.

Method 1: We take moments about the y-axis and so we'll need to re-express the expressions x2 and x1 as functions of y.

\large{\bar{y}=\frac{\text{total\ moments}}{\text{total\ area}}=\frac{1}{A}\int_c^dy(x_2-x_1)dy}

Method 2: We can also keep everything in terms of x by extending the "Alternate Method" given above:

\large{\bar{y}=\frac{\text{total\ moments}}{\text{total\ area}}=\frac{1}{A}\int_a^b\frac{[y_2]^2-[y_1]^2}{2}dx}

Example

Find the centroid of the area bounded by y = x3, x = 2 and the x-axis.

Didn't find what you are looking for on this page? Try search:

Find your integral using Wolfram|Alpha!

This next search box allows you to enter your math problem and Mathematica solves it for you. (It's free!)

See how to enter math.

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Online algebra solver - can solve your math problems step-by-step

Calculus Lessons on DVD

get MathTutorDVDs

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

Ready for a break?

shadow factory

Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

This page has

  • 1 tweets
  • 25 Facebook likes & comments

Short URL for this Page

Save typing! You can use this URL to reach this page:

intmath.com/centroid

axs