NotesAtHKU

Centroids

Expression of centroids

We use $\bar{x}$ and $\bar{y}$ to denote the centroid of a shape. They are the horizontal and vertical distance about a given reference axis.

The centroid is the centre of mass of the shape. For an axis that the shape is symmetric about, the centroid will be on the axis.

The centroid is also known as the first moment of area.

Centroid of composite shapes

The centroid of a shape can be found by:

\bar{x}=\frac{\sum (A_i x_i)}{\sum A_i}, \bar{y}=\frac{\sum (A_i y_i)}{\sum A_i}

Where $A_i$ is the area of the composition shape, and $x_i, y_i$ are the distances of the composition shape's centroid from the reference axes.

We measure the moment of inertia about a reference axis. This is because at different points, the moment of inertia will be different. (Unlike the use of reference axes in centroids, where the reference axes is just relative).

For finding the moment of inertia at the centroid, we simply set the reference axis at the centroid.

Moment of inertia / Second moment of area

The moment of inertia is a measure of an object's resistance to changes in its rotation, about a reference axis. It is given by:

I_x=\int y^2\:dA, I_y=\int x^2\:dA

Where $x$ and $y$ are the distances from the axis. This units are $m^4$ .

Moment of inertia for rectangles

The moment of inertia for a rectangle is given by:

I_x=\frac{bh^3}{12}, I_y=\frac{hb^3}{12}

Where $b$ is the width of the rectangle, and $h$ is the height of the rectangle. This can be derived by the formulas above.

It is important that the direction where $b$ extends is parallel to the reference axis (so width $\ne$ longest length)

Moment of inertia for circles

The moment of inertia for a circle is given by:

I_x=I_y=\frac{\pi r^4}{4}

Where $r$ is the radius of the circle.

Parallel axis theorem

The moment of inertia about an axis parallel to a reference axis for a shape is given by:

I_{x'}=I_{x}+Ad^2

Where $A$ is the area of the shape, and $d$ is the distance between the reference axis and the parallel axis.

Moment of inertia for composite shapes

The moment of inertia for a composite shape is simply the sum of the moments of inertia of the individual shapes about the same axis. This can be expressed as:

I_x=I_{x1\to\bar{y}} + I_{x2\to\bar{y}}, I_y=I_{y1\to\bar{x}} + I_{y2\to\bar{x}}

Related: Bending stress

Centroids and moment of inertia