Engineering Statics

F-8


Determine the centroid of the area shown in terms of 'a'.

The first thing we do is draw a diagram of the shape and break the area into 
two components, a rectangle A<sub>1</sub> and the area under the curve, A<sub>2</sub>.
The area  A<sub>1</sub> is the area of a rectangle of base 1/a and height a. 
Hence the area is 1.  <BR>The centroid of the area in the x direction is in the 
middle of <I>1/2 * 1/a</I> or <I>1/(2a)</I>.
The second area must be found by integraion.  <BR>A vertical slice of <I>dA=ydx</I> has 
been chosen.
The value for <I>1/x</I> has been substituted for y.
The integral will be performed between the limits of <I>1/a</I> (left side) and <I>a</I>.
The integration has been performed.
The limits have been substituted.
Simplifying:
ln(a<sup>-1</sup> = -1(lna).
Combining terms:
The centroid in the x direction is calculated using <I>dA=(1/x)dx</I>. 
<BR>Therefore, <I>xdA = x(1/x)dx = dx</I>.
Simplification sets up the integral.
Integrating:
Substituting limits:
A table is now set up to find the combined centroid.  <BR>The technique used is 
identical to that used when standard bodies are analyzed.  <BR>The total area 
is found and the total value of the first moment is found.
The first moment terms are placed over a common denominator.
Simplifying:
Simplifying:
Dividing by the area gives the final answer.