Engineering Statics

F-5


Locate by direct integration the centroid of the area shown.

The first thing we do is draw a diagram of the shape and the slice we will use during integration.
We know the area of the triangle is 1/2 hb.
The first moment of the area can be written either as the distance from 
the y axis to the centroid of the area times the area or the 
integral of the distance to each area dA times the distance to the 
centroid of the slice.
The area of the slice is the height times the width or ydx.
In order to integrate we must express y=f(x).   We know that the equation 
of a line is y = mx+b.  The slope of the given line is h/b and since the 
line passes through the origin, the intercept is, b, is 0.  Hence the 
functional relationship is y = (h/b)x. 
The slices are added from 0 to b, hence the limits of integration.
The constant (h/b) can be factored out of the integral sign.
The integration is performed.
The upper and lower limits are substituted in.
Simplifying:
The centroid is defined as the first moment of the area divided by the area.
The expressions for the numerator and denominator are substituted.
Simplification yields the location of the centroid in the x-direction.
The centroid in the y direction is defined as the first moment of the area 
about the x axis, divided by the area.  The equation of the line y=(h/b)x 
can be solved for easier substitution.
A horizontal slice is taken of the triangle.<BR> 
The horizontal slice will extend from <BR> 
the vertical line at x=b to the diagonal line defining the triangle.<BR> 
Hence dA=(b-x)dy, where (b-x) is the length of the slice.
Substituting in for x yields an expression in terms of y and dy for dA.
Substituting into the integral and adding the limits of integration in the 
y direction, from 0 to h:
For ease of integration the integral is split into two integrals.
Constant are factored out of the integral sign.
The integration is performed and the limits of integration are 
substituted for y.
The expressions arae simplified.
The two terms are combined (1/2 - 1/3 = 1/6).
The value of A is substituted into the denominator.
The location of the centroid in the y-direction is determined.<BR>Notice that 
for a triangle, the centroid in both directions is located one third of the 
distance from the base.