Class Today
• Print notes and integration examples
• Print composites examples
• Centroids
– Defined
– Finding Centroids
• Using single integration
• Using double integration
• Example Problems
• Group Work Time
1
Recall working with distributed loads …
• Distributed loads are sometimes
reduced to a single resultant force
at a particular location.
• The moment of a distributed load
is calculated using the single,
concentrated resultant force.
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Recall working with distributed loads …
The moment calculated
using the resultant force
equals the summation of
the moments for each
differential area
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Moments of …
The analysis of many
engineering problems involves
using the moments of quantities
such as masses, forces, volumes,
areas, or lines which, by nature,
are not concentrated values.
The moment of an area
5
Center of Gravity / Mass Defined
• CENTER OF MASS –
locates the point in a system
where the resultant mass can
be concentrated so that the
moment of the concentrated
mass with respect to any axis
equals the moment of the
distributed mass with respect
to the same axis.
• CENTER OF GRAVITY –
locates where the resultant,
concentrated weight acts on
a body.
6
Finding Centroids
Calculate as a weighted average:
1. Compute the “moment” of each differential element
[weight, mass, volume, area, length] about an axis
2. Divide by total [weight, mass, volume, area, length]
7
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Centroids: Using Single Integration
1)
2)
3)
4)
5)
DRAW a differential element on the graph.
~ ~
Label the centroid (x,
y) of the differential element.
Label the point where the element intersects the curve (x, y)
Write down the appropriate general equation to use.
Express each term in the general equation using the coordinates
describing the curve or function.
6) Determine the limits of integration
7) Integrate
8
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Centroids: Using Single Integration
1)
2)
3)
4)
5)
DRAW a differential element on the graph.
~ ~
Label the centroid (x,
y) of the differential element.
Label the point where the element intersects the curve (x, y)
Write down the appropriate general equation to use.
Express each term in the general equation using the coordinates
describing the curve or function.
6) Determine the limits of integration
7) Integrate
9
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Centroids: Using Single Integration
1)
2)
3)
4)
5)
DRAW a differential element on the graph.
~ ~
Label the centroid (x,
y) of the differential element.
Label the point where the element intersects the curve (x, y)
Write down the appropriate general equation to use.
Express each term in the general equation using the coordinates
describing the curve or function.
6) Determine the limits of integration
7) Integrate
10
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Centroids: Using Single Integration
1)
2)
3)
4)
5)
DRAW a differential element on the graph.
~ ~
Label the centroid (x,
y) of the differential element.
Label the point where the element intersects the curve (x, y)
Write down the appropriate general equation to use.
Express each term in the general equation using the coordinates
describing the curve or function.
6) Determine the limits of integration
7) Integrate
11
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Using Double Integration
1)
Determine whether you will integrate using dxdy
or dydx. (This will make a difference in how you define your
limits of integration.)
DRAW BOTH dx and dy ‘elements’ on the graph
~ y)
~
Label the centroid (x,
Write down the general equation
Define each term according to the problem
statement
6) Determine limits of integration (be careful here)
7) Integrate
2)
3)
4)
5)
12
Finding Centroids of Composite Shapes
1)
2)
3)
4)
5)
Divide the object into simple shapes.
Establish a coordinate axis system on the sketch
~ ~
Label the centroid (x, y) of each simple shape
Set up a table as shown below to calculate values
Subtract empty areas instead of adding them.
2
6)
Keep track of negative
coordinates and carry
signs through
3
1
13
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Finding Centroids of Composite Shapes
1)
2)
3)
4)
5)
Divide the object into simple shapes.
Establish a coordinate axis system on the sketch
~ ~
Label the centroid (x, y) of each simple shape
Set up a table as shown below to calculate values
Subtract empty areas instead of adding them.
y
2
3
6)
Keep track of negative
coordinates and carry
signs through
1
14
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x
Finding Centroids of Composite Shapes
1)
2)
3)
4)
5)
Divide the object into simple shapes.
Establish a coordinate axis system on the sketch
~ ~
Label the centroid (x, y) of each simple shape
Set up a table as shown below to calculate values
Subtract empty areas instead of adding them.
2
3
6)
Keep track of negative
coordinates and carry
signs through
1
15
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Finding Centroids of Composite Shapes
1)
2)
3)
4)
5)
Divide the object into simple shapes.
Establish a coordinate axis system on the sketch
~ ~
Label the centroid (x, y) of each simple shape
Set up a table as shown below to calculate values
Subtract empty areas instead of adding them.
2
3
6)
Keep track of negative
coordinates and carry
signs through
1
16
Image copyright 2013, Pearson Education, publishing as Prentice Hall