powerpoint presentation - 21 centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_centroids.pdf · 1...
TRANSCRIPT
![Page 1: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/1.jpg)
1
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
![Page 2: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/2.jpg)
• 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.
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Recall working with distributed loads …
![Page 3: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/3.jpg)
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Recall working with distributed loads …
The moment calculated
using the resultant force
equals the summation of
the moments for each
differential area
![Page 4: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/4.jpg)
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.
![Page 5: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/5.jpg)
5
The moment of an area
![Page 6: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/6.jpg)
6
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.
![Page 7: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/7.jpg)
7
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]
Image copyright 2013, Pearson Education, publishing as Prentice Hall
![Page 8: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/8.jpg)
8
Centroids: Using Single Integration 1) DRAW a differential element on the graph.
2) Label the centroid (x, y) of the differential element.
3) Label the point where the element intersects the curve (x, y)
4) Write down the appropriate general equation to use.
5) Express each term in the general equation using the coordinates
describing the curve or function.
6) Determine the limits of integration
7) Integrate
~ ~
Image copyright 2013, Pearson Education, publishing as Prentice Hall
![Page 9: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/9.jpg)
9
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Centroids: Using Single Integration 1) DRAW a differential element on the graph.
2) Label the centroid (x, y) of the differential element.
3) Label the point where the element intersects the curve (x, y)
4) Write down the appropriate general equation to use.
5) Express each term in the general equation using the coordinates
describing the curve or function.
6) Determine the limits of integration
7) Integrate
~ ~
![Page 10: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/10.jpg)
1) DRAW a differential element on the graph.
2) Label the centroid (x, y) of the differential element.
3) Label the point where the element intersects the curve (x, y)
4) Write down the appropriate general equation to use.
5) 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
~ ~
![Page 11: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/11.jpg)
11
Image copyright 2013, Pearson Education, publishing as Prentice Hall
Centroids: Using Single Integration 1) DRAW a differential element on the graph.
2) Label the centroid (x, y) of the differential element.
3) Label the point where the element intersects the curve (x, y)
4) Write down the appropriate general equation to use.
5) Express each term in the general equation using the coordinates
describing the curve or function.
6) Determine the limits of integration
7) Integrate
~ ~
![Page 12: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/12.jpg)
12
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.)
2) DRAW BOTH dx and dy ‘elements’ on the graph
3) Label the centroid (x, y)
4) Write down the general equation
5) Define each term according to the problem
statement
6) Determine limits of integration (be careful here)
7) Integrate
~ ~
![Page 13: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/13.jpg)
13
Finding Centroids of Composite Shapes 1) Divide the object into simple shapes.
2) Establish a coordinate axis system on the sketch
3) Label the centroid (x, y) of each simple shape
4) Set up a table as shown below to calculate values
5) Subtract empty areas instead of adding them.
6) Keep track of negative
coordinates and carry
signs through
~ ~
Image copyright 2013, Pearson Education, publishing as Prentice Hall
3
2
1
![Page 14: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/14.jpg)
14
Finding Centroids of Composite Shapes 1) Divide the object into simple shapes.
2) Establish a coordinate axis system on the sketch
3) Label the centroid (x, y) of each simple shape
4) Set up a table as shown below to calculate values
5) Subtract empty areas instead of adding them.
6) Keep track of negative
coordinates and carry
signs through
~ ~
Image copyright 2013, Pearson Education, publishing as Prentice Hall
3
2
1
y
x
![Page 15: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/15.jpg)
15
Finding Centroids of Composite Shapes 1) Divide the object into simple shapes.
2) Establish a coordinate axis system on the sketch
3) Label the centroid (x, y) of each simple shape
4) Set up a table as shown below to calculate values
5) Subtract empty areas instead of adding them.
6) Keep track of negative
coordinates and carry
signs through
~ ~
Image copyright 2013, Pearson Education, publishing as Prentice Hall
3
2
1
![Page 16: PowerPoint Presentation - 21 Centroidsweb.mst.edu/~ide50-3/schedule/lessons/21_Centroids.pdf · 1 Class Today •Print notes and integration examples •Print composites examples](https://reader035.vdocument.in/reader035/viewer/2022062401/5a9db12f7f8b9a42488c4dc1/html5/thumbnails/16.jpg)
16
Finding Centroids of Composite Shapes 1) Divide the object into simple shapes.
2) Establish a coordinate axis system on the sketch
3) Label the centroid (x, y) of each simple shape
4) Set up a table as shown below to calculate values
5) Subtract empty areas instead of adding them.
6) Keep track of negative
coordinates and carry
signs through
~ ~
Image copyright 2013, Pearson Education, publishing as Prentice Hall
3
2
1