Chapter 5.1-5.5: Distributed Forces: Centroids and Centers of Gravity

Statics Lecture 29: Center of Gravity

Statics Lecture 30: Centroid of an Area

Statics Lecture 31: Centroid of composite areas

Statics Lecture 32: Mass Moment of Inertia and Area Moment of Inertia

 

 Statics Lecture 33: Area Moment of Inertia -- Calculation

Statics Lecture 34: Area Moment of Inertia -- Composite Areas

wiki list of centroids:

http://en.wikipedia.org/wiki/List_of_centroids

http://www.engineering.com/Library/ArticlesPage/tabid/85/articleType/ArticleView/articleId/109/Centroids-of-Common-Shapes.aspx

centroids:

http://en.wikipedia.org/wiki/Centroid

 

5.2: Start in 2D

Divide plate into n small elements, each with weight "W".

element n = 1 (x1, y1)

element n = 2 (x2, y2) etc.

Total weight:

Moments about x and y axis:

 or: 

Center of Gravity of a wire:

** Center of gravity "G" is not usually located on the wire.

Similar to:


Just divide above by thickness and specific weight:

Centroid of an Area
Define coordinates for the center of gravity of a homogeneous plate:

For homogeneous wire:

5.4  First Moments of Areas and Lines

   

Symmetrical objects:

5.5  Composite Plates and Wires

 1st moment of composite area:

Pay attention to your signs!

Probs 5.1-5.33

Examples:

Find the centroids of the areas:

HW Problems:

x=19.27, y=26.6

x=92, y=23.3

x=3.2, y=2.0

56.7°

Calc Review

Youtubes:

First Fundamental Theorem:

http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-19-first-fundamental-theorem/

Second Fundamental Theorem

http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-20-second-fundamental-theorem/

When in doubt:
use http://integrals.wolfram.com/index.jsp

Q#2 review

From here on out, we will have a quiz every couple of chapters, so each will not cover too much material.

The chap 3-4 quiz will have:

• 2 2D problems
• 2 3D problems

Each problem will have

• 5 points for FBD's
• 15-25 points solving for M's and F's

What you need to know:

Forces at supports - don't memorize these, just think about what types of forces they can and cannot support!

2D examples:

3D examples:

Divide and Conquer!
Just take it one step at a time...


Steps for solving statics problems:
1. Draw a nice big FBD
2. List knowns
3. List unknowns
4. Figure out the geometry

• Find and label Θ's, lengths, etc.
• 2 sides + one Θ? Use law of Cosines
• 1 side and 3 Θ's? use Sine law
• Find unit vectors, split F's up into Fx + Fy + Fz

5. Apply equilibrium equ's:
                          Σ M = 0,  ΣF = 0

• Take your M about the point with the most unknowns to eliminate those unknowns from the M balance.
• Solve  Σ's with the fewest unknowns first.

6.  Check your answer

• Do the signs make sense?
• Do the magnitudes make sense?

Steps for Drawing a good FBD:

1.  Identify the object you want to isolate
2. Draw a sketch of the object isolated from it's surroundings
3. Draw vectors representing all of the external forces acting on the isolated object, and label them. 
          Understand reactions @ your supports and connections! 
4. Choose how to orient your coordinate system.
5.  Label all of your angles and distances.


2D Free Body Diagram Practice Problems:

 


 



3D Free Body Diagram Practice Problems:

2D Practice Problems:

 Start thinking about bridge designs...
What are the most effective ways to support weight?



Which of the below configurations puts the least stress on A?

3D Practice Problems: