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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

• 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 …

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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

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.

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The moment of an area

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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.

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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]

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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

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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

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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

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Image copyright 2013, Pearson Education, publishing as Prentice Hall

Centroids: Using Single Integration

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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

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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.)

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

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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

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Image copyright 2013, Pearson Education, publishing as Prentice Hall

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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

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2

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y

x

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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

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2

1

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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

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