Chapter one

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Chapter one. Center of Gravity and Centroid. Chapter Objectives. To discuss the concept of center of gravity, center of mass and the centroid. To show how to determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape. z. z.

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Presentation Transcript

Chapter one

Center of Gravity and

Centroid

Chapter Objectives
• To discuss the concept of center of gravity, center of mass and the centroid.
• To show how to determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape.

z

z

W2

z

Wn

W1

WR

WR

My

Mx

zn

G

z1

y

y

y

z

xn

x1

x

_

z

_

x

y

y1

_

y

yn

x

x

x

Center of Gravity, Center of Mass, and Centroid of a Body.Center of Gravity of a System of Particles

=

=

The unique point G where the resultant weight of a n-particle system is reduced to a single force (no moments) is called center of gravity.

Center of Gravity, Center of Mass, and Centroid of a Body. Center of Gravity – Center of Mass

A rigid body can be considered as made up of an infinite number of particles. Hence, using the same principles as in the previous slides, we get the coordinates of G by simply replacing

 , W

(discrete summation)

 , dW

(continuous summation)

Considering that the weight acts in the z-direction, then the equivalence between the particle and reduced systems shows that

 Fz,

 My,

The above formula can be generalized to

z

z

W2

z

Wn

W1

WR

WR

My

Mx

zn

G

z1

y

y

y

z

xn

x1

x

_

z

_

x

y

y1

_

y

yn

x

x

x

9.1 Center of Gravity, Center of Mass, and Centroid of a Body.Center of Mass of a System of Particles

=

=

The center of gravity (G) is a point which locates the resultant weight of a system of particles or body.

Similarly, the center of mass is a point which locates the resultant mass of a system of particles or body. Generally, its location is the same as that of G.

Since the acceleration due to gravity g is constant for every particle is constant, then,

W = mg

Substituting into the center of gravity formula and canceling g, yields to

The center of mass is obtained when the density,  =g, is considered, therefore

The center of mass is obtained when the density,  =g, is considered, therefore

The center of mass is obtained when the density,  =g, is considered, therefore

Since the specific weight is defined as =W/V or =dW/dV the center or gravity is expressed as

The centroidC is a point which defines the geometric center of an object.

The centroid coincides with the center of mass or the center of gravity only if the material of the body is homogenous (density or specific weight is constant throughout the body).

Center of Gravity, Center of Mass, and Centroid of a Body.Centroid Symmetry

If an object has an axis of symmetry, then the centroid of object lies on that axis.

Not necessarily, the centroid is located on the object.

3. Determine coordinates ( , ) of the centroid of the differential element in terms of the general point (x,y).

x

y

Center of Gravity, Center of Mass, and Centroid of a Body.Steps for Determining Area Centroid

1. Choose an appropriate differential element dA at a general point (x,y). Rectangular of polar coordinates could be used. Hint: Generally, if y is easily expressed in terms of x (e.g., y = x2 + 1), use a vertical rectangular element. If the converse is true, then use a horizontal rectangular element.

2. Express dA in terms of the differentiating element dx (or dy).

4. Express all the variables and integral limits in the formula using either x or y depending on whether the differential element is in terms of dx or dy, respectively, and integrate.

Note: Similar steps are used for determining CG, CM, etc. These steps will become clearer by doing a few examples.

Composite Bodies
• First moments of areas, like moments of forces, can be positive or negative.
• For example: an area whose centroid is located to the left of Y axis will have a negative first moment with respect to that axis.
• Also, the area of a hole should be assigned a negative sign.

y

y

Example 1

y

20 mm

30 mm

Locate the centroid of the plane

area shown.

36 mm

24 mm

x

Example 1

y

20 mm

30 mm

Locate the centroid of the plane area shown.

36 mm

Several points should be emphasized when solving these types of problems.

24 mm

x

1. Decide how to construct the given area from common shapes.

2. It is strongly recommended that you construct a table containing areas or length and the respective coordinates of the centroids.

3. When possible, use symmetry to help locate the centroid.

Example 1

y

20 + 10

Decide how to construct the given area from common shapes.

C1

C2

24 + 12

30

x

10

Dimensions in mm

Example 1

y

20 + 10

Construct a table containing areas and respective coordinates of the centroids.

C1

C2

Dimensions in mm

24 + 12

30

x

10

A, mm2 x, mm y, mm xA, mm3 yA, mm3

1 20 x 60 =1200 10 30 12,000 36,000

2 (1/2) x 30 x 36 =540 30 36 16,200 19,440

S 1740 28,200 55,440

Example 1

y

20 + 10

XSA = S xA

Then

X (1740) = 28,200

X = 16.21 mm

or

C1

C2

YSA = S yA

and

24 + 12

30

Y (1740) = 55,440

x

10

Y = 31.9 mm

or

Dimensions in mm

Example 2

For the plane area shown, determine (a) the first moments with respect to the X and Y axes, (b) the location of the centroid

120 mm

Y

60 mm

40 mm

80 mm

X

60 mm

Example 2

1. The area of the circle is negative, since it is to be subtracted from the other areas.

2. The coordinate y of the centroid of the triangle is negative for the axes shown

Example 2

• First moments of the area:
• SxA = 757700 mm3
• SyA = 506200 mm3
• (b) Location of Centroid (C):
• XSA = SxA and YSA = SyA
• X(13828 mm2) = 757700 mm3
• X = 54.8 mm
• Y(13828 mm2) = 506200 mm3
• Y = 36.6 mm

Y

C

36.6 mm

X

54.8 mm

Example 3

Find the centroid of the given body

Example 3

To find the centroid,

Determine the area of the components

Example 3

The centroid of each area

Example 3

The total area

Compute the x centroid

Example 3

Compute the y centroid

Example 3

The problem can be done using a table to represent the composite body.

An alternative method of computing the centroid is to subtract areas from a total area.

Assume that area is a large square and subtract the small triangular area.

Example 3

Example 3

The problem can be done using a table to represent the composite body.

Example 4

Find the centroid of the given body

Example 4

Determine the area of the components

Example 4

The centroid of each area

Example 4

An Alternative Method would be to subtract areas

Example 5

Find the centroid of the body

Example 6

Find the centroid of the body