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Contents

Introduction

Center of Gravity of a 2D Body

Centroids and First Moments of Areas and Lines

Centroids of Common Shapes of Areas

Centroids of Common Shapes of Lines

Composite Plates and Areas

Sample Problem 7.1

Determination of Centroids by Integration

Sample Problem 7.2

Theorems of Pappus-Guldinus

Sample Problem 7.3

Distributed Loads on Beams

Sample Problem 7.4

Center of Gravity of a 3D Body: Centroid of a Volume

Centroids of Common 3D Shapes

Composite 3D Bodies

Sample Problem 7.5

Introduction

- The earth exerts a gravitational force on each of the particles forming a body. These forces can be replace by a single equivalent force equal to the weight of the body and applied at the center of gravity for the body.

- The centroid of an area is analogous to the center of gravity of a body. The concept of the first moment of an area is used to locate the centroid.

- Determination of the area of a surface of revolution and the volume of a body of revolution are accomplished with the Theorems of Pappus-Guldinus.

- An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and is divided into two equal parts by BB’.

- If an area possesses two lines of symmetry, its centroid lies at their intersection.

- An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (-x,-y).

- The first moment of an area with respect to a line of symmetry is zero.

- If an area possesses a line of symmetry, its centroid lies on that axis

- The centroid of the area coincides with the center of symmetry.

Sample Problem 7.1

- SOLUTION:
- Divide the area into a triangle, rectangle, and semicircle with a circular cutout.

- Calculate the first moments of each area with respect to the axes.

- Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.

For the plane area shown, determine the first moments with respect to the x and y axes and the location of the centroid.

- Compute the coordinates of the area centroid by dividing the first moments by the total area.

Sample Problem 7.1

- Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.

Sample Problem 7.1

- Compute the coordinates of the area centroid by dividing the first moments by the total area.

- Double integration to find the first moment may be avoided by defining dA as a thin rectangle or strip.

Sample Problem 7.2 by defining

- SOLUTION:
- Determine the constant k.

- Evaluate the total area.

- Using either vertical or horizontal strips, perform a single integration to find the first moments.

Determine by direct integration the location of the centroid of a parabolic spandrel.

- Evaluate the centroid coordinates.

Sample Problem 7.2 by defining

- Using vertical strips, perform a single integration to find the first moments.

Sample Problem 7.2 by defining

- Or, using horizontal strips, perform a single integration to find the first moments.

- Evaluate the centroid coordinates. by defining

- Surface of revolution is generated by rotating a plane curve about a fixed axis.

- Area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the rotation.

- Body of revolution is generated by rotating a plane area about a fixed axis.

- Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the rotation.

The outside diameter of a pulley is 0.8 m, and the cross section of its rim is as shown. Knowing that the pulley is made of steel and that the density of steel is determine the mass and weight of the rim.

Sample Problem 7.3- SOLUTION:
- Apply the theorem of Pappus-Guldinus to evaluate the volumes or revolution for the rectangular rim section and the inner cutout section.

- Multiply by density and acceleration to get the mass and acceleration.

- SOLUTION section of its rim is as shown. Knowing that the pulley is made of steel and that the density of steel is :
- Apply the theorem of Pappus-Guldinus to evaluate the volumes or revolution for the rectangular rim section and the inner cutout section.

- Multiply by density and acceleration to get the mass and acceleration.

- A distributed load is represented by plotting the load per unit length, w (N/m) . The total load is equal to the area under the load curve (dW = wdx).

- A distributed load can be replace by a concentrated load with a magnitude equal to the area under the load curve and a line of action passing through the area centroid.

Sample Problem 7.4 unit length,

- SOLUTION:
- The magnitude of the concentrated load is equal to the total load or the area under the curve.

- The line of action of the concentrated load passes through the centroid of the area under the curve.

- Determine the support reactions by summing moments about the beam ends.

A beam supports a distributed load as shown. Determine the equivalent concentrated load and the reactions at the supports.

- SOLUTION unit length, :
- The magnitude of the concentrated load is equal to the total load or the area under the curve.

- The line of action of the concentrated load passes through the centroid of the area under the curve.

Sample Problem 7.4 unit length,

- Determine the support reactions by summing moments about the beam ends.

- Center of gravity unit length, G

- Results are independent of body orientation,

- For homogeneous bodies,

Centroids of Common 3D Shapes unit length,

- Moment of the total weight concentrated at the center of gravity G is equal to the sum of the moments of the weights of the component parts.

- For homogeneous bodies,

- SOLUTION gravity G is equal to the sum of the moments of the weights of the component parts.:
- Form the machine element from a rectangular parallelepiped and a quarter cylinder and then subtracting two 1-in. diameter cylinders.

Locate the center of gravity of the steel machine element. The diameter of each hole is 1 in.

Sample Problem 7.5 gravity G is equal to the sum of the moments of the weights of the component parts.

Sample Problem 7.5 gravity G is equal to the sum of the moments of the weights of the component parts.

Problem 7.6 gravity G is equal to the sum of the moments of the weights of the component parts.

y

20 mm

30 mm

Locate the centroid of the plane

area shown.

36 mm

24 mm

x

y gravity G is equal to the sum of the moments of the weights of the component parts.

Problem 7.6

20 mm

30 mm

Solving Problems on Your Own

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.

Problem 7.6 Solution gravity G is equal to the sum of the moments of the weights of the component parts.

y

20 + 10

Decide how to construct the given area from common shapes.

C1

C2

24 + 12

30

x

10

Dimensions in mm

Problem 7.6 Solution gravity G is equal to the sum of the moments of the weights of the component parts.

y

20 + 10

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

C1

C2

24 + 12

30

x

10

Dimensions in mm

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

Problem 7.6 Solution gravity G is equal to the sum of the moments of the weights of the component parts.

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

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

Problem 7.7 gravity G is equal to the sum of the moments of the weights of the component parts.

The beam AB supports two concentrated loads and rests on soil which exerts a linearly distributed upward load as shown. Determine (a) the distance a for which wA = 20 kN/m, (b) the corresponding value wB.

24 kN

30 kN

a

0.3 m

A

B

wA

wB

1.8 m

Problem 7.7 gravity G is equal to the sum of the moments of the weights of the component parts.

Solving Problems on Your Own

24 kN

30 kN

a

0.3 m

The beam AB supports two concentrated loads and rests on soil which exerts a linearly distributed upward load as shown. Determine (a) the distance a for which wA = 20 kN/m, (b) the corresponding value wB.

A

B

wA

wB

1.8 m

1. Replace the distributed load by a single equivalent force.

The magnitude of this force is equal to the area under the

distributed load curve and its line of action passes through

the centroid of the area.

2. When possible, complex distributed loads should be

divided into common shape areas.

1 gravity G is equal to the sum of the moments of the weights of the component parts.

2

1

2

Problem 7.7 Solution

24 kN

30 kN

a

0.3 m

Replace the distributed

load by a pair of

equivalent forces.

C

A

B

20 kN/m

wB

0.6 m

0.6 m

RI

RII

RI = (1.8 m)(20 kN/m) = 18 kN

We have

RII = (1.8 m)(wB kN/m) = 0.9 wB kN

+ gravity G is equal to the sum of the moments of the weights of the component parts.

+

Problem 7.7 Solution

24 kN

30 kN

a

0.3 m

C

A

B

wB

0.6 m

0.6 m

RII = 0.9 wB kN

RI = 18 kN

(a)

SMC = 0: (1.2 - a)m x 24 kN - 0.6 m x 18 kN

- 0.3m x 30 kN = 0

or a = 0.375 m

(b)

SFy = 0: -24 kN + 18 kN + (0.9 wB) kN - 30 kN= 0

or wB = 40 kN/m

y gravity G is equal to the sum of the moments of the weights of the component parts.

2 in

1 in

2 in

3 in

r = 1.25 in

x

z

0.75 in

2 in

r = 1.25 in

2 in

Problem 7.8

For the machine element

shown, locate the z coordinate

of the center of gravity.

y gravity G is equal to the sum of the moments of the weights of the component parts.

Problem 7.8

2 in

1 in

2 in

3 in

r = 1.25 in

x

z

0.75 in

2 in

r = 1.25 in

2 in

Solving Problems on Your Own

For the machine element

shown, locate the z coordinate

of the center of gravity.

Determine the center of gravity of composite body. For a homogeneous body

the center of gravity coincides

with the centroid of its volume. For this case the center of gravity can be determined by

X S V = S x V Y S V = S y V Z S V = S z V

where X, Y, Z and x, y, z are the coordinates of the centroid of the body and the components, respectively.

y gravity G is equal to the sum of the moments of the weights of the component parts.

Problem 7.8 Solution

2 in

1 in

2 in

3 in

r = 1.25 in

x

z

0.75 in

2 in

r = 1.25 in

2 in

y

V

IV

I

III

x

II

z

Determine the center of gravity

of composite body.

First assume that the machine

element is homogeneous so

that its center of gravity will

coincide with the centroid of

the corresponding volume.

Divide the body into five common shapes.

y gravity G is equal to the sum of the moments of the weights of the component parts.

y

2 in

1 in

V

2 in

IV

3 in

r = 1.25 in

I

III

x

II

x

z

0.75 in

z

2 in

r = 1.25 in

2 in

V, in3z, in. z V, in4

I (4)(0.75)(7) = 21 3.5 73.5

II (p/2)(2)2 (0.75) = 4.7124 7+ [(4)(2)/(3p)] = 7.8488 36.987

III -p(11.25)2 (0.75)= -3.6816 7 -25.771

IV (1)(2)(4) = 8 2 16

V -(p/2)(1.25)2 (1) = -2.4533 2 -4.9088

S 27.576 95.807

Z S V = S z V : Z (27.576 in3 ) = 95.807 in4Z = 3.47 in

Problem 7.9 gravity G is equal to the sum of the moments of the weights of the component parts.

y

y = kx1/3

Locate the centroid of the volume obtained by rotating the shaded area about the x axis.

a

x

h

y gravity G is equal to the sum of the moments of the weights of the component parts.

Problem 7.9

y = kx1/3

Solving Problems on Your Own

Locate the centroid of the volume obtained by rotating the shaded area about the x axis.

a

The procedure for locating the

centroids of volumes by direct

integration can be simplified:

x

h

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

2. If possible, identify an element of volume dV which produces

a single or double integral, which are easier to compute.

3. After setting up an expression for dV, integrate and determine

the centroid.

Problem 7.9 Solution gravity G is equal to the sum of the moments of the weights of the component parts.

y

Use symmetry to help locate the centroid. Symmetry implies

x

dx

y = 0 z = 0

x

z

r

Identify an element of volume dV which produces a single or double integral.

y = kx1/3

Choose as the element of volume a disk or radius r and thickness dx. Then

dV = p r2 dx xel = x

Problem 7.9 Solution gravity G is equal to the sum of the moments of the weights of the component parts.

y

x

Identify an element of volume dV which produces a single or double integral.

dx

dV = p r2 dx xel = x

x

z

r

r = kx1/3

so that

Now

dV = p k2 x2/3dx

y = kx1/3

a = kh1/3

k = a/h1/3

or

At x = h, y = a :

a2

h2/3

dV = p x2/3dx

Then

a gravity G is equal to the sum of the moments of the weights of the component parts.2

h2/3

dV = p x2/3dx

3

5

h

ò

0

3

8

= p a2h2

Problem 7.9 Solution

y

Integrate and determine the centroid.

x

dx

a2

h2/3

h

ò

V = p x2/3dx

x

z

r

0

a2

h2/3

h

[ ]

3

5

= p x5/3

y = kx1/3

0

= p a2h

a2

h2/3

a2

h2/3

3

8

ò

xel dV = x (p x2/3 dx) = p [ x8/3 ]

Also

3 gravity G is equal to the sum of the moments of the weights of the component parts.

5

3

5

5

8

x = h

Problem 7.9 Solution

y

Integrate and determine the centroid.

x

dx

V = p a2h

3

8

ò

xel dV = p a2h2

x

z

r

y = kx1/3

3

8

ò

xV = xdV:

x ( p a2h) = p a2h2

Now

y = 0 z = 0

Problem 7.10 gravity G is equal to the sum of the moments of the weights of the component parts.

The square gate AB is held in the position shown by hinges along its top edge A and by a shear pin at B. For a depth of water d = 3.5 ft, determine the force exerted on the gate by the shear pin.

A

d

1.8 ft

B

30o

Problem 7.10 gravity G is equal to the sum of the moments of the weights of the component parts.

Solving Problems on Your Own

The square gate AB is held in the position shown by hinges along its top edge A and by a shear pin at B. For a depth of water d = 3.5 ft, determine the force exerted on the gate by the shear pin.

A

d

1.8 ft

B

30o

Assuming the submerged body has a width b, the load per unit length is w = brgh, where h is the distance below the surface of the fluid.

1. First, determine the pressure distributionacting perpendicular

the surface of the submerged body. The pressure distribution

will be either triangular or trapezoidal.

Problem 7.10 gravity G is equal to the sum of the moments of the weights of the component parts.

Solving Problems on Your Own

The square gate AB is held in the position shown by hinges along its top edge A and by a shear pin at B. For a depth of water d = 3.5 ft, determine the force exerted on the gate by the shear pin.

A

d

1.8 ft

B

30o

2. Replace the pressure distribution with a resultant force, and

construct the free-body diagram.

3. Write the equations of static equilibrium for the problem, and

solve them.

Problem 7.10 Solution gravity G is equal to the sum of the moments of the weights of the component parts.

Determine the pressure distribution

acting perpendicular the surface of the submerged body.

1.7 ft

A

PA

PA = 1.7 rg

PB = (1.7 + 1.8 cos 30o)rg

(1.8 ft) cos 30o

B

PB

1 gravity G is equal to the sum of the moments of the weights of the component parts.

2

1

2

1

2

1

2

Problem 7.10 Solution

Ay

A

Replace the pressure distribution with a resultant force, and construct the free-body diagram.

Ax

1.7 rg

(1.8 ft) cos 30o

LAB/3

P1

FB

The force of the water on the gate is

LAB/3

P2

B

LAB/3

(1.7 + 1.8 cos 30o)rg

P = Ap = A(rgh)

P1 = (1.8 ft)2(62.4 lb/ft3)(1.7 ft) = 171.85 lb

P2 = (1.8 ft)2(62.4 lb/ft3)(1.7 + 1.8 cos 30o)ft = 329.43 lb

30 gravity G is equal to the sum of the moments of the weights of the component parts.o

Problem 7.10 Solution

Ay

A

Ax

Write the equations of

static equilibrium for the

problem, and solve them.

1.7 rg

(1.8 ft) cos 30o

LAB/3

P1

FB

S MA = 0:

+

LAB/3

P2

B

2

3

1

3

( LAB)P1 + ( LAB)P2

LAB/3

(1.7 + 1.8 cos 30o)rg

- LABFB = 0

P1 = 171.85 lb

P2 = 329.43 lb

2

3

1

3

(171.85 lb) + (329.43 lb) - FB = 0

FB = 276.90 lb

FB = 277 lb

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