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Assignment 8 Key Solution Chapter 5, Problem 7 Locate the centroid of the plane area shown. Solution 7. 1 1365. 32 2 Then and

Assignment 8

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Page 1: Assignment 8

Assignment 8 Key Solution

Chapter 5, Problem 7

Locate the centroid of the plane area shown.

Solution 7.

1 1365.32

2

Then

and

Page 2: Assignment 8

Chapter 5, Problem 10.

Show that as approaches the location of the centroid approaches that

of a circular arc of radius

Chapter 5, Solution 10.

First, determine the location of the centroid.

Then

and

Now

Page 3: Assignment 8

Chapter 5, Problem 18.

The horizontal x axis is drawn through the centroid C of the area shown and divides the area into two component areas Determine the first moment of each component area with respect to the x axis, and explain the results obtained.

Solution 18.

Then

Now, for the first moments about the x-axis:

Area I

Page 4: Assignment 8

Area II

Note that which is expected as since x is a centroidal axis.

Page 5: Assignment 8

Chapter 5, Problem 26.

The homogeneous wire ABCD is bent as shown and is supported by a pin at B. Knowing that determine the angle for which portion BC of the wire is horizontal.

Chapter 5, Solution 26.

The wire supported only by the pin at B is a two-force body. For equilibrium the center of gravity of the wire must lie directly under B. Also, because the wire is homogeneous the center of gravity will coincide with the centroid. In other words,

or

or

Page 6: Assignment 8

Chapter 5, Problem 37.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and b.

Chapter 5, Solution 37.

or

Then

Hence

Page 7: Assignment 8

Chapter 5, Problem 43.

A homogeneous wire is bent into the shape shown. Determine by direct integration the x coordinate of its centroid.

Solution 43.

First note that because the wire is homogeneous, its center of gravity coincides with the centroid of the corresponding line

Now and

Hence

Alternative solution

Chapter 5, Problem 91.

Page 8: Assignment 8

Determine the y coordinate of the centroid of the body shown.

Chapter 5, Solution 91. Labeling the two parts of the body as follows:

V

1

2

Then or

Chapter 5, Problem 100.

Page 9: Assignment 8

Sheet metal of uniform thickness is used to fabricate a portion of the flashing for a roof. Locate the center of gravity of the flashing knowing that it is composed of the three elements shown.

Chapter 5, Solution 100.

Labeling the five parts of the body as follows, and noting that the center of gravity coincides with the centroid of the area due to the uniform thickness.

1 300 200 0 72 48 0

2 600 200 72 24

3 600 140 8.0640

4 300 400 54 72

5 240 400 2.6588

471 057 169.358 125.159

Page 10: Assignment 8

Therefore:

or

or

or

Chapter 5, Problem 109.

A thin steel wire of uniform cross section is bent into the shape shown, where arc BC is a quarter circle of radius R. Locate its center of gravity.

Chapter 5, Solution 109.

Page 11: Assignment 8

Assume that the body is homogeneous so that its center of gravity coincides with the centroid of the line

1 300 0 150 0 0 45 000 0

2 280 140 0 0 39 200 0 0

3 260 230 0 120 59 800 0 31 200

4 54 000 90 000 72 000

1311.24 153 000 135 000 103 200

Then