Vector Mechanics for Engineers: Statics, 11th Edition
Vector Mechanics for Engineers: Statics, 11th Edition
11th Edition
ISBN: 9780077687304
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek
Publisher: McGraw-Hill Education
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Chapter 5, Problem 5.137RP

5.137 and 5.138 Locate the centroid of the plane area shown.

Chapter 5, Problem 5.137RP, 5.137 and 5.138 Locate the centroid of the plane area shown. Fig. P5.137

Fig. P5.137

Expert Solution & Answer
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To determine

The centroid of the plane shown.

Answer to Problem 5.137RP

The centroid of the plane area (X¯,Y¯) is (5.67in, 5.17in).

Explanation of Solution

Refer Figure 1.

Vector Mechanics for Engineers: Statics, 11th Edition, Chapter 5, Problem 5.137RP

The plane is considered as two separate sections as in figure 1. Section 1 is a perpendicular triangle and section 2 is a rectangle.

Write an expression to calculate the area of section 1.

A1=12bh (I)

Here, A1 is the area of section 1, b is the base of the triangle and h is the height of the triangle.

Write an expression to calculate the area of section 2.

A2=lw (II)

Here, A2 is the area of section 2, l is the length of the rectangle and w is the width of the rectangle.

Write an expression to calculate the area of the plane.

A=A1+A2 (III)

Here, A is the area of the plane.

Write an expression to calculate the x component of the centroid of the plane.

X¯=1n(x¯iAi)A (IV)

Here, X¯ is the x component of the centroid of the plane, Ai is the area of each section and x¯i is the centroid of each section.

There are two sections in the plane. Rewrite equation (IV) according to the plane.

X¯=x1¯A1+x2¯A2A (V)

Here, x1¯ is the x component of the centroid of the triangle and x2¯ is the x component of the centroid of the rectangle.

Write an expression to calculate the y component of the centroid of the plane.

Y¯=1n(y¯iAi)A (VI)

Here, Y¯ is the y component of the centroid of the plane and y¯i is the centroid of each section.

There are two sections in the plane. Rewrite equation (VI) according to the plane.

Y¯=y1¯A1+y2¯A2A (VII)

Here, y1¯ is the y component of the centroid of the triangle and y1¯ is the y component of the centroid of the rectangle.

Conclusion:

Substitute 12in for b, and 6in for h in equation (I) to find A1.

A1=12(12in)(6in)=36in2

Substitute 6in for l and, 3in for w in equation (II) to find A2.

A2=(6in)(3in)=18in2

Substitute 36in2 for A1, and 18in2 for A2 in equation (III) to find A.

A=36in2+18in2=54in2

Substitute 4in for x1¯, 36in2 for A1, 18in2 for A2, 9in for x2¯, and 54in2 for A in equation (V) to find X¯.

X¯=(4in)(36in2)+(9in)(18in2)54in2=144in3+162in354in2=306in354in2=5.67in

Substitute 4in for y1¯, 36in2 for A1, 18in2 for A2, 7.5in for y2¯, and 54in2 for A in equation (VII) to find Y¯.

X¯=(4in)(36in2)+(7.5in)(18in2)54in2=144in3+135in354in2=279in354in2=5.17inY¯=y1¯A1+y2¯A2A

Thus, the centroid of the plane area (X¯,Y¯) is (5.67in, 5.17in).

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Chapter 5 Solutions

Vector Mechanics for Engineers: Statics, 11th Edition

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Determine (a)...Ch. 5.4 - Prob. 5.97PCh. 5.4 - Prob. 5.98PCh. 5.4 - Prob. 5.99PCh. 5.4 - Prob. 5.100PCh. 5.4 - Prob. 5.101PCh. 5.4 - Prob. 5.102PCh. 5.4 - Prob. 5.103PCh. 5.4 - For the machine element shown, locate the y...Ch. 5.4 - For the machine element shown, locate the x...Ch. 5.4 - 5.106 and 5.107 Locate the center of gravity of...Ch. 5.4 - 5.106 and 5.107 Locate the center of gravity of...Ch. 5.4 - A corner reflector for tracking by radar has two...Ch. 5.4 - A wastebasket, designed to fit in the corner of a...Ch. 5.4 - An elbow for the duct of a ventilating system is...Ch. 5.4 - A window awning is fabricated from sheet metal...Ch. 5.4 - Prob. 5.112PCh. 5.4 - Prob. 5.113PCh. 5.4 - A thin steel wire with a uniform cross section is...Ch. 5.4 - The frame of a greenhouse is constructed from...Ch. 5.4 - Locate the center of gravity of the figure shown,...Ch. 5.4 - Prob. 5.117PCh. 5.4 - A scratch awl has a plastic handle and a steel...Ch. 5.4 - PROBLEM 5.117 A bronze bushing is mounted inside a...Ch. 5.4 - PROBLEM 5.120 A brass collar, of length 2.5 in.,...Ch. 5.4 - PROBLEM 5.121 The three legs of a small...Ch. 5.4 - Prob. 5.122PCh. 5.4 - Determine by direct integration the values of x...Ch. 5.4 - Prob. 5.124PCh. 5.4 - PROBLEM 5.125 Locate the centroid of the volume...Ch. 5.4 - Prob. 5.126PCh. 5.4 - Prob. 5.127PCh. 5.4 - PROBLEM 5.128 Locate the centroid of the volume...Ch. 5.4 - PROBLEM 5.129 Locate the centroid of the volume...Ch. 5.4 - Show that for a regular pyramid of height h and n...Ch. 5.4 - PROBLEM 5.131 Determine by direct integration the...Ch. 5.4 - PROBLEM 5.132 The sides and the base of a punch...Ch. 5.4 - Locate the centroid of the section shown, which...Ch. 5.4 - Locate the centroid of the section shown, which...Ch. 5.4 - Determine by direct integration the location of...Ch. 5.4 - Alter grading a lot, a builder places four stakes...Ch. 5 - 5.137 and 5.138 Locate the centroid of the plane...Ch. 5 - 5.137 and 5.138 Locate the centroid of the plane...Ch. 5 - Prob. 5.139RPCh. 5 - Determine by direct integration the centroid of...Ch. 5 - Determine by direct integration the centroid of...Ch. 5 - The escutcheon (a decorative plate placed on a...Ch. 5 - Determine the reactions at the supports for the...Ch. 5 - A beam is subjected to a linearly distributed...Ch. 5 - A tank is divided into two sections by a 1 1-m...Ch. 5 - Determine the y coordinate of the centroid of the...Ch. 5 - An 8-in.-diameter cylindrical duct and a 4 8-in....Ch. 5 - Three brass plates are brazed to a steel pipe to...
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