Engineering Mechanics - Statics Chapter 9 b 4in= c 6in= Solution: x c a− b b 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ab 1 2 ac+ = x c 1.143− in= y c ab a 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1 2 ac 2a 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ab 1 2 ac+ = y c 1.714 in= z c 1− 2 ac c 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ab 1 2 ac+ = z c 0.857− in= Problem 9-72 The sheet metal part has a weight per unit area of and is supported by the smooth rod and at C. If the cord is cut, the part will rotate about the y axis until it reaches equilibrium. Determine the equilibrium angle of tilt, measured downward from the negative x axis, that AD makes with the -x axis. Given: a 3in= b 4in= c 6in= 946 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Solution: x c ab b 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ab 1 2 ac+ = x c 1.143 in= z c 1 2 ac c 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ab 1 2 ac+ = z c 0.857 in= θ atan x c z c ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = θ 53.13 deg= Problem 9-73 A toy skyrocket consists of a solid conical top of density ρ t , a hollow cylinder of density ρ c , and a stick having a circular cross section of density ρ s . Determine the length of the stick, x, so that the center of gravity G of the skyrocket is located along line aa. Given: a 3mm= ρ t 600 kg m 3 = b 10 mm= c 5mm= ρ c 400 kg m 3 = d 100 mm= ρ s 300 kg m 3 = e 20 mm= Solution: Guess x 200 mm= Given ρ t π b 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 e 3 d e 4 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ρ c π 4 b 2 c 2 − () d d 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ρ s π a 2 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ xd x 2 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 0= x Find x()= x 490 mm= 947 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-74 Determine the location (x c , y c ) of the center of mass of the turbine and compressor assembly. The mass and the center of mass of each of the various components are indicated below. Given: a 0.75 m= M 1 25 kg= b 1.25 m= M 2 80 kg= c 0.5 m= M 3 30 kg= d 0.75 m= M 4 105 kg= e 0.85 m= f 1.30 m= g 0.95 m= Solution: MM 1 M 2 + M 3 + M 4 += x c 1 M M 2 aM 3 ab+()+ M 4 ab+ c+()+ ⎡ ⎣ ⎤ ⎦ = x c 1.594 m= y c 1 M M 1 dM 2 e+ M 3 f+ M 4 g+()= y c 0.940 m= Problem 9-75 The solid is formed by boring a conical hole into the hemisphere. Determine the distance z c to the center of gravity. 948 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Solution: V 2 3 πa 3 π 3 a 2 a−= π 3 a 3 = z c 1 V 5a 8 2 3 πa 3⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 3 4 a π 3 a 3⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = z c a 2 = Problem 9-76 Determine the location x c of the centroid of the solid made from a hemisphere, cylinder, and cone. Given: a 80 mm= b 60 mm= c 30 mm= d 30 mm= Solution: V 1 3 πd 2 a πd 2 b+ 2 3 πd 3 += x c 1 V 1 3 πd 2 a 3a 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ πd 2 ba b 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 2 3 πd 3 ab+ 3c 8 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = x c 105.2 mm= 949 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-77 The buoy is made from two homogeneous cones each having radius r. Find the distance z c to the buoy's center of gravity G. Given: r 1.5 ft= h 1.2 ft= a 4ft= Solution: z c π 3 r 2 a a 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ π 3 r 2 h h 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − π 3 r 2 ah+() = z c 0.7 ft= Problem 9-78 The buoy is made from two homogeneous cones each having radius r. If it is required that the buoy's center of gravity G be located at z c ,determine the height h of the top cone. Given: z c 0.5 ft= r 1.5 ft= a 4ft= Solution: Guess h 1ft= 950 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Given z c π 3 r 2 a a 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ π 3 r 2 h h 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − π 3 r 2 ah+() = h Find h()= h 2ft= Problem 9-79 Locate the center of mass z c of the forked lever, which is made from a homogeneous material and has the dimensions shown. Given: a 0.5 in= b 2.5 in= c 2in= d 3in= e 0.5 in= Solution: Vba 2 2ead+ π 2 ce+() 2 c 2 − ⎡ ⎣ ⎤ ⎦ a+= z c 1 V ba 2 b 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2ead b e+ c+ d 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + πa 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ce+() 2 bc+ e+ 4 ce+ 3π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ + π− a 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ c 2 bc+ e+ 4 c 3π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ + ... ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ = z c 4.32 in= 951 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-80 A triangular plate made of homogeneous material has a constant thickness which is very small. If it is folded over as shown, determine the location y c of the plate's center of gravity G. Given: a 6in= b 3in= c 1in= d 3in= e 1in= f 3in= Solution: y c 2db b 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1 2 2cb() 2b 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 1 2 2ef() f 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 2db 1 2 2cb()+ 1 2 2d()af+()+ = y c 0.75 in= Problem 9-81 A triangular plate made of homogeneous material has a constant thickness which is very small. If it is folded over as shown, determine the location z c of the plate's center of gravity G. Given a 6in= b 3in= c 1in= 952 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 d 3in= e 1in= f 3in= Solution: z c 1 2 2efa()2ea a 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 1 2 2 de−()a a 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 2db 1 2 2cb()+ 1 2 2da f+()+ = z c 1.625 in= Problem 9-82 Each of the three homogeneous plates welded to the rod has a density ρ and a thickness a. Determine the length l of plate C and the angle of placement, θ, so that the center of mass of the assembly lies on the y axis. Plates A and B lie in the x–y and z–y planes, respectively. Units Used: Mg 1000 kg= Given: a 10 mm= f 100 mm= b 200 mm= g 150 mm= c 250 mm= e 150 mm= ρ 6 Mg m 3 = Solution: The thickness and density are uniform Guesses θ 10 deg= l 10 mm= 953 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Given bf f 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ gl g 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ cos θ()− 0= c− e e 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ gl g 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ sin θ()+ 0= l θ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ Find l θ,()= l 265 mm= θ 70.4 deg= Problem 9-83 The assembly consists of a wooden dowel rod of length L and a tight-fitting steel collar. Determine the distance x c to its center of gravity if the specific weights of the materials are γ w and γ st .The radii of the dowel and collar are shown. Given: L 20 in= γ w 150 lb ft 3 = γ st 490 lb ft 3 = a 5in= b 5in= r 1 1in= r 2 2in= Solution: x c γ w πr 1 2 L L 2 γ st π r 2 2 r 1 2 −()ba b 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + γ w πr 1 2 L γ st π r 2 2 r 1 2 −()b+ = x c 8.225 in= Problem 9-84 Determine the surface area and the volume of the ring formed by rotating the square about the vertical axis. 954 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Given: θ 45 deg= Solution: A 22π b a 2 sin θ()− ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ a ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 22π b a 2 sin θ()+ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ a ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ + ...= A 8πba= V 2π ba 2 = Problem 9-85 The anchor ring is made of steel having specific weight γ st . Determine the surface area of the ring. The cross section is circular as shown. Given: γ st 490 lb ft 3 = a 4in= b 8in= 955 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Solution: A 2π a 2 ba− 4 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2π ba− 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = A 118 in 2 = Problem 9-86 Using integration, determine both the area and the distance y c to the centroid of the shaded area. Then using the second theorem of PappusGuldinus, determine the volume of the solid generated by revolving the shaded area about the x axis. Given: a 1ft= b 2ft= c 2ft= Solution: A 0 c ya y c ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 b+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= A 3.333 ft 2 = y c 1 A 0 c yya y c ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 b+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= y c 1.2 ft= V 2πy c A= V 25.1 ft 3 = 956 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-87 The grain bin of the type shown is manufactured by Grain Systems, Inc. Determine the required square footage of the sheet metal needed to form it, and also the maximum storage capacity (volume) within it. Given: a 30 ft= b 20 ft= c 45 ft= Solution: A 2πac 2π a 2 a 2 b 2 ++= A 11.9 10 3 × ft 2 = V 2πac a 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2π a 3 1 2 ab ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ += V 146 10 3 × ft 3 = Problem 9-88 Determine the surface area and the volume of the conical solid. 957 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Solution: A 2a 3 2 a 2 2π= A 3πa 2 = V 2 1 2 a 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 3 2 a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 3 6 a2π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = V π 4 a 3 = Problem 9-89 Sand is piled between two walls as shown. Assume the pile to be a quarter section of a cone and that ratio p of this volume is voids (air space). Use the second theorem of Pappus-Guldinus to determine the volume of sand. Given: r 3m= h 2m= p 0.26= Solution: V 1 p−() π 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ r 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ hr 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = V 3.487 m 3 = 958 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-90 The rim of a flywheel has the cross section A-A shown. Determine the volume of material needed for its construction. Given: r 300 mm= a 20 mm= b 40 mm= c 20 mm= d 60 mm= Solution: V 2π rb+ c 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ dc 2π r b 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ba+= V 4.25 10 6 × mm 3 = Problem 9-91 The Gates Manufacturing Co. produces pulley wheels such as the one shown. Determine the weight of the wheel if it is made from steel having a specific weight γ. Given: a 1in= c 0.5 in= d 1in= e 1in= f 0.25 in= b 2 cd+ e+()= γ 490 lb ft 3 = 959 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Solution: W γ2π da c d 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ cd+ a 3 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ af− 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ e+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = W 3.01 lb= Problem 9-92 The Gates Manufacturing Co. produces pulley wheels such as the one shown. Determine the total surface area of the wheel in order to estimate the amount of paint needed to protect its surface from rust. Given: a 1in= c 0.5 in= d 1in= e 1in= f 0.25 in= b 2 cd+ e+()= Solution: A 2π fc d+()ac+ 2 de+()c de+ 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 2 e 2 af− 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + cd+ e 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = A 70 in 2 = Problem 9-93 Determine the volume of material needed to make the casting. Given: r 1 4in= r 2 6in= 960 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 r 3 r 2 r 1 −= Solution: V 2π 2 π 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ r 2 2 4r 2 3π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2r 2 2r 3 () r 2 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 2 π 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ r 3 2 r 2 4r 3 3π − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = V 1.40 10 3 × in 3 = Problem 9-94 A circular sea wall is made of concrete. Determine the total weight of the wall if the concrete has a specific weight γ c . Given: γ c 150 lb ft 3 = a 60 ft= b 15 ft= c 8ft= d 30 ft= θ 50 deg= Solution: W γ c θ a 1 2 db c−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 3 bc−() 1 2 db c−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ + ab+ c 2 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ dc+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = W 3.12 10 6 × lb= Problem 9-95 Determine the surface area of the tank, which consists of a cylinder and hemispherical cap. 961 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Given: a 4m= b 8m= Solution: A 2π ab 2a π πa 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = A 302 m 2 = Problem 9-96 Determine the volume of the tank, which consists of a cylinder and hemispherical cap. Given: a 4m= b 8m= Solution: V 2π 4a 3π πa 2 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ a 2 ba()+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = V 536 m 3 = 962 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-97 Determine the surface area of the silo which consists of a cylinder and hemispherical cap. Neglect the thickness of the plates. Given: a 10 ft= b 10 ft= c 80 ft= Solution: A 2π 2a π πa 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ac+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = A 5.65 10 3 × ft 2 = Problem 9-98 Determine the volume of the silo which consists of a cylinder and hemispherical cap. Neglect the thickness of the plates. Given: a 10 ft= b 10 ft= c 80 ft= Solution: V 2π 4a 3π πa 2 4 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ca a 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = V 27.2 10 3 × ft 3 = 963 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-99 The process tank is used to store liquids during manufacturing. Estimate both the volume of the tank and its surface area. The tank has a flat top and the plates from which the tank is made have negligible thickness. Given: a 4m= b 6m= c 3m= Solution: V 2π c 3 ca 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ c 2 cb()+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = V 207 m 3 = A 2π c 2 ccb+ c 2 a 2 c 2 ++ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = A 188 m 2 = Problem 9-100 Determine the height h to which liquid should be poured into the cup so that it contacts half the surface area on the inside of the cup. Neglect the cup's thickness for the calculation. Given: a 30 mm= b 50 mm= c 10 mm= Solution: Total area A total 2π c c 2 ac+ 2 b 2 ac−() 2 ++ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = Guess h 1mm= e 1mm= 964 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Given ac− b ec− h = A total 2 2π c c 2 ec+ 2 h 2 ec−() 2 ++ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = e h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ Find eh,()= e 21.942 mm= h 29.9 mm= Problem 9-101 Using integration, compute both the area and the centroidal distance x c of the shaded region. Then, using the second theorem of Pappus–Guldinus, compute the volume of the solid generated by revolving the shaded area about the aa axis. Given: a 8in= b 8in= Solution: A 0 a xb x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d= x c 2a 1 A 0 a xxb x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d−= A 21.333 in 2 = x c 10 in= V 2πAx c = V 1.34 10 3 × in 3 = 965 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-102 Using integration, determine the area and the centroidal distance y c of the shaded area. Then, using the second theorem of Pappus–Guldinus, determine the volume of a solid formed by revolving the area about the x axis. Given: a 0.5 ft= b 2ft= c 1ft= Solution: A a b x c 2 x ⌠ ⎮ ⎮ ⌡ d= A 1.386 ft 2 = y c 1 A a b x 1 2 c 2 x ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⎮ ⌡ d= y c 0.541 ft= V 2πAy c = V 4.71 ft 3 = Problem 9-103 Determine the surface area of the roof of the structure if it is formed by rotating the parabola about the y axis. Given: a 16 m= b 16 m= 966 © 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. dhanesh_h Mathcad - CombinedMathcads