Engineering Mechanics - Statics Chapter 9 Problem 9-37 Locate the center of mass x c of the hemisphere. The density of the material varies linearly from zero at the origin O to ρ o at the surface. Hint: Choose a hemispherical shell element for integration Solution: for a spherical shell x c x 2 = ρρ 0 x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = dV 2 π x 2 dx= x c 0 a x ρ 0 x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ x 2 2 π x 2 ⌠ ⎮ ⎮ ⌡ d 0 a x ρ 0 x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 π x 2 ⌠ ⎮ ⎮ ⌡ d = 2 5 a⋅= x c 2 5 a= Problem 9-38 Locate the centroid z c of the right-elliptical cone. Given: a 3ft= b 4ft= c 10 ft= x b ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 y a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + 1= 921 © 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: Volume and Moment Arm : From the geometry, x cz− b c = x b c cz−()= y cz− a c = y a c cz−()= The volume of the thin disk differential element is dV π b c cz−() a c cz−()dz= z c 0 c zz π b c cz−() a c cz−() ⌠ ⎮ ⎮ ⌡ d 0 c z π b c cz−() a c cz−() ⌠ ⎮ ⎮ ⌡ d = z c 2.5 ft= Problem 9-39 Locate the center of gravity z c of the frustum of the paraboloid.The material is homogeneous. Given: a 1m= b 0.5 m= c 0.3 m= Solution V 0 a z π b 2 z a b 2 c 2 − () − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= 922 © 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 z c 1 V 0 a zz π b 2 z a b 2 c 2 − () − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= z c 0.422 m= Problem 9-40 Locate the center of gravity y c of the volume. The material is homogeneous. Given: a 25 mm= c 50 mm= d 50 mm= Solution: V c cd+ y π a y c ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⎮ ⌡ d= y c 1 V c cd+ yy π a y c ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⎮ ⌡ d= y c 84.7 mm= Problem 9-41 Locate the center of gravity for the homogeneous half-cone. 923 © 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 0 h y π 2 ay h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d= 1 6 h π a 2 = y c 6 ha 2 π 0 h yy π 2 ay h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d= 3 4 h= y c 3 4 h= z c 6 ha 2 π 0 h y 4ay 3h π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ π 2 ay h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d= 1 π a= z c a π = x c 6 ha 2 π 0 h y0 π 2 ay h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d= x c 0= Problem 9-42 Locate the centroid z c of the spherical segment . 924 © 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 a 2 a z π a 2 z 2 − () ⌠ ⎮ ⎮ ⌡ d= 5 24 a 3 π = z c 24 5 π a 3 a 2 a zz π a 2 z 2 − () ⌠ ⎮ ⎮ ⌡ d= 27 40 a= z c 27 40 a= Problem 9-43 Determine the location z c of the centroid for the tetrahedron. Suggestion: Use a triangular "plate" element parallel to the x-y plane and of thickness dz. 925 © 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: cz− c x a = y b = x a c cz−()= y b c cz−()= z c 0 c zz ab c 2 cz−() 2 ⌠ ⎮ ⎮ ⌡ d 0 c z ab c 2 cz−() 2 ⌠ ⎮ ⎮ ⌡ d = 1 4 c= z c 1 4 c= Problem 9-44 Determine the location (x, y) of the particle M 1 so that the three particles, which lie in the x–y plane, have a center of mass located at the origin O. Given: M 1 7kg= M 2 3kg= M 3 5kg= a 2m= b 3m= c 4m= Solution: Guesses x 1m= y 1m= Given M 1 xM 2 b+ M 3 c− 0= M 1 yM 2 a− M 3 a− 0= x y ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ Find xy,()= x y ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1.57 2.29 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ m= 926 © 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 9-45 Locate the center of gravity (x c , y c , z c ) of the four particles. Given: M 1 2lb= a 2ft= M 2 3lb= b 3ft= M 3 1lb= c 1− ft= M 4 1lb= d 1ft= f 4ft= e 4ft= h 2− ft= g 2ft= i 2ft= Solution: x c M 1 0ft M 2 a+ M 3 d+ M 4 g+ M 1 M 2 + M 3 + M 4 + = x c 1.29 ft= y c M 1 0ft M 2 b+ M 3 e+ M 4 h+ M 1 M 2 + M 3 + M 4 + = y c 1.57 ft= z c M 1 0ft M 2 c+ M 3 f+ M 4 i+ M 1 M 2 + M 3 + M 4 + = z c 0.429 ft= Problem 9-46 A rack is made from roll-formed sheet steel and has the cross section shown. Determine the location (x c , y c ) of the centroid of the cross section. The dimensions are indicated at the center thickness of each segment. Given: a 15 mm= c 80 mm= 927 © 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 50 mm= e 30 mm= Solution: L 3a 2c+ e+= x c 2a a 2 ae a 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ca e+()+ ea e 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + cd−()a+ L = x c 24.4 mm= y c d d 2 c c 2 + cd−() dc+ 2 + ad+ ec+ L = y c 40.6 mm= Problem 9-47 The steel and aluminum plate assembly is bolted together and fastened to the wall. Each plate has a constant width w in the z direction and thickness t . If the density of A and B is ρ s , and the density of C is ρ al , determine the location x c , the center of mass. Neglect the size of the bolts. Units Used: Mg 10 3 kg= Given: w 200 mm= a 300 mm= t 20 mm= b 100 mm= ρ s 7.85 Mg m 3 = c 200 mm= ρ al 2.71 Mg m 3 = 928 © 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 2 ρ s atw () a 2 ρ al bc+()tw ⎡ ⎣ ⎤ ⎦ ab− bc+ 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 2 ρ s atw ρ al bc+()tw+ = x c 179mm= Problem 9-48 The truss is made from five members, each having a length L and a mass density ρ . If the mass of the gusset plates at the joints and the thickness of the members can be neglected, determine the distance d to where the hoisting cable must be attached, so that the truss does not tip (rotate) when it is lifted. Given: L 4m= ρ 7 kg m = Solution: d ρ L L 2 L 4 + 3L 4 + L+ 5L 4 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 5 ρ L = d 3m= Problem 9-49 Locate the center of gravity (x c , y c , z c ) of the homogeneous wire. 929 © 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 300 mm= b 400 mm= Solution: L π a 2 2 a 2 b 2 ++= x c 1 L a 2 b 2 + a 2 π a 2 2a π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = y c 1 L a 2 b 2 + a 2 π a 2 2a π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = z c 1 L 2 a 2 b 2 + b 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = x c y c z c ⎛ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎠ 112.2 112.2 135.9 ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ mm= Problem 9-50 Determine the location (x c , y c ) of the center of gravity of the homogeneous wire bent in the form of a triangle. Neglect any slight bends at the corners. If the wire is suspended using a thread T attached to it at C, determine the angle of tilt AB makes with the horizontal when the wire is in equilibrium. Given: a 5in= b 9in= c 12 in= 930 © 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. [...]... publisher Engineering Mechanics - Statics Chapter 9 Solution: V= π 3 2 3 π 2 πa − a a = a 3 3 3 zc = 1 ⎡ 5a ⎛ 2 3 3 ⎛ π 3 ⎤ ⎢ ⎜ π a ⎟ − a ⎜ a ⎟⎥ V⎣ 8 3 ⎠ 4 ⎝ 3 ⎠⎦ zc = a 2 Problem 9-7 6 Determine the location xc 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 2 2 3 2 πd a + πd b + πd 3 3 xc = 1 ⎡ 1 2 ⎛ 3a ⎞ b ⎞ 2 3 3c... writing from the publisher Engineering Mechanics - Statics Chapter 9 xc = 1 ⎡ b πd ⎢b d − A⎣ 2 4 yc = 1 ⎡ ⎛ d⎞ πd ⎢b d⎜ ⎟ + A ⎣ ⎝ 2⎠ 4 2 ⎛ 4d ⎞ + 1 d c⎛ b + ⎜ ⎟ 2 ⎜ ⎝ ⎝ 3 ⎠ 2 c ⎞⎤ ⎟⎥ 3 ⎠⎦ xc = 2. 732 in 2 ⎤ ⎛ 4d ⎞ − π a ⎛ 4a ⎞ + 1 d c⎛ d ⎞⎥ ⎜ ⎟ ⎜ ⎟ 2 ⎜ 3 2 ⎝ 3 ⎠ ⎝ ⎠⎦ ⎝ 3 ⎠ yc = 1 .4 23 in Problem 9-5 7 Determine the location yc of the centroidal axis xcxc of the beam's cross-sectional area Neglect the... publisher Engineering Mechanics - Statics Chapter 9 Problem 9-7 0 Determine the distance to the centroid of the shape which consists of a cone with a hole of height h bored into its base Given: d = 100 mm h = 50 mm ρ = 8 mg m 3 a = 150 mm b = 500 mm Solution: 2 d⎞ b⎞ h⎞ π a b⎛ ⎟ − π ⎛ ⎟ h⎛ ⎟ ⎜ ⎜ ⎜ 3 ⎝ 4 ⎝ 2⎠ ⎝ 2⎠ zc = 2 d⎞ 1 2 πa b − π ⎛ ⎟ h ⎜ 3 ⎝ 2⎠ 1 2 zc = 128 .4 mm Problem 9-7 1 The sheet metal part has... publisher Engineering Mechanics - Statics Chapter 9 Problem 9-7 4 Determine the location (xc, yc) 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 M1 = 25 kg b = 1.25 m M2 = 80 kg c = 0.5 m M3 = 30 kg d = 0.75 m M4 = 105 kg e = 0.85 m f = 1 .30 m g = 0.95 m Solution: M = M1 + M2 + M3 + M4 xc... publisher Engineering Mechanics - Statics Chapter 9 c = 3 in d = 6 in Solution: 1 ⎛ c⎞ 1 2 ⎟ + a c⎜ b + ⎟ + ( b + c)d ( b + c) 3 ⎝ 2⎠ 2 ⎝ 3 2 a b⎛ ⎜ xc = b⎞ ab + 2 ca + 1 2 xc = 4. 625 in ( b + c)d 1 ⎛ a⎞ 1 ⎛ d⎞ ⎟ + a c⎜ ⎟ − ( b + c)d⎜ ⎟ ⎝ 2⎠ 2 ⎝ 3 2 ⎝ 3 a b⎛ ⎜ yc = 1 a⎞ ab + 1 2 ca + 1 2 yc = 1 in ( b + c)d Problem 9-5 9 Determine the location yc of the centroid C for a beam having the cross-sectional... permission in writing from the publisher Engineering Mechanics - Statics Chapter 9 Given: a = 1 in b = 3 in c = 1 in d = 1 in e = 1 in Solution: 2 A = ( a + b) ( a + e) − πa − 4 1 ( a + b − d) ( a + e − c) 2 1 ⎡ ( a + b) πa xc = ⎢ ( a + e) − A⎣ 2 4 2 yc = 2 1⎡ ( a + e) ⎢( a + b) A⎣ 2 2 ⎛ 4a ⎞ − 1 ( a + b − d) ( a + e − c) ⎛ a + b − a + b − d ⎞⎤ ⎜ ⎟⎥ ⎜ ⎟ 2 3 ⎝ ⎠⎦ ⎝ 3 ⎠ π a ⎛ 4a ⎞ 1 a + e − c ⎞⎤ ⎛ ⎟⎥ ⎜ ⎟ − 2... skyrocket is located along line aa Given: a = 3 mm ρ t = 600 m b = 10 mm ρ c = 40 0 c = 5 mm 3 kg m d = 100 mm e = 20 mm kg ρ s = 30 0 3 kg m 3 Solution: Guess Given x = Find ( x) x = 200 mm 2 ⎛ a2 ⎞ b⎞ e π 2 e d⎞ x 2 ρ tπ ⎛ ⎟ ⎛ d + ⎞ + ρ c b − c d⎛ ⎟ + ρ sπ ⎜ ⎟ x⎛ d − ⎞ = 0 ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎝ 2⎠ 3 ⎝ 4 4 ( ) ⎝ 2⎠ 4 ⎝ 2⎠ x = 49 0 mm 947 © 2007 R C Hibbeler Published by Pearson Education, Inc., Upper Saddle River,... xc = 3 sin ( α ) α− 3 sin ( 2α ) 2 Problem 9-6 3 Locate the centroid yc for the strut’s cross-sectional area Given: a = 40 mm b = 120 mm c = 60 mm Solution: A = πb 2 2 − 2a c 1 ⎡π b yc = ⎢ A⎣ 2 2 ⎛ 4b ⎞ − 2a c⎛ c ⎞⎤ ⎜ ⎟⎥ ⎜ ⎟ ⎝ 2 ⎠⎦ ⎝ 3 ⎠ yc = 56.6 mm Problem 9-6 4 The “New Jersey” concrete barrier is commonly used during highway construction Determine the location yc of its centroid Given: a = 4 in... 2 2 3 3 ⎛ xc ⎞ ⎛ 152.8 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ yc ⎟ = ⎜ −15.0 ⎟ mm ⎜ z ⎟ ⎝ 111.5 ⎠ ⎝ c⎠ M = 16 . 34 7 kg Problem 9-6 6 Locate the centroid yc of the concrete beam having the tapered cross section shown Given: a = 100 mm b = 36 0 mm c = 80 mm d = 30 0 mm e = 30 0 mm Solution: 1 ⎛ b⎞ ⎛ b⎞ ⎟ + ( d − a)b⎜c + ⎟ + a b⎜ c + ⎟ ⎝ 2⎠ 2 ⎝ 3 ⎝ 2⎠ ( d + 2e)c⎛ ⎜ yc = c⎞ ( d + 2e)c + 1 2 yc = 135 mm ( d − a)b + a b Problem 9-6 7... permission in writing from the publisher Engineering Mechanics - Statics Chapter 9 π 2 ⎛ a⎞ π 2 ⎛ h⎞ r a⎜ ⎟ − r h⎜ ⎟ zc = Given 3 ⎝ 4 3 ⎝ 4 π 2 3 h = Find ( h) h = 2 ft r ( a + h) Problem 9-7 9 Locate the center of mass zc 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 = 2 in d = 3 in e = 0.5 in Solution: 2 V = b a + 2e a . publisher. Engineering Mechanics - Statics Chapter 9 Solution: V a 2 a z π a 2 z 2 − () ⌠ ⎮ ⎮ ⌡ d= 5 24 a 3 π = z c 24 5 π a 3 a 2 a zz π a 2 z 2 − () ⌠ ⎮ ⎮ ⌡ d= 27 40 a= z c 27 40 a= Problem 9 -4 3 Determine. publisher. Engineering Mechanics - Statics Chapter 9 x c 1 A bd b 2 π d 2 4 4d 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − 1 2 dc b c 3 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = x c 2. 732 in= y c 1 A bd d 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ π d 2 4 4d 3 π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + π a 2 2 4a 3 π ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − 1 2 dc d 3 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ =. publisher. Engineering Mechanics - Statics Chapter 9 Problem 9 -4 5 Locate the center of gravity (x c , y c , z c ) of the four particles. Given: M 1 2lb= a 2ft= M 2 3lb= b 3ft= M 3 1lb= c 1− ft= M 4 1lb=