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Engineering Mechanics - Statics Chapter 10 I E 162 10 6 × mm 4 = Problem 10-30 Locate the centroid y c of the cross-sectional area for the angle. Then find the moment of inertia I x' about the x' centroidal axis. Given: a 2in= b 6in= c 6in= d 2in= Solution: y c ac c 2 ? ? ? ? ? ? bd d 2 ? ? ? ? ? ? + ac bd+ = y c 2.00 in= I x' 1 12 ac 3 ac c 2 y c ? ? ? ? ? ? ? 2 + 1 12 bd 3 + bd y c d 2 ? ? ? ? ? ? ? 2 += I x' 64.00 in 4 = Problem 10-31 Locate the centroid x c of the cross-sectional area for the angle. Then find the moment 1009 © 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 10 of inertia I y' about the centroidal y' axis. Given: a 2in= b 6in= c 6in= d 2in= Solution: x c ac a 2 ? ? ? ? ? ? bd a b 2 + ? ? ? ? ? ? + ac bd+ = x c 3.00 in= I y' 1 12 ca 3 ca x c a 2 ? ? ? ? ? ? ? 2 + 1 12 db 3 + db a b 2 + x c ? ? ? ? ? ? ? 2 += I y' 136.00 in 4 = Problem 10-32 Determine the distance x c to the centroid of the beam's cross-sectional area: then find the moment of inertia about the y' axis. 1010 © 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 10 Given: a 40 mm= b 120 mm= c 40 mm= d 40 mm= Solution: x c 2 ab+()c ab+ 2 ? ? ? ? ? ? 2ad a 2 + 2 ab+()c 2da+ = x c 68.00 mm= I y' 2 1 12 ca b+() 3 ca b+() ab+ 2 x c ? ? ? ? ? ? ? 2 + ? ? ? ? ? ? 1 12 2da 3 + 2da x c a 2 ? ? ? ? ? ? ? 2 += I y' 36.9 10 6 × mm 4 = Problem 10-33 Determine the moment of inertia of the beam's cross-sectional area about the x' axis. 1011 © 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 10 Given: a 40 mm= b 120 mm= c 40 mm= d 40 mm= Solution: I x' 1 12 ab+()2c 2d+() 3 1 12 b 2d() 3 ?= I x' 49.5 10 6 × mm 4 = Problem 10-34 Determine the moments of inertia for the shaded area about the x and y axes. Given: a 3in= b 3in= c 6in= d 4in= r 2in= Solution: I x 1 3 ab+()cd+() 3 1 36 bc 3 1 2 bc d 2c 3 + ? ? ? ? ? ? 2 + ? ? ? ? ? ? ? ?r 4 4 ?r 2 d 2 + ? ? ? ? ? ? ?= 1012 © 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 10 I x 1192 in 4 = I y 1 3 cd+()ab+() 3 1 36 cb 3 1 2 bc a 2b 3 + ? ? ? ? ? ? 2 + ? ? ? ? ? ? ? ?r 4 4 ?r 2 a 2 + ? ? ? ? ? ? ?= I y 364.84 in 4 = Problem 10-35 Determine the location of the centroid y' of the beam constructed from the two channels and the cover plate. If each channel has a cross-sectional area A c and a moment of inertia about a horizontal axis passing through its own centroid C c , of I x'c , determine the moment of inertia of the beam?s cross-sectional area about the x' axis. Given: a 18 in= b 1.5 in= c 20 in= d 10 in= A c 11.8 in 2 = I x'c 349 in 4 = Solution: y c 2A c dabc b 2 + ? ? ? ? ? ? + 2A c ab+ = y c 15.74 in= I x' I x'c A c y c d?() 2 + ? ? ? ? 2 1 12 ab 3 + ab c b 2 + y c ? ? ? ? ? ? ? 2 += I x' 2158 in 4 = Problem 10-36 Compute the moments of inertia I x and I y for the beam's cross-sectional area about 1013 © 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 10 the x and y axes. Given: a 30 mm= b 170 mm= c 30 mm= d 140 mm= e 30 mm= f 30 mm= g 70 mm= Solution: I x 1 3 ac d+ e+() 3 1 3 bc 3 + 1 12 ge 3 + ge c d+ e 2 + ? ? ? ? ? ? 2 += I x 154 10 6 × mm 4 = I y 1 3 ca b+() 3 1 3 df 3 + 1 3 cf g+() 3 += I y 91.3 10 6 × mm 4 = Problem 10-37 Determine the distance y c to the centroid C of the beam's cross-sectional area and then compute the moment of inertia I cx' about the x' axis. Given: a 30 mm= e 30 mm= b 170 mm= f 30 mm= c 30 mm= g 70 mm= d 140 mm= 1014 © 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 10 y c ab+()c c 2 ? ? ? ? ? ? df c d 2 + ? ? ? ? ? ? + fg+()ec d+ e 2 + ? ? ? ? ? ? + ab+()cdf+ fg+()e+ = y c 80.7 mm= I x' 1 12 ab+()c 3 ab+()cy c c 2 ? ? ? ? ? ? ? 2 + 1 12 fd 3 + fd c d 2 + y c ? ? ? ? ? ? ? 2 + 1 12 fg+()e 3 fg+()ec d+ e 2 + y c ? ? ? ? ? ? ? 2 ++ ...= I x' 67.6 10 6 × mm 4 = Problem 10-38 Determine the distance x c to the centroid C of the beam's cross-sectional area and then compute the moment of inertia I y' about the y' axis. Given: a 30 mm= b 170 mm= c 30 mm= d 140 mm= e 30 mm= f 30 mm= g 70 mm= Solution: x c bc b 2 a+ ? ? ? ? ? ? cd+()f f 2 ? ? ? ? ? ? + fg+()e fg+ 2 + bc bc+ fg+()e+ = x c 61.6 mm= 1015 Solution: © 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 10 I y' 1 12 ca b+() 3 ca b+() ab+ 2 x c ? ? ? ? ? ? ? 2 + 1 12 df 3 + df x c f 2 ? ? ? ? ? ? ? 2 + 1 12 ef g+() 3 ef g+()x c fg+ 2 ? ? ? ? ? ? ? 2 ++ ...= I y' 41.2 10 6 × mm 4 = Problem 10-39 Determine the location y c of the centroid C of the beam?s cross-sectional area. Then compute the moment of inertia of the area about the x' axis Given: a 20 mm= b 125 mm= c 20 mm= f 120 mm= g 20 mm= d fc? 2 = e fc? 2 = Solution: y c ag+()f ag+ 2 ? ? ? ? ? ? cb a g+ b 2 + ? ? ? ? ? ? + ag+()fcb+ = y c 48.25 mm= I x' 1 12 fa g+() 3 f()ag+()y c ag+ 2 ? ? ? ? ? ? ? 2 + 1 12 cb 3 + cb b 2 a+ g+ y c ? ? ? ? ? ? ? 2 += I x' 15.1 10 6 × mm 4 = 1016 © 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 10 Problem 10-40 Determine y c , which locates the centroidal axis x' for the cross-sectional area of the T-beam, and then find the moments of inertia I x' and I y' . Given: a 25 mm= b 250 mm= c 50 mm= d 150 mm= Solutuion: y c b 2 ? ? ? ? ? ? b2ab c 2 + ? ? ? ? ? ? 2dc+ b2ac2d+ = y c 207 mm= I x' 1 12 2ab 3 2ab y c b 2 ? ? ? ? ? ? ? 2 + 1 12 2dc 3 + c2db c 2 + y c ? ? ? ? ? ? ? 2 += I x' 222 10 6 × mm 4 = I y' 1 12 b 2a() 3 1 12 c 2d() 3 += I y' 115 10 6 × mm 4 = Problem 10-41 Determine the centroid y' for the beam?s cross-sectional area; then find I x' . Given: a 25 mm= 1017 © 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 10 b 100 mm= c 25 mm= d 50 mm= e 75 mm= Solution: y c 2 ae+ d+()c c 2 ? ? ? ? ? ? 2ab c b 2 + ? ? ? ? ? ? + 2 ae+ d+()c 2ab+ = y c 37.50 mm= I x' 2 12 ae+ d+()c 3 2 ae+ d+()cy c c 2 ? ? ? ? ? ? ? 2 + 2 1 12 ab 3 ab c b 2 + y c ? ? ? ? ? ? ? 2 + ? ? ? ? ? ? + ...= I x' 16.3 10 6 × mm 4 = Problem 10-42 Determine the moment of inertia for the beam's cross-sectional area about the y axis. Given: a 25 mm= b 100 mm= c 25 mm= d 50 mm= e 75 mm= 1018 © 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 10 Solution: l y 1 12 2 3 ad+ e+() 3 c 2 1 12 ba 3 ab e a 2 + ? ? ? ? ? ? 2 + ? ? ? ? ? ? += l y 94.8 10 6 × mm 4 = Problem 10-43 Determine the moment for inertia I x of the shaded area about the x axis. Given: a 6in= b 6in= c 3in= d 6in= Solution: I x ba 3 3 1 12 ca 3 + 1 12 bc+()d 3 += I x 648 in 4 = Problem 10-44 Determine the moment for inertia I y of the shaded area about the y axis. Given: a 6in= b 6in= c 3in= d 6in= 1019 © 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 10 Solution: I y ab 3 3 1 36 ac 3 + 1 2 ac b c 3 + ? ? ? ? ? ? 2 + 1 36 db c+() 3 + 1 2 db c+() 2 bc+() 3 ? ? ? ? ? ? 2 += I y 1971 in 4 = Problem 10-45 Locate the centroid y c of the channel's cross-sectional area, and then determine the moment of inertia with respect to the x' axis passing through the centroid. Given: a 2in= b 12 in= c 2in= d 4in= Solution: y c c 2 bc 2 cd+ 2 ? ? ? ? ? ? cd+()a+ bc 2 cd+()a+ = y c 2in= I x 1 12 bc 3 bc y c c 2 ? ? ? ? ? ? ? 2 + 2 12 ac d+() 3 + 2ac d+() cd+ 2 y c ? ? ? ? ? ? ? 2 += I x 128 in 4 = Problem 10-46 Determine the moments for inertia I x and I y of the shaded area. Given: r 1 2in= 1020 © 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 10 r 2 6in= Solution: I x ?r 2 4 8 ?r 1 4 8 ? ? ? ? ? ? ? = I x 503 in 4 = I y ?r 2 4 8 ?r 1 4 8 ? ? ? ? ? ? ? = I y 503 in 4 = Problem 10-47 Determine the moment of inertia for the parallelogram about the x' axis, which passes through the centroid C of the area. Solution: ha( )sin ?()= I xc 1 12 bh 3 = 1 12 ba( )sin ?() ?? ?? 3 = 1 12 a 3 b sin ?() 3 = I xc 1 12 a 3 b sin ?() 3 = Problem 10-48 Determine the moment of inertia for the parallelogram about the y ' axis, which passes through the centroid C of the area. 1021 © 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 10 Solution: Aba( ) sin ?()= x c 1 ba( ) sin ?() ba( ) sin ?() b 2 1 2 a( ) cos ?()a( ) sin ?() a( )cos ?() 3 ? ? ? ? ? ? ? 1 2 a( ) cos ?()a( ) sin ?()b a( )cos ?() 3 + ? ? ? ? ? ? + ... ? ? ? ? ? ? ? ? ? ? = ba( )cos ?()+ 2 = I y' 1 12 a( ) sin ?()b 3 a( )sin ?()b b 2 x c ? ? ? ? ? ? ? 2 + 1 36 a( ) sin ?() a( )cos ?() ?? ?? 3 1 2 a( ) sin ?()a( ) cos ?()x c a( )cos ?() 3 ? ? ? ? ? ? ? 2 + ? ? ? ? ? ? ?+ ... 1 36 a( ) sin ?() a( )cos ?() ?? ?? 3 1 2 a( ) sin ?()a( ) cos ?()b a( )cos ?() 3 + x c ? ? ? ? ? ? ? 2 ++ ... = Simplifying we find. I y' ab 12 b 2 a 2 cos ?() 2 + () sin ?()= Problem 10-49 Determine the moments of inertia for the triangular area about the x' and y' axes, which pass through the centroid C of the area. 1022 © 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 10 Solution: I x' 1 36 bh 3 = x c 2 3 a 1 2 ha a ba? 3 + ? ? ? ? ? ? 1 2 hb a?()+ 1 2 ha 1 2 hb a?()+ = ba+ 3 = I y' 1 36 ha 3 1 2 ha ba+ 3 2 3 a? ? ? ? ? ? ? 2 + 1 36 hb a?() 3 + 1 2 hb a?()a ba? 3 + ba+ 3 ? ? ? ? ? ? ? 2 += I y' 1 36 hb b 2 ab? a 2 + () = Problem 10-50 Determine the moment of inertia for the beam?s cross-sectional area about the x' axis passing through the centroid C of the cross section. Given: a 100 mm= b 25 mm= c 200 mm= ? 45 deg= 1023 © 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 10 Solution: I x' 1 12 2a 2 c sin ?() b+() ?? ?? 3 ? ? ? ? 4 1 12 c cos ?()()c sin ?()() 3 ? ? ? ? ? ? 2 1 4 c 4 ? 1 2 sin 2?()? ? ? ? ? ? ? ? ? ? ? ? ? ?+ ...= I x' 520 10 6 × mm 4 = Problem 10-51 Determine the moment of inertia of the composite area about the x axis. Given: a 2in= b 4in= c 1in= d 4in= Solution: I x 1 3 ab+()2a() 3 ?c 4 4 ?c 2 a 2 + ? ? ? ? ? ? ? 0 d x 1 3 2a 1 x d ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? ? ? 3 ? ? ? ? ? d+= I x 153.7 in 4 = 1024 © 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 10 Problem 10-52 Determine the moment of inertia of the composite area about the y axis. Given: a 2in= b 4in= c 1in= d 4in= Solution: I y 1 3 2a()ab+() 3 ?c 4 4 ?c 2 b 2 + ? ? ? ? ? ? ? 0 d xx 2 2a 1 x d ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ? ? ? d+= I y 271.1 in 4 = Problem 10-53 Determine the radius of gyration k x for the column's cross-sectional area. Given: a 200 mm= b 100 mm= Solution: I x 1 12 2ab+()b 3 2 1 12 ba 3 ba a 2 b 2 + ? ? ? ? ? ? 2 + ? ? ? ? ? ? += k x I x b 2ab+()2a b+ = k x 109 mm= 1025 © 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 10 Problem 10-54 Determine the product of inertia for the shaded portion of the parabola with respect to the x and y axes. Given: a 2in= b 1in= I xy a? a x b x a ? ? ? ? ? ? 2 b yxy ? ? ? ? d ? ? ? ? d= I xy 0.00 m 4 = Also because the area is symmetric about the y axis, the product of inertia must be zero. Problem 10-55 Determine the product of inertia for the shaded area with respect to the x and y axes. Solution: I xy 0 b x 0 h x b ? ? ? ? ? ? 1 3 yxy ? ? ? ? ? d ? ? ? ? d= 3 16 b 2 h 2 = I xy 3 16 b 2 h 2 = 1026 © 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 10 Problem 10-56 Determine the product of inertia of the shaded area of the ellipse with respect to the x and y axes. Given: a 4in= b 2in= Solution: I xy 0 a xx b 1 x a ? ? ? ? ? ? 2 ? 2 ? ? ? ? ? ? ? ? ? ? b 1 x a ? ? ? ? ? ? 2 ? ? ? ? ? ? ? d= I xy 8.00 in 4 = Problem 10-57 Determine the product of inertia of the parabolic area with respect to the x and y axes. 1027 © 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 10 Solution: I xy 0 a xx b x a 2 ? ? ? ? ? ? ? ? b x a ? ? ? ? ? d= 1 6 a 3 b 2 a = I xy 1 6 a 2 b 2 = Problem 10-58 Determine the product of inertia for the shaded area with respect to the x and y axes. Given: a 8in= b 2in= Solution: I xy 0 a xx b x a ? ? ? ? ? ? 1 3 2 b x a ? ? ? ? ? ? 1 3 ? ? ? ? ? ? ? d= I xy 48.00 in 4 = Problem 10-59 Determine the product of inertia for the shaded parabolic area with respect to the x and y axes. Given: a 4in= b 2in= Solution: I xy 0 a xx b 2 x a b x a ? ? ? ? d= 1028 © 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 10 I xy 10.67 in 4 = Problem 10-60 Determine the product of inertia for the shaded area with respect to the x and y axes. Given: a 2m= b 1m= Solution: I xy 0 a xx b 2 1 x a ? ? ? ? ? ? ? b 1 x a ? ? ? ? ? d= I xy 0.333 m 4 = Problem 10-61 Determine the product of inertia for the shaded area with respect to the x and y axes. 1029 © 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

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