Engineering Mechanics - Statics Episode 3 Part 6 pps

Engineering Mechanics - Statics Chapter 10 Problem 10-16 Determine the moment of inertia of the shaded area about the x axis. Given: a 2in= b 4in= Solution: I x a− a x 1 3 bcos π x 2a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 3 ⌠ ⎮ ⎮ ⌡ d= I x 36.2 in 4 = Problem 10-17 Determine the moment of inertia for the shaded area about the y axis. Given: a 2in= b 4in= Solution: I y a− a xx 2 bcos π x 2a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= I y 7.72 in 4 = Problem 10-18 Determine the moment of inertia for the shaded area about the x axis. 1001 © 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 4in= b 2in= Solution: Solution I x a− a x bcos π x 2a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 3 3 ⌠ ⎮ ⎮ ⎮ ⎮ ⌡ d= I x 9.05 in 4 = Problem 10-19 Determine the moment of inertia for the shaded area about the y axis. Given: a 4in= b 2in= Solution: I y a− a xx 2 bcos π x 2a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= I y 30.9 in 4 = Problem 10-20 Determine the moment for inertia of the shaded area about the x axis. 1002 © 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 2in= b 4in= c 12 in= Solution: I x a ab+ x 1 3 c 2 x ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 3 ⌠ ⎮ ⎮ ⎮ ⌡ d= I x 64.0 in 4 = Problem 10-21 Determine the moment of inertia of the shaded area about the y axis. Given: a 2in= b 4in= c 12 in= Solution: I y a ab+ xx 2 c 2 x ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= 1003 © 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 192.00 in 4 = Problem 10-22 Determine the moment of inertia for the shaded area about the x axis. Given: a 2m= b 2m= Solution: I x 0 b yy 2 a y 2 b 2 ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ ⌠ ⎮ ⎮ ⎮ ⌡ d= I x 3.20 m 4 = Problem 10-23 Determine the moment of inertia for the shaded area about the y axis. Use Simpson's rule to evaluate the integral. Given: a 1m= b 1m= 1004 © 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 0 a xx 2 be x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⎮ ⌡ d= I y 0.628 m 4 = Problem 10-24 Determine the moment of inertia for the shaded area about the x axis. Use Simpson's rule to evaluate the integral. Given: a 1m= b 1m= Solution: I y 0 a x be x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ 3 3 ⌠ ⎮ ⎮ ⎮ ⎮ ⎮ ⌡ d= I y 1.41 m 4 = Problem 10-25 The polar moment of inertia for the area is I C about the z axis passing through the centroid C. The moment of inertia about the x axis is I x and the moment of inertia about the y' axis is I y' . Determine the area A. Given: I C 28 in 4 = 1005 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 x 17 in 4 = I y' 56 in 4 = a 3in= Solution: I C I x I y += I y I C I x −= I y' I y Aa 2 += A I y' I y − a 2 = A 5.00 in 2 = Problem 10-26 The polar moment of inertia for the area is J cc about the z' axis passing through the centroid C. If the moment of inertia about the y' axis is I y' and the moment of inertia about the x axis is I x . Determine the area A. Given: J cc 548 10 6 × mm 4 = I y' 383 10 6 × mm 4 = I x 856 10 6 × mm 4 = h 250 mm= Solution: I x' I x Ah 2 −= J cc I x' I y' += J cc I x Ah 2 − I y' += 1006 © 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 A I x I y' + J cc − h 2 = A 11.1 10 3 × mm 2 = Problem 10-27 Determine the radius of gyration k x of the column’s cross-sectional area. Given: a 100 mm= b 75 mm= c 90 mm= d 65 mm= Solution: Cross-sectional area: A 2b()2a() 2d()2c()−= Moment of inertia about the x axis: I x 1 12 2b()2a() 3 1 12 2d()2c() 3 −= Radius of gyration about the x axis: k x I x A = k x 74.7 mm= Problem 10-28 Determine the radius of gyration k y of the column’s cross-sectional area. Given: a 100 mm= b 75 mm= 1007 © 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 c 90 mm= d 65 mm= Solution: Cross-sectional area: A 2b()2a() 2d()2c()−= Moment of inertia about the y axis: I y 1 12 2a()2b() 3 1 12 2c()2d() 3 −= Radius of gyration about the y axis: k y I y A = k y 59.4 mm= Problem 10-29 Determine the moment of inertia for the beam's cross-sectional area with respect to the x' centroidal axis. Neglect the size of all the rivet heads, R, for the calculation. Handbook values for the area, moment of inertia, and location of the centroid C of one of the angles are listed in the figure. Solution: I E 1 12 15 mm( ) 275 mm() 3 4 1.32 10 6 () mm 4 1.36 10 3 () mm 2 275 mm 2 28 mm− ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ + 2 1 12 75 mm( ) 20 mm() 3 75 mm( ) 20 mm() 275 mm 2 10mm+ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ + = 1008 © 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 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. [...]... publisher Engineering Mechanics - Statics Chapter 10 Solution: Ix' = 36 3 bh 2 1 ⎛ b − a ⎞ 1 h ( b − a) a h a + ⎜a + ⎟ 3 2 3 ⎠2 ⎝ xc = = 1 1 h a + h ( b − a) 2 2 Iy' = Iy' = 1 1 36 1 36 3 ha + (2 1 2 ha⎛ ⎜ b+a ⎝ 3 hb b − ab + a 2 − 2 3 2 a⎞ + ⎟ ⎠ 1 36 b+a 3 3 h ( b − a) + 1 2 h( b − a) ⎛ a + ⎜ ⎝ b−a 3 − b + a⎞ 3 2 ⎟ ⎠ ) Problem 1 0-5 0 Determine the moment of inertia for the beam’s cross-sectional area... 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: Ix = 1 1 e⎞ 3 1 3 3 ⎛ a ( c + d + e) + b c + g e + g e ⎜c + d + ⎟ 3 3 12 2⎠ ⎝ Iy = 1 3 1 3 1 3 c ( a + b ) + d f + c ( f + g) 3 3 3 2 6 6 Determine the distance yc to the centroid C of the beam's cross-sectional area and then compute... from the publisher Engineering Mechanics - Statics Chapter 10 Solution: ly = ⎡1 3 1 3 a 3 2 ( a + d + e) c + 2⎢ b a + a b ⎛ e + ⎞ ⎜ ⎟ 12 ⎣12 ⎝ 2⎠ 6 2⎤ ⎥ ⎦ 4 ly = 94.8 × 10 mm Problem 1 0-4 3 Determine the moment for inertia Ix of the shaded area about the x axis Given: a = 6 in b = 6 in c = 3 in d = 6 in Solution: 3 Ix = ba 3 + 1 12 3 ca + 1 12 ( b + c) d 3 Ix = 64 8 in 4 Problem 1 0-4 4 Determine the moment.. .Engineering Mechanics - Statics Chapter 10 Given: a = 40 mm b = 120 mm c = 40 mm d = 40 mm Solution: ⎛ a + b ⎞ + 2a d a ⎟ 2 ⎝ 2 ⎠ 2( a + b)c⎜ xc = xc = 68 .00 mm 2( a + b)c + 2d a ⎡1 Iy' = 2⎢ ⎣ 12 c ( a + b) + c( a + b) ⎛ ⎜ 3 a+b ⎝ 2 6 − xc⎞ ⎟ 2⎤ 1 ⎥ + 2d a3 + 2d a ⎛xc − ⎜ ⎠ ⎦ 12 ⎝ a⎞ 2 ⎟ 2⎠ 4 Iy' = 36 .9 × 10 mm Problem 1 0 -3 3 Determine the moment of inertia of the beam's cross-sectional area... publisher Engineering Mechanics - Statics Chapter 10 Solution: h ⌠ 2 1⎤ ⎮ ⎡ ⎮ ⎢ 3 ⎮ y 1 ⎢b ⎛ y ⎞ ⎥ d y = 3 b2 h2 Ixy = ⎜ ⎟ ⎮ 2 ⎣ ⎝ h⎠ ⎦ 16 ⌡0 Ixy = 3 16 2 2 h b Problem 1 0 -6 2 Determine the product of inertia of the shaded area with respect to the x and y axes Given: a = 4 in b = 2 in Solution: a ⌠ 3 3 ⎮ ⎛ b⎞ ⎛ x ⎞ ⎛ x ⎞ Ixy = ⎮ x⎜ ⎟ ⎜ ⎟ b ⎜ ⎟ dx 2 a a ⌡0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Ixy = 4.00 in 4 1 030 © 2007 R... 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: 1 Ix' = 12 3 ( a + b) ( 2c + 2d) − 1 12 b ( 2d) 3 6 4 Ix' = 49.5 × 10 mm Problem 1 0 -3 4 Determine the moments of inertia for the shaded area about the x and y axes Given: a = 3 in b = 3 in c = 6 in d = 4 in r = 2 in Solution: ⎡ 1 3 1 ⎛ 2c ⎞ 2⎤ ⎛ π r4 2 2⎞ Ix = ( a +... publisher Engineering Mechanics - Statics Chapter 10 r2 = 6 in Solution: ⎛ π r24 π r14 ⎞ ⎟ Ix = ⎜ − 8 ⎠ ⎝ 8 Ix = 5 03 in 4 ⎛ π r24 π r14 ⎞ ⎟ Iy = ⎜ − 8 ⎠ ⎝ 8 Iy = 5 03 in 4 Problem 1 0-4 7 Determine the moment of inertia for the parallelogram about the x' axis, which passes through the centroid C of the area Solution: h = ( a)sin ( θ ) Ixc = Ixc = 1 12 3 bh = 1 12 b ⎡( a)sin ( θ )⎤ = ⎣ ⎦ 3 1 3 3 a b sin... Engineering Mechanics - Statics Chapter 10 b = 100 mm c = 25 mm d = 50 mm e = 75 mm Solution: ⎛ c ⎞ + 2a b⎛c + b ⎞ ⎟ ⎜ ⎟ ⎝ 2⎠ ⎝ 2⎠ 2( a + e + d)c⎜ yc = Ix' = yc = 37 .50 mm 2( a + e + d)c + 2a b 2 12 ( a + e + d) c + 2( a + e + d)c ⎛ yc − ⎜ 3 ⎡1 + 2⎢ ⎣ 12 a b + a b ⎛c + ⎜ 3 ⎝ 6 b 2 − yc⎞ ⎟ 2⎤ ⎝ c⎞ 2 ⎟ 2⎠ ⎥ ⎠⎦ 4 Ix' = 16 .3 × 10 mm Problem 1 0-4 2 Determine the moment of inertia for the beam's cross-sectional... or by any means, without permission in writing from the publisher Engineering Mechanics - Statics Ixy = 10 .67 in Chapter 10 4 Problem 1 0 -6 0 Determine the product of inertia for the shaded area with respect to the x and y axes Given: a = 2m b = 1m Solution: a ⌠ ⎛b Ixy = ⎮ x⎜ ⎮ ⎝2 ⌡0 1− x⎞ x ⎟ b 1 − dx a⎠ a Ixy = 0 .33 3 m 4 Problem 1 0 -6 1 Determine the product of inertia for the shaded area with respect... the publisher Engineering Mechanics - Statics Chapter 10 Solution: ( a + b)c⎛ ⎜ ⎛ ⎟ + d f⎜ c + ⎝ 2⎠ ⎝ yc = c⎞ d⎞ ⎛ ⎟ + ( f + g)e⎜ c + d + 2⎠ ⎝ e⎞ ⎟ 2⎠ ( a + b)c + d f + ( f + g)e yc = 80.7 mm 2 Ix' = 2 1 c⎞ 1 3 3 ⎞ ⎛ ⎛ d ( a + b) c + ( a + b)c ⎜ yc − ⎟ + f d + f d ⎜ c + − yc⎟ 12 2⎠ 12 ⎝ ⎝ 2 ⎠ e 1 3 ⎞ ⎛ ( f + g) e + ( f + g)e ⎜ c + d + − yc⎟ 2 12 ⎝ ⎠ + 6 2 4 Ix' = 67 .6 × 10 mm Problem 1 0 -3 8 Determine . 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 1971in 4 = Problem 1 0-4 5 Locate. 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 +. 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 1 0 -3 7 Determine the distance y c to the centroid C of the beam's cross-sectional area