4 Solutions 44918 1/23/09 12:03 PM Page 190 © 2010 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 •4–1 If A, B, and D are given vectors, prove the distributive law for the vector cross product, i.e., A : (B + D) = (A : B) + (A : D) 190 Solutions 44918 1/23/09 12:03 PM Page 191 © 2010 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 4–2 Prove the triple A # B : C = A : B # C scalar product identity 191 Solutions 44918 1/23/09 12:03 PM Page 192 © 2010 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 4–3 Given the three nonzero vectors A, B, and C, show that if A # (B : C) = 0, the three vectors must lie in the same plane *4–4 Two men exert forces of F = 80 lb and P = 50 lb on the ropes Determine the moment of each force about A Which way will the pole rotate, clockwise or counterclockwise? ft P F 45Њ 12 ft B C A •4–5 If the man at B exerts a force of P = 30 lb on his rope, determine the magnitude of the force F the man at C must exert to prevent the pole from rotating, i.e., so the resultant moment about A of both forces is zero ft P F 45Њ B 12 ft C A 192 Solutions 44918 1/23/09 12:03 PM Page 193 © 2010 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 4–6 If u = 45°, determine the moment produced by the 4-kN force about point A 3m A 0.45 m u kN 193 Solutions 44918 1/23/09 12:03 PM Page 194 © 2010 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 4–7 If the moment produced by the 4-kN force about point A is 10 kN # m clockwise, determine the angle u, where 0° … u … 90° 3m A 0.45 m u kN 194 Solutions 44918 1/23/09 12:03 PM Page 195 © 2010 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 *4–8 The handle of the hammer is subjected to the force of F = 20 lb Determine the moment of this force about the point A F 30Њ in 18 in A B 195 Solutions 44918 1/23/09 12:03 PM Page 196 © 2010 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 •4–9 In order to pull out the nail at B, the force F exerted on the handle of the hammer must produce a clockwise moment of 500 lb # in about point A Determine the required magnitude of force F F 30Њ in 18 in A B 196 Solutions 44918 1/23/09 12:03 PM Page 197 © 2010 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 4–10 The hub of the wheel can be attached to the axle either with negative offset (left) or with positive offset (right) If the tire is subjected to both a normal and radial load as shown, determine the resultant moment of these loads about point O on the axle for both cases 0.05 m O 0.05 m O 0.4 m 0.4 m 800 N 197 800 N kN kN Case Case Solutions 44918 1/23/09 12:03 PM Page 198 © 2010 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 4–11 The member is subjected to a force of F = kN If u = 45°, determine the moment produced by F about point A 1.5 m u F ϭ kN 6m A 198 Solutions 44918 1/23/09 12:03 PM Page 199 © 2010 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 *4–12 Determine the angle u (0° … u … 180°) of the force F so that it produces a maximum moment and a minimum moment about point A Also, what are the magnitudes of these maximum and minimum moments? 1.5 m u F ϭ kN 6m A 199 Solutions 44918 1/23/09 12:04 PM Page 305 © 2010 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 4–150 The beam is subjected to the distributed loading Determine the length b of the uniform load and its position a on the beam such that the resultant force and couple moment acting on the beam are zero b 40 lb/ft a 60 lb/ft 10 ft 4–151 Currently eighty-five percent of all neck injuries are caused by rear-end car collisions To alleviate this problem, an automobile seat restraint has been developed that provides additional pressure contact with the cranium During dynamic tests the distribution of load on the cranium has been plotted and shown to be parabolic Determine the equivalent resultant force and its location, measured from point A ft A 12 lb/ft 0.5 ft w w ϭ 12(1 ϩ 2x2) lb/ft B 18 lb/ft x 305 Solutions 44918 1/23/09 12:04 PM Page 306 © 2010 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 *4–152 Wind has blown sand over a platform such that the intensity of the load can be approximated by the function w = 10.5x32 N>m Simplify this distributed loading to an equivalent resultant force and specify its magnitude and location measured from A w 500 N/m w ϭ (0.5x3) N/m x A 10 m •4–153 Wet concrete exerts a pressure distribution along the wall of the form Determine the resultant force of this distribution and specify the height h where the bracing strut should be placed so that it lies through the line of action of the resultant force The wall has a width of m p / p ϭ (4 z 2) kPa 4m h kPa z 306 Solutions 44918 1/23/09 12:04 PM Page 307 © 2010 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 4–154 Replace the distributed loading with an equivalent resultant force, and specify its location on the beam measured from point A w kN/m (4 Ϫ x)2 w ϭ –– A x B 4m 307 Solutions 44918 1/23/09 12:04 PM Page 308 © 2010 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 4–155 Replace the loading by an equivalent resultant force and couple moment at point A 50 lb/ft 50 lb/ft B ft ft 100 lb/ft 60Њ A 308 Solutions 44918 1/23/09 12:04 PM Page 309 © 2010 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 *4–156 Replace the loading by an equivalent resultant force and couple moment acting at point B 50 lb/ft 50 lb/ft B ft ft 100 lb/ft 60Њ A 309 Solutions 44918 1/23/09 12:04 PM Page 310 © 2010 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 •4–157 The lifting force along the wing of a jet aircraft consists of a uniform distribution along AB, and a semiparabolic distribution along BC with origin at B Replace this loading by a single resultant force and specify its location measured from point A w w ϭ (2880 Ϫ 5x2) lb/ft 2880 lb/ft B A C 12 ft 4–158 The distributed load acts on the beam as shown Determine the magnitude of the equivalent resultant force and specify where it acts, measured from point A x 24 ft w w ϭ (Ϫ2x ϩ 4x ϩ 16) lb/ft A B ft 310 x Solutions 44918 1/23/09 12:04 PM Page 311 © 2010 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 4–159 The distributed load acts on the beam as shown Determine the maximum intensity wmax What is the magnitude of the equivalent resultant force? Specify where it acts, measured from point B w w ϭ (Ϫ2x ϩ 4x ϩ 16) lb/ft A B ft 311 x Solutions 44918 1/23/09 12:04 PM Page 312 © 2010 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 *4–160 The distributed load acts on the beam as shown Determine the magnitude of the equivalent resultant force and specify its location, measured from point A w x2 ϩ 17 x ϩ 4) lb/ft w ϭ (Ϫ15 15 lb/ft lb/ft B A 10 ft 312 x Solutions 44918 1/23/09 12:04 PM Page 313 © 2010 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 •4–161 If the distribution of the ground reaction on the pipe per foot of length can be approximated as shown, determine the magnitude of the resultant force due to this loading 25 lb/ft 2.5 ft u w ϭ 25 (1 ϩ cos u) lb/ft 50 lb/ft 313 Solutions 44918 1/23/09 12:04 PM Page 314 © 2010 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 4–162 The beam is subjected to the parabolic loading Determine an equivalent force and couple system at point A w 400 lb/ft w ϭ (25 x2)lb/ft A O x ft 4–163 Two couples act on the frame If the resultant couple moment is to be zero, determine the distance d between the 100-lb couple forces 100 lb 30° 150 lb d ft ft B A ft 30° 100 lb 150 lb 314 3 Solutions 44918 1/23/09 12:04 PM Page 315 © 2010 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 F ϭ 20 lb *4–164 Determine the coordinate direction angles a, b , g of F, which is applied to the end of the pipe assembly, so that the moment of F about O is zero z O y 10 in y 10 in in in in x F ϭ 20 lb •4–165 Determine the moment of the force F about point O The force has coordinate direction angles of a = 60°, b = 120°, g = 45° Express the result as a Cartesian vector z O in in x 315 in Solutions 44918 1/23/09 12:04 PM Page 316 © 2010 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 4–166 The snorkel boom lift is extended into the position shown If the worker weighs 160 lb, determine the moment of this force about the connection at A ft 25 ft 50Њ A z 4–167 Determine the moment of the force FC about the door hinge at A Express the result as a Cartesian vector C 1.5 m 2.5 m FC ϭ 250 N 30Њ a A B a 1m 0.5 m y x z *4–168 Determine the magnitude of the moment of the force FC about the hinged axis aa of the door C 1.5 m 2.5 m FC ϭ 250 N 30Њ A B a 1m x 316 a 0.5 m y Solutions 44918 1/23/09 12:04 PM Page 317 © 2010 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 z •4–169 Express the moment of the couple acting on the pipe assembly in Cartesian vector form Solve the problem (a) using Eq 4–13 and (b) summing the moment of each force about point O Take F = 525k6 N O 300 mm 200 mm y F 150 mm B x –F 400 mm 200 mm A z 4–170 If the couple moment acting on the pipe has a magnitude of 400 N # m, determine the magnitude F of the vertical force applied to each wrench O 300 mm 200 mm y F 150 mm B x –F 400 mm 200 mm A 317 Solutions 44918 1/23/09 12:04 PM Page 318 © 2010 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 z 4–171 Replace the force at A by an equivalent resultant force and couple moment at point P Express the results in Cartesian vector form P ft 10 ft ft y F ϭ 120 lb ft ft A ft x z *4–172 The horizontal 30-N force acts on the handle of the wrench Determine the moment of this force about point O Specify the coordinate direction angles a, b , g of the moment axis 200 mm B 10 mm 50 mm O x 318 30 N 45Њ 45Њ y Solutions 44918 1/23/09 12:04 PM Page 319 © 2010 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 z •4–173 The horizontal 30-N force acts on the handle of the wrench What is the magnitude of the moment of this force about the z axis? 200 mm B 10 mm 50 mm O x 319 30 N 45Њ 45Њ y ... writing from the publisher 4 6 If u = 45 °, determine the moment produced by the 4- kN force about point A 3m A 0 .45 m u kN 193 Solutions 44 918 1/23/09 12:03 PM Page 1 94 © 2010 Pearson Education,... from the publisher 4 7 If the moment produced by the 4- kN force about point A is 10 kN # m clockwise, determine the angle u, where 0° … u … 90° 3m A 0 .45 m u kN 1 94 Solutions 44 918 1/23/09 12:03... publisher 4 21 Determine the direction u for 0° … u … 180° of the force F so that it produces the maximum moment about point A Calculate this moment F ϭ 40 0 N u 2m A 3m 2 04 Solutions 44 918 1/23/09