Engineering Mechanics - Statics Chapter 9 g 9.81 m s 2 = θ atan b c ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = Solution: A 0 c xa x c bx c + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= A 4.667 m 2 = x c 1 A 0 c xxa x c bx c + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= x c 1.257 m= y c 1 A 0 c x 1 2 a x c bx c + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ a x c bx c − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= y c 0.143 m= Equilibrium W ρ Atg= Guesses A x 1N= A y 1N= N B 1N= Given A y W− N B cos θ () + 0= A x − N B sin θ () + 0= N B b 2 c 2 + Wx c − 0= A x A y N B ⎛ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎠ Find A x A y , N B , () = A x A y N B ⎛ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎠ 33.9 73.9 47.9 ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ kN= Problem 9-23 Locate the centroid x c of the shaded area. Given: a 4ft= b 4ft= 911 © 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 0 a x bx a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= x c 1 A 0 a xx bx a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= x c 2.00 ft= Problem 9-24 Locate the centroid y c of the shaded area. Given: a 4ft= b 4ft= Solution: A 0 a x bx a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= y c 1 A 0 a x 1 2 b x a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ bx a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= y c 1.60 ft= Problem 9-25 Locate the centroid x c of the shaded area. Given: a 4m= 912 © 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 b 4m= Solution: A 0 a xb x a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= x c 1 A 0 a xxb x a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= x c 1.80 m= Problem 9-26 Locate the centroid y c of the shaded area. Given: a 4m= b 4m= Solution: A 0 a xb x a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= y c 1 A 0 a x 1 2 b x a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ b x a b x a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= y c 1.80 m= 913 © 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 9-27 Locate the centroid x c of the shaded area. Given: a 1in= b 3in= c 2in= Solution: A a ab+ xc x ab+ ⌠ ⎮ ⎮ ⌡ d= x c 1 A a ab+ xxc x ab+ ⌠ ⎮ ⎮ ⌡ d= x c 2.66 in= Problem 9-28 Locate the centroid y c of the shaded area. Given: a 1in= b 3in= c 2in= 914 © 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 a ab+ xc x ab+ ⌠ ⎮ ⎮ ⌡ d= y c 1 A a ab+ x 1 2 c x ab+ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d= y c 0.804 in= Problem 9-29 Locate the centroid x c of the shaded area. Given: a 4in= b 2in= c 3in= Solution: A b ab+ y bc y ⌠ ⎮ ⎮ ⌡ d= A 6.592 in 2 = x c 1 A b ab+ y 1 2 bc y ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⌠ ⎮ ⎮ ⌡ d= x c 0.910 in= Problem 9-30 Locate the centroid y c of the shaded area. 915 © 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 4in= b 2in= c 3in= Solution: A b ab+ y bc y ⌠ ⎮ ⎮ ⌡ d= A 6.592 in 2 = y c 1 A b ab+ yy bc y ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= y c 3.64 in= Problem 9-31 Determine the location r c of the centroid C of the cardioid, r = a(1 − cos θ ). Solution: A 0 2 π θ 0 a 1 cos θ () − () rr ⌠ ⎮ ⌡ d ⌠ ⎮ ⌡ d= 3 2 a 2 π = 916 © 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 x c 2 3a 2 π 0 2 π θ 0 a 1 cos θ () − () rrcos θ () r ⌠ ⎮ ⌡ d ⌠ ⎮ ⌡ d= 5− 6 a= r c 5a 6 = Problem 9-32 Locate the centroid of the ellipsoid of revolution. Solution: dV π z 2 dy= z 2 a 2 1 y 2 b 2 − ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ = V 0 b y π a 2 1 y 2 b 2 − ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ ⌠ ⎮ ⎮ ⎮ ⌡ d= 1 3 b 3 b 2 b 2 − b 2 a 2 π = y c 3 2ba 2 π 0 b yy π a 2 1 y 2 b 2 − ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ ⌠ ⎮ ⎮ ⎮ ⌡ d= 3 8 b b 2 = y c 3b 8 = By symmetry x c z c = 0= Problem 9-33 Locate the centroid z c of the very thin conical shell. Hint: Use thin ring elements having a center at (0, 0, z), radius y, and width dL dy() 2 dz() 2 += 917 © 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: r az h = dL dy 2 dz 2 += 1 dy dz ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + dz= 1 a h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + dz= A 0 h z2 π az h 1 a h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + ⌠ ⎮ ⎮ ⌡ d= h 2 π a h h 2 a 2 + h 2 ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ = z c 1 π aa 2 h 2 + 0 h zz2 π az h 1 a h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 + ⌠ ⎮ ⎮ ⌡ d= 2 3 π ah 2 a 2 + () h 2 π a h 2 a 2 + h 2 ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ = z c 2h 3 = Problem 9-34 Locate the centroid z c of the volume. 918 © 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 2ft= b 2ft= Solution: V 0 b z π a 2 z b ⌠ ⎮ ⎮ ⌡ d= V 12.566 ft 3 = z c 1 V 0 b zz π a 2 z b ⌠ ⎮ ⎮ ⌡ d= z c 1.333 ft= Problem 9-35 Locate the centroid of the solid. Solution: z 2 h 2 a y= ya z h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 = x c y c = 0= By symmetry z c 0 h zz π a z h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⎮ ⌡ d 0 h z π a z h ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⎮ ⌡ d = 5 6 h 6 h 5 = z c 5h 6 = 919 © 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 9-36 Locate the centroid of the quarter-cone. Solution: r a h hz−()= z c z= x c y c = 4r 3 π = V 0 h z π 4 a h hz−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⌡ d= 1 12 ha 2 π = z c 12 ha 2 π 0 h zz π 4 a h hz−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⌡ d= 1 4 h= x c 12 ha 2 π 0 h z 4 3 π a h hz−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ π 4 a h h z−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⌡ d ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ = a π = x c y c = a π = z c h 4 = 920 © 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 b = 4 in c = 6 in Solution: − a b⎛ ⎜ b⎞ ⎟ ⎝ 2⎠ xc = ab + 1 2 ac 1 ⎛ 2a ⎞ ⎟ + a c⎜ ⎟ ⎝ 2⎠ 2 ⎝ 3 ⎠ a b⎛ ⎜ yc = a⎞ ab + −1 zc = xc = −1 .143 in 2 a c⎛ ⎜ ab + 2 2 yc = 1. 714 in ac c⎞ ⎟ ⎝ 3⎠ 1 1 zc = −0.857 in ac Problem 9-7 2 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... the publisher Engineering Mechanics - Statics Chapter 9 ⎞ sin ( α ) ⎛ 1 − cos ( α ) ⎜ ⎟ 3 ⎝ α − sin ( α ) cos ( α ) ⎠ 2 xc = 2r 2r 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 ⎠⎦ ⎝ 3π ⎠ yc = 56.6 mm Problem 9-6 4 The “New... 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... by any means, without permission in writing from the publisher Engineering Mechanics - Statics Chapter 9 Solution: c−z x y = = c a b x= a ( c − z) c y= b ( c − z) c c zc = ⌠ ab 2 ⎮ z ( c − z) dz 2 ⎮ c ⌡ 0 c ⌠ ab 2 ⎮ ( c − z) dz ⎮ c2 ⌡ = 1 c 4 zc = 1 c 4 0 Problem 9-4 4 Determine the location (x, y) of the particle M1 so that the three particles, which lie in the x–y plane, have a center of mass located... publisher Engineering Mechanics - Statics a Chapter 9 1⌠ ⎮ zπ ⎡b2 − z b2 − c2 ⎤ dz zc = ⎢ ⎥ V⎮ a ⎣ ⎦ ⌡ ( ) 0 zc = 0.422 m Problem 9-4 0 Locate the center of gravity yc of the volume The material is homogeneous Given: a = 25 mm c = 50 mm d = 50 mm Solution: ⌠ ⎮ V = ⎮ ⎮ ⌡ c+ d 2 ⎡ y ⎞ 2⎤ π ⎢a ⎛ ⎟ ⎥ d y ⎜ ⎣ ⎝c⎠ ⎦ c ⌠ ⎮ 1⎮ yc = V⎮ ⌡ c+ d 2 ⎡ ⎛ y ⎞ 2⎤ yπ ⎢a ⎜ ⎟ ⎥ d y ⎣ ⎝c⎠ ⎦ yc = 84.7 mm c Problem 9-4 1 Locate... 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 9-4 5 Locate the center of gravity (xc, yc, zc) of the four particles Given: M1 = 2 lb a = 2 ft M2 = 3 lb b = 3 ft M3 = 1 lb c = −1 ft M4 = 1 lb d = 1 ft f = 4 ft e = 4 ft h = −2 ft g = 2 ft i = 2 ft Solution:... 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 ⎜ ⎝ ⎝ 3π ⎠ 2 c ⎞⎤ ⎟⎥ 3 ⎠⎦ xc = 2.732 in 2 ⎤ ⎛ 4d ⎞ − π a ⎛ 4a ⎞ + 1 d c⎛ d ⎞⎥ ⎜ ⎟ ⎜ ⎟ 2 ⎜ 3⎟ 2 ⎝ 3π ⎠ ⎝ ⎠⎦ ⎝ 3π ⎠ yc = 1.423 in Problem 9-5 7 Determine the location yc of the centroidal axis xcxc of the beam's cross-sectional area Neglect the... 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 2a xc = 2 + a⎛ e + ⎜ ⎝ a⎞ ⎛ e⎞ ⎟ + c( a + e) + e⎜a + ⎟ + ( c − d)a 2⎠ ⎝ 2⎠ L d yc = a d 2 +c c + ( c − d) 2 d+c 2 xc = 24.4 mm + ad + ec yc = 40.6 mm L Problem 9-4 7 The steel and aluminum plate assembly is bolted together and fastened to... 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: 2( ρ s a t w) xc = a 2 + ⎡ρ al( b + c)t w ⎛ a − b + ⎤⎜ ⎣ ⎦ ⎝ b + c⎞ 2 ⎟ ⎠ 2ρ s a t w + ρ al( b + c)t w xc = 179 mm Problem 9-4 8 The truss is made from five members, each having a length L and a mass density ρ If the mass of the gusset plates... any means, without permission in writing from the publisher Engineering Mechanics - Statics Chapter 9 Solution: L = a+b+ 2 2 2 2 2 2a a +c + b +c xc = 1⎡ a+b + ⎢( a + b) L⎣ 2 a +c yc = 1⎛ 2 2c ⎜ a +c + L⎝ 2 2 2 + 2⎛ 2 b + c ⎜a + ⎝ 2 c⎞ b +c ⎟ b ⎞⎤ ⎟⎥ 2 ⎠⎦ xc = 6.50 in yc = 4.00 in 2⎠ ⎛ xc − a ⎞ ⎟ ⎝ c − yc ⎠ θ = atan ⎜ θ = 10.6 deg Problem 9-5 1 The three members of the frame each have weight density . means, without permission in writing from the publisher. Engineering Mechanics - Statics Chapter 9 Problem 9-3 6 Locate the centroid of the quarter-cone. Solution: r a h hz−()= z c z= x c y c = 4r 3 π = V 0 h z π 4 a h hz−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⌡ d= 1 12 ha 2 π = z c 12 ha 2 π 0 h zz π 4 a h hz−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ 2 ⌠ ⎮ ⎮ ⌡ d= 1 4 h= x c 12 ha 2 π 0 h z 4 3 π a h hz−() ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ π 4 a h h. permission in writing from the 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=. publisher. Engineering Mechanics - Statics Chapter 9 Given: a 4in= b 2in= c 3in= Solution: A b ab+ y bc y ⌠ ⎮ ⎮ ⌡ d= A 6.592 in 2 = y c 1 A b ab+ yy bc y ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= y c 3.64 in= Problem 9-3 1 Determine