Engineering Mechanics - Statics Chapter 9 F Rv γ π r 2 4 = F Rv 196.0 lb ft = Problem 9-119 The load over the plate varies linearly along the sides of the plate such that p = k y (a-x). Determine the magnitude of the resultant force and the coordinates (x c , y c ) of the point where the line of action of the force intersects the plate. Given: a 2ft= b 6ft= k 10 lb ft 4 = Solution: pxy,()ky a x−()= F R 0 a x 0 b ypxy,() ⌠ ⎮ ⌡ d ⌠ ⎮ ⌡ d= F R 360lb= x c 1 F R 0 a x 0 b yxp x y,() ⌠ ⎮ ⌡ d ⌠ ⎮ ⌡ d= x c 0.667 ft= y c 1 F R 0 a x 0 b yyp x y,() ⌠ ⎮ ⌡ d ⌠ ⎮ ⌡ d= y c 4ft= Problem 9-120 The drum is filled to its top (y = a) with oil having a density γ . Determine the resultant force of the oil pressure acting on the flat end of plate A of the drum and specify its location measured from the top of the drum. 981 © 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 1.5 ft= γ 55 lb ft 3 = Solution: F R a− a y γ 2 a 2 y 2 − ay−() ⌠ ⎮ ⌡ d= F R 583lb= da 1 F R a− a yy γ 2 a 2 y 2 − ay−() ⌠ ⎮ ⌡ d−= d 1.875 ft= Problem 9-121 The gasoline tank is constructed with elliptical ends on each side of the tank. Determine the resultant force and its location on these ends if the tank is half full. Given: a 3ft= b 4ft= γ 41 lb ft 3 = Solution: F R a− 0 y γ − y2 b a a 2 y 2 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d= F R 984lb= 982 © 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 y c 1 F R a− 0 yy γ − y2 b a a 2 y 2 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⌠ ⎮ ⎮ ⌡ d= y c 1.767− ft= x c 0ft= Problem 9-122 The loading acting on a square plate is represented by a parabolic pressure distribution. Determine the magnitude of the resultant force and the coordinates (x c , y c ) of the point where the line of action of the force intersects the plate. Also, what are the reactions at the rollers B and C and the ball-and-socket joint A? Neglect the weight of the plate. Units Used: kPa 10 3 Pa= kN 10 3 N= Given: a 4m= p 0 4 kPa= Solution: Due to symmetry x c 0= F R 0 a yp 0 y a a ⌠ ⎮ ⎮ ⌡ d= F R 42.667 kN= y c 1 F R 0 a yyp 0 y a a ⌠ ⎮ ⎮ ⌡ d= y c 2.4 m= Equilibrium Guesses A y 1kN= B y 1kN= C y 1kN= Given A y B y + C y + F R − 0= B y C y + () aF R y c − 0= B y a 2 C y a 2 − 0= 983 © 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 A y B y C y ⎛ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎠ Find A y B y , C y , () = A y B y C y ⎛ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎠ 17.067 12.8 12.8 ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ kN= Problem 9-123 The tank is filled with a liquid which has density . ρ Determine the resultant force that it exerts on the elliptical end plate, and the location of the center of pressure, measured from the x axis. Units Used: kN 10 3 N= Given: a 1m= b 0.5 m= ρ 900 kg m 3 = g 9.81 m s 2 = Solution: F R b− b y ρ g2a 1 y b ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − by−() ⌠ ⎮ ⎮ ⌡ d= F R 6.934 kN= y c 1 F R b− b yy ρ g2a 1 y b ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 − by−() ⌠ ⎮ ⎮ ⌡ d= y c 0.125− m= Problem 9-124 A circular V-belt has an inner radius r and a cross-sectional area as shown. Determine the volume of material required to make the belt. 984 © 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: r 600 mm= a 25 mm= b 50 mm= c 75 mm= Solution: V 2 π r c 3 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 2 1 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ac r c 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ bc+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = V 22.4 10 3− × m 3 = Problem 9-125 A circular V-belt has an inner radius r and a cross-sectional area as shown. Determine the surface area of the belt. Given: r 600 mm= a 25 mm= b 50 mm= c 75 mm= Solution: A 2 π rb 2 r c 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ a 2 c 2 ++ rc+()b 2a+()+ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = A 1.246 m 2 = 985 © 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 9-126 Locate the center of mass of the homogeneous rod. Given: a 200 mm= b 600 mm= c 100 mm= d 200 mm= θ 45 deg= Solution: Lab+ c+ d+= x c 1 L b b 2 sin θ () cbsin θ () + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = x c 154.3 mm= y c 1 L d d 2 b b 2 cos θ () + cbcos θ () + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = y c 172.5 mm= z c 1 L a a 2 da+ c c 2 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = z c 50.0 mm= Problem 9-127 Locate the centroid of the solid 986 © 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: y c 0 2a zz π aa z 2 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d 0 2a z π aa z 2 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⌠ ⎮ ⎮ ⌡ d = 2 a 4 π a 3 π z− π a 3 = y c 2 3 a= Problem 9-128 Locate the centroid (x c , y c ) of the thin plate. Given: a 6in= Solution: A 4a 2 a 2 2 − π a 2 4 −= A 97.7 in 2 = x c 1 A a 2 − 2 2− 3 a ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ π a 2 4 a 4a 3 π − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = x c 0.262− in= y c 1 A a 2 − 2 2a 3 π a 2 4 4a 3 π a− ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = y c 0.262 in= 987 © 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 9-129 Determine the weight and location (x c , y c ) of the center of gravity G of the concrete retaining wall. The wall has a length L, and concrete has a specific gravity of γ. Units Used: kip 10 3 lb= Given: a 12 ft= f 1ft= b 9ft= g 2ft= c 1.5 ft= L 10 ft= d 5.5 ft= γ 150 lb ft 3 = e 1.5 ft= Solution: Abcaf+ 1 2 ae f−()+= W γ AL= W 42.8 kip= x c 1 A bc b 2 af g f 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 1 2 ae f−()gf+ ef− 3 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = x c 3.52 ft= y c 1 A bc c 2 af c a 2 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + 1 2 ae f−()c a 3 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = y c 4.09 ft= Problem 9-130 The hopper is filled to its top with coal. Determine the volume of coal if the voids (air space) are a fraction p of the volume of the hopper. Given: a 1.5 m= b 4m= c 1.2 m= 988 © 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 0.2 m= p 0.35= Solution: V 1 p−()2 π dc d 2 1 2 ca d−()d ad− 3 + ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ + ab a 2 + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ = V 20.5 m 3 = Problem 9-131 Locate the centroid (x c , y c ) of the shaded area. Given: a 16 ft= b 4ft= cab− () 2 = Solution: A 0 b xa x− () 2 ⌠ ⎮ ⌡ d= A 29.3 ft 2 = 989 © 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 1 A 0 b xxa x− () 2 ⌠ ⎮ ⌡ d= x c 1.6 ft= y c 1 A 0 b x a x− () 2 ⎡ ⎣ ⎤ ⎦ 2 2 ⌠ ⎮ ⎮ ⎮ ⌡ d= y c 4.15 ft= Problem 9-132 The rectangular bin is filled with coal, which creates a pressure distribution along wall A that varies as shown, i.e., p = p 0 (z/b) 1/3 . Compute the resultant force created by the coal, and its location, measured from the top surface of the coal. Given: p 0 8 lb ft 2 = a 3ft= b 8ft= Solution: F 0 b zp 0 z b ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1 3 a ⌠ ⎮ ⎮ ⎮ ⎮ ⌡ d= F 144lb= z c 1 F 0 b zzp 0 z b ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1 3 a ⌠ ⎮ ⎮ ⎮ ⎮ ⌡ d= z c 4.57 ft= 990 © 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. [...]... the publisher Engineering Mechanics - Statics Chapter 10 Problem 1 0-5 Determine the moment for inertia of the shaded area about the y axis Given: a = 4 in b = 2 in Solution: a ⌠ 3 ⎮ 2 ⎛x⎞ I y = ⎮ x b ⎜ ⎟ dx ⎝ a⎠ ⌡ 0 Iy = 21.33 in 4 Problem 1 0-6 Determine the moment of inertia for the shaded area about the x axis Solution: ⌠ ⎮ ⎮ Ix = ⎮ ⎮ ⌡ b 3 ⎛ x⎞ ⎜h ⎟ ⎝ b ⎠ dx = 2 b h3 3 15 Ix = 2 3 bh 15 0 995 © 2007... permission in writing from the publisher Engineering Mechanics - Statics Chapter 10 Alternatively h ⌠ 2⎞ ⎮ 2⎛ ⎜ b − b y ⎟ d y = 2 b h3 Ix = ⎮ y 2⎟ ⎜ 15 ⎮ h ⎠ ⎝ ⌡0 Ix = 2 3 bh 15 Ix = ab 3( 1 + 3n) Problem 1 0-7 Determine the moment of inertia for the shaded area about the x axis Solution: b ⌠ 1⎤ ⎮ ⎡ ⎢ ⎮ n⎥ ⎞ ⎮ A y2 ⎢a − a ⎛ y ⎟ ⎥ d y Ix = ⎜ ⎮ ⎣ ⎝ b⎠ ⎦ ⌡ 3 0 Problem 1 0-8 Determine the moment of inertia for... reproduced, in any form or by any means, without permission in writing from the publisher Engineering Mechanics - Statics A = Chapter 10 Ix + Iy' − Jcc 2 h 3 2 A = 11.1 × 10 mm Problem 1 0-2 7 Determine the radius of gyration kx 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:... 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: Iy = 1 12 1 3 ( 2a) ( 2b) − 12 ( 2c) ( 2d) 3 Radius of gyration about the y axis: ky = Iy A ky = 59.4 mm Problem 1 0-2 9 Determine the moment of inertia for the beam's cross-sectional area with respect to the... publisher Engineering Mechanics - Statics Chapter 10 Solution: a ⌠ ⌠ 2 ⎮ 2 I y = ⎮ x d A = ⎮ x y dx ⌡0 ⌡ a b ⌠ n+ 2 ⎮ x dx = Iy = n⌡ a 0 ⎡⎛ b ⎞ xn + 3 ⎤ ⎢ ⎥ ⎢⎜ an ⎟ n + 3⎥ ⎣⎝ ⎠ ⎦ a 0 3 ba Iy = n+3 Problem 1 0-9 Determine the moment of inertia for the shaded area about the x axis Given: a = 4 in b = 2 in Solution: b ⌠ 2 y⎞ ⎤ ⎮ 2⎡ ⎢a − a ⎛ ⎟ ⎥ d y Ix = ⎮ y ⎜ ⎣ ⎝ b⎠ ⎦ ⌡ 0 Ix = 4.27 in 4 Problem 1 0-1 0 Determine... permission in writing from the publisher Engineering Mechanics - Statics Chapter 10 Given: a = 4 in b = 2 in Solution: a ⌠ 2 Iy = ⎮ x b ⎮ ⌡ x dx a 0 Iy = 36.6 in 4 Problem 1 0-1 1 Determine the moment of inertia for the shaded area about the x axis Given: a = 8 in b = 2 in Solution: b ⌠ 3 ⎮ 2⎛ ⎜a − a y ⎞ d y ⎟ Ix = ⎮ y 3⎟ ⎜ ⎮ b ⎠ ⎝ ⌡0 Ix = 10.67 in 4 Problem 1 0-1 2 Determine the moment of inertia for the... without permission in writing from the publisher Engineering Mechanics - Statics Chapter 10 Given: a = 2m b = 1m Solution: b ⌠ ⎮ 2 ⎛ Ix = ⎮ y a⎜ 1 − ⎜ ⎮ ⎝ ⌡− b 2⎞ y b ⎟ dy 2⎟ ⎠ Ix = 0.53 m 4 Problem 1 0-1 3 Determine the moment of inertia for the shaded area about the y axis Given: a = 2m b = 1m Solution: a ⌠ x 2 Iy = ⎮ x 2b 1 − dx ⎮ a ⌡ Iy = 2.44 m 4 0 Problem 1 0-1 4 Determine the moment of inertia for the... 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: b ⌠ 2⎤ ⎮ 2⎡ ⎛ y⎞ ⎥ dy Ix = ⎮ y ⎢a − a ⎜ ⎟ ⎣ ⎝ b⎠ ⎦ ⌡ 0 Ix = 34.1 in 4 Problem 1 0-1 5 Determine the moment of inertia for the shaded area about the y axis Given: a = 4 in b = 4 in Solution: a ⌠ 2 Iy = ⎮ x b ⎮ ⌡ x dx a... in any form or by any means, without permission in writing from the publisher Engineering Mechanics - Statics Chapter 10 Problem 1 0-1 6 Determine the moment of inertia of the shaded area about the x axis Given: a = 2 in b = 4 in Solution: ⌠ ⎮ Ix = ⎮ ⌡ a 3 1⎛ ⎛ π x ⎞⎞ ⎜ b cos ⎜ ⎟ ⎟ dx 3⎝ ⎝ 2a ⎠ ⎠ −a Ix = 36.2 in 4 Problem 1 0-1 7 Determine the moment of inertia for the shaded area about the y axis Given:... any form or by any means, without permission in writing from the publisher Engineering Mechanics - Statics Iy = 192.00 in Chapter 10 4 Problem 1 0-2 2 Determine the moment of inertia for the shaded area about the x axis Given: a = 2m b = 2m Solution: b ⌠ ⎮ 2 ⎛ y2 ⎞ I x = ⎮ y a⎜ ⎟ d y ⎜ b2 ⎟ ⎮ ⎝ ⎠ ⌡0 Ix = 3.20 m 4 Problem 1 0-2 3 Determine the moment of inertia for the shaded area about the y axis Use . from the publisher. Engineering Mechanics - Statics Chapter 10 Alternatively I x 0 h yy 2 bb y 2 h 2 − ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ ⌠ ⎮ ⎮ ⎮ ⌡ d= 2 15 bh 3 = I x 2 15 bh 3 = Problem 1 0-7 Determine the moment. from the publisher. Engineering Mechanics - Statics Chapter 9 x c 1 A 0 b xxa x− () 2 ⌠ ⎮ ⌡ d= x c 1.6 ft= y c 1 A 0 b x a x− () 2 ⎡ ⎣ ⎤ ⎦ 2 2 ⌠ ⎮ ⎮ ⎮ ⌡ d= y c 4 .15 ft= Problem 9-1 32 The rectangular. Engineering Mechanics - Statics Chapter 9 F Rv γ π r 2 4 = F Rv 196.0 lb ft = Problem 9-1 19 The load over the plate varies linearly along the sides of the plate such that p = k y (a-x).