Explanation Given information: The thickness of the rectangular sheet is 2 mm , thickness of the wall of the drum is 1.8 mm , density of the trough is 785 0 kg / m 3 , length of A G is 21 0 mm , length of B C is 57 0 mm , length of A B is 84 0 mm . Figure (1) Write the expression for mass of the section (1) of the trough. m 1 = ρ t 1 ( A G × A B ) ..... (I) Here, density of the trough is ρ , and the thickness of the rectangular piece is t 1 . Write the expression for mass of the section (2) of the trough. m 2 = ρ t 2 π × B C 2 × A B ..... (II) Here, thickness of the wall of the drum is t 2 . Write the expression for mass of the section (3) of the trough. m 3 = ρ t 2 π 2 ( B C 2 ) 2 ..... (III) Write the expression for mass of the section (4) of the trough. m 4 = m 3 Write the expression for the mass moment of inertia of the section (1) of the trough about the x axis. ( I 1 ) x = m 1 [ ( A G 12 ) 2 − ( B C − A G 2 ) 2 ] ..... (IV) Write the expression for the mass moment of inertia of the section (2) of the trough about the x axis. ( I 2 ) x = m 2 × ( B C 2 ) 2 ..... (V) Write the expression for the mass moment of inertia of the section (3) of the trough about the x axis. ( I 3 ) x = m 3 2 ( B C 2 ) 2 ..... (VI) Write the expression for the mass moment of inertia of the section (4) of the trough about the x axis. ( I 4 ) x = ( I 3 ) x ..... (VII) Write the expression for the mass moment of inertia of the section (1) of the trough about the y axis. ( I 1 ) y = m 1 [ ( A B 2 + A G 2 12 ) + ( A B 2 ) 2 + ( B C − A G 2 ) 2 ] ..... (VIII) Write the expression for the mass moment of inertia of the section (2) of the trough about the y axis. ( I 2 ) y = m 2 [ 1 12 ( ( A B 2 ) + 6 × ( B C 2 ) 2 ) + ( A B 2 ) 2 ] = m 2 [ ( A B 2 12 ) + 1 2 × ( B C 2 ) 2 + A B 2 4 ] = m 2 [ ( A B 2 3 ) + B C 2 8 ] ..... (IX) Write the expression for the mass moment of inertia of the section (3) of the trough about the y axis. ( I 3 ) y = m 3 4 ( B C 2 ) 2 ..... (X) Write the expression for the mass moment of inertia of the section (4) of the trough about the y axis. ( I 4 ) y = m 3 [ 1 4 ( B C 2 ) 2 + A B 2 ] ..... (XI) Write the expression for the mass moment of inertia of the section (1) of the trough about the z axis. ( I 1 ) z = m 1 [ ( A B 2 3 ) ] ..... (XII) Write the expression for the mass moment of inertia of the section (2) of the trough about the z axis. ( I 2 ) z = m 2 [ 1 12 ( ( A B 2 ) + 6 × ( B C 2 ) 2 ) + ( A B 2 ) 2 ] = m 2 [ ( A B 2 12 ) + 1 2 × ( B C 2 ) 2 + A B 2 4 ] = m 2 [ ( A B 2 3 ) + B C 2 8 ] ..... (XIII) Write the expression for the mass moment of inertia of the section (3) of the trough about the z axis.. ( I 3 ) z = m 3 4 ( B C 2 ) 2 ..... (XIV) Write the expression for the mass moment of inertia of the section (4) of the trough about the z axis. ( I 4 ) z = [ 1 4 ( B C 2 ) 2 + A B 2 ] ..... (XV) Write the expression for the mass moment of inertia of the trough about the x axis. I x = ( I 1 ) x + ( I 2 ) x + ( I 3 ) x + ( I 4 ) x ..... (XVI) Write the expression for the mass moment of inertia of the trough about the y axis. I y = ( I 1 ) y + ( I 2 ) y + ( I 3 ) y + ( I 4 ) y ..... (XVII) Write the expression for the mass moment of inertia of the trough about the z axis. I z = ( I 1 ) z + ( I 2 ) z + ( I 3 ) z + ( I 4 ) z ..... (XVIII) Calculation: The thickness of the trough is 2 mm , density of the trough is 785 0 kg / m 3 , length of A G is 21 0 mm , length of B C is 57 0 mm , length of A B is 84 0 mm . Substitute 785 0 kg / m 3 for ρ , 2 mm for t 1 , 21 0 mm for A G and 84 0 mm for A B in Equation (I). m 1 = ( 785 0 kg / m 3 ) ( 2 mm ) ( 21 0 mm × 84 0 mm ) = ( 785 0 kg / m 3 ) ( 3528 0 0 mm 3 ) ( 1 m 3 1 0 9 mm 3 ) = 2.77 kg Substitute 785 0 kg / m 3 for ρ , 1.8 mm for t 2 , 57 0 mm for B C , and 84 0 mm for A B in Equation (II). m 2 = ( 785 0 kg / m 3 ) ( 1.8 mm ) π × 57 0 mm 2 × 84 0 mm = ( 785 0 kg / m 3 ) ( 15 0 4194.563 mm 3 ) ( 1 m 3 1 0 9 mm 3 ) =1 0 . 6 2 kg Substitute 785 0 kg / m 3 for ρ , 1.8 mm for t 2 , 57 0 mm for B C , and in Equation (III). m 3 = ( 785 0 kg / m 3 ) ( 1.8 mm ) π 2 ( 57 0 mm 2 ) 2 = ( 785 0 kg / m 3 ) ( 459316.55 mm 3 ) ( 1 m 3 1 0 9 mm 3 ) = 1.8 0 kg Substitute 2.77 kg for m 1 , 57 0 mm for B C , and 21 0 mm for A G in Equation (IV). ( I 1 ) x = ( 2.77 kg ) [ ( 21 0 mm 12 ) 2 + ( 57 0 mm − 21 0 mm 2 ) 2 ] = ( 2.77 kg ) ( 36 0 75 mm 2 ) ( 1 m 2 1 0 6 mm 2 ) = 0 . 1 kg ⋅ m 2 Substitute 1 0 .62 kg for m 2 , and 57 0 mm for B C in Equation (V). ( I 2 ) x = ( 1 0 .62 kg ) × ( 57 0 mm 2 ) 2 = ( 1 0 .62 kg ) ( 81225 mm 2 ) ( 1 m 2 1 0 6 mm 2 ) = 0 .86 kg ⋅ m 2 Substitute 1

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

See similar textbooks

Videos

Question

Chapter B, Problem B.30P

To determine

The mass moment of inertia of the trough about each coordinate axis.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter B, Problem B.29P

Chapter B, Problem B.31P

Students have asked these similar questions

For items 4 and 5. A thin rectangular plate is welded on the center of the top of a circular cylinder. The mass of the cylinder is 5kg and the mass of the triangular plate is 2kg. Note that the x-axis is located on the centroidal x-axis of the triangular plate. 30cm 26cm 0.1725 m diam. d=7cm X 10cm 4. Which of the following is closest to the moment of inertia of the composite object about the x- axis? 0.208 kg-m^2 5. Which of the following is closest to the radius of gyration of the composite object about the x- axis?

Determine the moments of inertia about the centroidal x-axes of the trapezoidal area. a=147 mm; b=294 mm; h=441 mm. Answer the question in mm4. Yanıt: b b Yanıt: Answer the question in mm4. h Determine the moments of inertia about the centroidal y-axes of the trapezoidal area. X W

For the entire section shown, the moments of inertia with respect to the centroidal x and y axes at point C are Ix = 0.162(106) mm4 and Iy = 0.454(106) mm4, respectively. a. Determine the product of inertia with respect to the centroid at C, in mm4. b. Use a Mohr's Circle to determine the orientation (in degrees) of the principal axes of the section about the centroid C. c. Use the same Mohr's Circle to determine the values of the principal moments of inertia about the centroid C, in mm4.

Chapter B Solutions

Vector Mechanics For Engineers

Ch. B - A thin plate with a mass m is cut in the shape of...Ch. B - Prob. B.2P Ch. B - Prob. B.3P Ch. B - Prob. B.4P Ch. B - A piece of thin, uniform sheet metal is cut to...Ch. B - Prob. B.6P Ch. B - Prob. B.7P Ch. B - Prob. B.8P Ch. B - Prob. B.9P Ch. B - Prob. B.10P

Ch. B - Prob. B.11P Ch. B - Prob. B.12P Ch. B - Determine by direct integration the mass moment of...Ch. B - Prob. B.14P Ch. B - A thin, rectangular plate with a mass m is welded...Ch. B - A thin steel wire is bent into the shape shown....Ch. B - Prob. B.17P Ch. B - Prob. B.18P Ch. B - Prob. B.19P Ch. B - Prob. B.20P Ch. B - Prob. B.21P Ch. B - Prob. B.22P Ch. B - Prob. B.23P Ch. B - Prob. B.24P Ch. B - Prob. B.25P Ch. B - Prob. B.26P Ch. B - Prob. B.27P Ch. B - Prob. B.28P Ch. B - Prob. B.29P Ch. B - Prob. B.30P Ch. B - Prob. B.31P Ch. B - Determine the mass moments of inertia and the...Ch. B - Prob. B.33P Ch. B - Prob. B.34P Ch. B - Prob. B.35P Ch. B - Prob. B.36P Ch. B - Prob. B.37P Ch. B - Prob. B.38P Ch. B - Prob. B.39P Ch. B - Prob. B.40P Ch. B - Prob. B.41P Ch. B - Prob. B.42P Ch. B - Prob. B.43P Ch. B - Prob. B.44P Ch. B - A section of sheet steel 2 mm thick is cut and...Ch. B - Prob. B.46P Ch. B - Prob. B.47P Ch. B - Prob. B.48P Ch. B - Prob. B.49P Ch. B - Prob. B.50P Ch. B - Prob. B.51P Ch. B - Prob. B.52P Ch. B - Prob. B.53P Ch. B - Prob. B.54P Ch. B - Prob. B.55P Ch. B - Determine the mass moment ofinertia of the steel...Ch. B - Prob. B.57P Ch. B - Prob. B.58P Ch. B - Determine the mass moment of inertia of the...Ch. B - Prob. B.60P Ch. B - Prob. B.61P Ch. B - Prob. B.62P Ch. B - Prob. B.63P Ch. B - Prob. B.64P Ch. B - Prob. B.65P Ch. B - Prob. B.66P Ch. B - Prob. B.67P Ch. B - Prob. B.68P Ch. B - Prob. B.69P Ch. B - Prob. B.70P Ch. B - For the component described in the problem...Ch. B - Prob. B.72P Ch. B - For the component described in the problem...Ch. B - Prob. B.74P

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The mass moment of inertia of the trough about each coordinate axis.

Solution Summary: The author explains the mass moment of inertia of the trough about each coordinate axis.

Videos

The mass moment of inertia of the trough about each coordinate axis.

Want to see the full answer?

Chapter B Solutions