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

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Chapter B, Problem B.30P

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