Explanation Given information: The density of the steel is 7850 kg / m 3 . The following figure represents the given system. Figure-(1) Write the expression for the mass of component 1. m 1 = ρ V 1 ...... (I) Here, the mass of component 1 is m 1 , the density of component ρ and the volume of component 1 is V 1 . Write the expression for the mass moment of inertia for component 1. ( I x y ) 1 = ( I ¯ x ′ y ′ ) 1 + m 1 ( x ¯ 1 ) ( y ¯ 1 ) ...... (II) Here, the products of the inertia of the body with respect to centroidal axis for component 1 is ( I ¯ x ′ y ′ ) 1 , the coordinates for the centre of gravity of the component 1 is ( x ¯ 1 ) and ( y ¯ 1 ) . Write the expression for the mass moment of inertia for component 1. ( I y z ) 1 = ( I ¯ y ′ z ′ ) 1 + m 1 ( z ¯ 1 ) ( y ¯ 1 ) ...... (III) Here, the products of the inertia of the body with respect to centroidal axis for component 1 is ( I ¯ y ′ z ′ ) 1 , the coordinates for the centre of gravity of the component 1 is ( z ¯ 1 ) and ( y ¯ 1 ) . Write the expression for the mass moment of inertia for component 1. ( I z x ) 1 = ( I ¯ z ′ x ′ ) 1 + m 1 ( z ¯ 1 ) ( x ¯ 1 ) ...... (IV) Here, the products of the inertia of the body with respect to centroidal axis for component 1 is ( I ¯ z ′ x ′ ) 1 , the coordinates for the centre of gravity of the component 1 is ( z ¯ 1 ) and ( x ¯ 1 ) . Write the expression for the mass of component 2. m 2 = ρ V 2 ...... (V) Here, the mass of component 2 is m 2 , the density of component ρ and the volume of component 2 is V 2 . Write the expression for the mass moment of inertia for component 2. ( I x y ) 2 = ( I ¯ x ′ y ′ ) 2 + m 2 ( x ¯ 2 ) ( y ¯ 2 ) ...... (VI) Here, the products of the inertia of the body with respect to centroidal axis for component 2 is ( I ¯ x ′ y ′ ) 2 , the coordinates for the centre of gravity of the component 2 is ( x ¯ 2 ) and ( y ¯ 2 ) . Write the expression for the mass moment of inertia for component 2. ( I y z ) 2 = ( I ¯ y ′ z ′ ) 2 + m 2 ( z ¯ 2 ) ( y ¯ 2 ) ...... (VII) Here, the products of the inertia of the body with respect to centroidal axis for component 2 is ( I ¯ y ′ z ′ ) 2 , the coordinates for the centre of gravity of the component 2 is ( z ¯ 2 ) and ( y ¯ 2 ) . Write the expression for the mass moment of inertia for component 2. ( I z x ) 2 = ( I ¯ z ′ x ′ ) 2 + m 2 ( z ¯ 2 ) ( x ¯ 2 ) ...... (VIII) Here, the products of the inertia of the body with respect to centroidal axis for component 2 is ( I ¯ z ′ x ′ ) 2 , the coordinates for the centre of gravity of the component 2 is ( z ¯ 2 ) and ( x ¯ 2 ) . Write the expression for the mass of component 3. m 3 = ρ V 3 ...... (IX) Here, the mass of component 3 is m 3 , the density of component ρ and the volume of component 3 is V 3 . Write the expression for the mass moment of inertia for component 3. ( I x y ) 3 = ( I ¯ x ′ y ′ ) 3 + m 3 ( x ¯ 3 ) ( y ¯ 3 ) ...... (X) Here, the products of the inertia of the body with respect to centroidal axis for component 3 is ( I ¯ x ′ y ′ ) 3 , the coordinates for the centre of gravity of the component 3 is ( x ¯ 3 ) and ( y ¯ 3 ) . Write the expression for the mass moment of inertia for component 3. ( I y z ) 3 = ( I ¯ y ′ z ′ ) 3 + m 3 ( z ¯ 3 ) ( y ¯ 3 ) ...... (XI) Here, the products of the inertia of the body with respect to centroidal axis for component 3 is ( I ¯ y ′ z ′ ) 3 , the coordinates for the centre of gravity of the component 3 is ( z ¯ 3 ) and ( y ¯ 3 ) . Write the expression for the mass moment of inertia for component 3. ( I z x ) 3 = ( I ¯ z ′ x ′ ) 3 + m 3 ( z ¯ 3 ) ( x ¯ 3 ) ...... (XII) Here, the products of the inertia of the body with respect to centroidal axis for component 3 is ( I ¯ z ′ x ′ ) 3 , the coordinates for the centre of gravity of the component 3 is ( z ¯ 3 ) and ( x ¯ 3 ) . Write the expression for the mass product of inertia. I x y = ( I x y ) 1 − ( I x y ) 2 − ( I x y ) 3 ...... (XIII) Here, the mass product of inertia is I x y . Write the expression for the mass product of inertia. I y z = ( I y z ) 1 − ( I y z ) 2 − ( I y z ) 3 ...... (XIV) Here, the mass product of inertia is I y z . Write the expression for the mass product of inertia. I z x = ( I z x ) 1 − ( I z x ) 2 − ( I z x ) 3 ...... (XV) Here, the mass product of inertia is I z x . Write the expression for the volume of the component 1. V 1 = a b c ..... (XVI) Here, the sides of the component is a or b and c . Write the expression for volume of the component 2. V 2 = b d e ....... (XVII) Here, the sides of the component is d or b and e . Write the expression for volume of the component 3. V 3 = π f 2 g 4 ....... (XVIII) Here, the diameter of the circle is f and the width the component 3 is g . Calculation: Substitute 160 mm for a , 80 mm for b and 50 mm for c in Equation (XVI). V 1 = ( 160 mm ) ( 80 mm ) ( 50 mm ) = 640000 mm 3 ( 1 m 3 10 9 mm 3 ) = 6.4 × 10 − 4 m 3 Substitute 7850 kg / m 3 for ρ and 6.4 × 10 − 4 m 3 for V 1 in Equation (I). m 1 = ( 6.4 × 10 − 4 m 3 ) ( 7850 kg / m 3 ) = 5.024 kg Substitute 5.024 kg for m 1 , 40 mm for x ¯ 1 , 25 mm for y ¯ 1 and 0 for ( I ¯ x ′ y ′ ) 1 in Equation (II). ( I x y ) 1 = 0 + 5.024 kg ( 40 mm ) ( 25 mm ) = 5024 kg ⋅ mm 2 ( 1 m 2 10 6 mm 2 ) = 5.024 × 10 − 3 kg ⋅ m 2 Substitute 5.024 kg for m 1 , 80 mm for z ¯ 1 , 25 mm for y ¯ 1 and 0 for ( I ¯ y ′ z ′ ) 1 in Equation (III). ( I y z ) 1 = 0 + 5.024 kg ( 80 mm ) ( 25 mm ) = 10048 kg ⋅ mm 2 ( 1 m 2 10 6 mm 2 ) = 0.010048 kg ⋅ m 2 Substitute 5.024 kg for m 1 , 80 mm for z ¯ 1 , 40 mm for x ¯ 1 , and 0 for ( I ¯ z ′ x ′ ) 1 in Equation (IV)

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,

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The mass product of the inertia I x y . The mass product of the inertia I y z . The mass product of the inertia I z x .

Solution Summary: The following figure represents the given system. Write the expression for the mass moment of inertia for component 1.

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The mass product of the inertia Ixy.

The mass product of the inertia Iyz.

The mass product of the inertia Izx.

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The mass product of the inertia I x y . The mass product of the inertia I y z . The mass product of the inertia I z x .

Solution Summary: The following figure represents the given system. Write the expression for the mass moment of inertia for component 1.

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The mass product of the inertia Ixy. The mass product of the inertia Iyz. The mass product of the inertia Izx.

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Chapter B Solutions

The mass product of the inertia Ixy.

The mass product of the inertia Iyz.

The mass product of the inertia Izx.