Explanation Write the expression of angular momentum along x ′ , y ′ , and z ′ direction. H O x ′ = − I x ′ y ′ ω x ′ H O y ′ = − I y ′ y ′ ω y ′ H O z ′ = I z ′ z ′ ω z ′ (I) Here, angular momentum along x ′ , y ′ , and z ′ are H O x ′ , H O y ′ , and H O z ′ , the moment of inertia along x ′ z ′ , y ′ z ′ , and z ′ z ′ are I x ′ z ′ , I y ′ z ′ , and I z ′ z ′ , and angular velocity along x ′ , y ′ , and z ′ direction are ω x ′ , ω y ′ , and ω z ′ . Draw the free body diagram of the rectangular plate as shown in Figure (1). Here, angular velocity is ω and distance between the half of the width is a . Write the expression of moment of inertia along x ′ z ′ direction. I x ′ z ′ = 0 (II) Write the expression of moment of inertia along y ′ y ′ direction. I y ′ y ′ = 1 12 m ( 2 a ) 2 = 1 3 m a 2 (III) Here, mass of the rectangular plate is m and half of the width of the rectangular plate is a . Write the expression of moment of inertia along z ′ z ′ direction. I z ′ z ′ = 1 12 m [ ( 2 a ) 2 + ( 2 b ) 2 ] = 1 3 m [ ( a ) 2 + ( b ) 2 ] (IV) Here, half of the breadth of the rectangular plate is b . Write the expression to determine the angular momentum of the system ( H O ) . H O = ( H O x ′ i ) + ( H O y ′ j ) + ( H O z ′ k ) (VI) Substitute I x ′ y ′ ω x ′ for H O x ′ , I x ′ y ′ ω y ′ for H O y ′ , I z ′ z ′ ω z ′ for H O z ′ in Equation (VI). H O = ( I x ′ y ′ ω x ′ i ′ ) + ( I y ′ y ′ ω y ′ j ′ ) + ( I z ′ z ′ ω z ′ k ′ ) (VII) Substitute 0 for I x ′ y ′ , 0 for ω x ′ , 1 3 m a 2 for I x ′ y ′ , ω / 2 for ω y ′ , and 1 3 m [ ( a ) 2 + ( b ) 2 ] I z ′ z ′ , ω / 2 for ω z ′ in Equation (IV)

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 7.8, Problem 64P

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The angular momentum about point O

The kinetic energy of the rod

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Chapter 7.8, Problem 63P

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The angular momentum about point O The kinetic energy of the rod