Explanation Given information: The mass is m , the angle is 45 ° , the angular velocity of the shaft ω = ω j and the angular acceleration α = α j . The below figure represent the angle of the axis. Figure-(1) Write the expression of angular velocity along y ′ axis. ω y ′ = ω cos θ ........ (I) Here, the angular velocity is ω and the angle is θ . Substitute 45 ° for θ in Equation (I). ω y ′ = ω cos ( 45 ° ) = ω × ( 1 2 ) = ω 2 Write the expression of angular velocity in z ′ direction. ω z ′ = ω cos θ ......... (II) Substitute 45 ° for θ in Equation (II). ω z ′ = ω cos ( 45 ° ) = ω × ( 1 2 ) = ω 2 Write the expression of moment of inertia about x ′ axis. I x ′ = 1 3 m a 2 ....... (III) Here, the mass is m and the side is a . Write the expression of moment of inertia about y ′ axis. I y ′ = 1 12 m a 2 ....... (IV) Write the expression of moment of inertia about z ′ axis. I z ′ = I x ′ + I y ′ ....... (IV) Substitute 1 3 m a 2 for I x ′ and 1 12 m a 2 for I y ′ in Equation (IV). I z ′ = ( 1 3 m a 2 ) + ( 1 12 m a 2 ) = 5 12 m a 2 Write the expression of angular momentum about A . H A = I x ′ ω x ′ i ′ + I y ′ ω y ′ j ′ + I z ′ ω z ′ k ′ ....... (V) Substitute 1 3 m a 2 for I x ′ , 1 12 m a 2 for I y ′ , 5 12 m a 2 for I z ′ , 0 for ω x ′ , ω 2 for ω y ′ and ω 2 for ω z ′ in Equation (V). H A = I x ′ ( 0 ) i ′ + ( 1 12 m a 2 ) ( ω 2 ) j ′ + ( 5 12 m a 2 ) ( ω 2 ) k ′ = ( 1 12 m a 2 ) ( ω 2 ) j ′ + ( 5 12 m a 2 ) ( ω 2 ) k ′ ........ (VI) Write the expression of unit vector in y ′ direction. j ′ = ( cos θ ) j − ( sin θ ) k ....... (VII) Substitute 45 ° for θ in Equation (VII). j ′ = ( cos 45 ° ) j − ( sin 45 ° ) k = ( 1 2 j − 1 2 k ) Write the expression of unit vector in z ′ direction. k ′ = ( sin θ ) j + ( cos θ ) k ....... (VIII) Substitute 45 ° for θ in Equation (VIII). k ′ = ( sin 45 ° ) j + ( cos 45 ° ) k = ( 1 2 j + 1 2 k ) Substitute ( 1 2 j + 1 2 k ) for k ′ and ( 1 2 j − 1 2 k ) for j ′ in Equation (VI). H A = I x ′ ( 0 ) i ′ + ( 1 12 m a 2 ) ( ω 2 ) j ′ + ( 5 12 m a 2 ) ( ω 2 ) k ′ = ( 1 12 m a 2 ) ( ω 2 ) ( 1 2 j − 1 2 k ) + ( 5 12 m a 2 ) ( ω 2 ) ( 1 2 j + 1 2 k ) = [ ( m a 2 ω 12 2 ) ( 1 2 j − 1 2 k ) + ( 5 m a 2 ω 12 2 ) ( 1 2 j + 1 2 k ) ] = ( m a 2 ω 24 ) j − ( m a 2 ω 24 ) k + ( 5 m a 2 ω 24 ) j + ( m a 2 ω 24 ) k H A = ( 6 m a 2 ω 24 ) j + ( 4 m a 2 ω 24 ) k = ( 3 m a 2 ω 12 ) j + ( 2 m a 2 ω 12 ) k = ( m a 2 ω 12 ) ( 3 j + 2 k ) ........ (IX) Write the expression of angular acceleration along y ′ axis. ω ˙ y = α cos θ ........ (X) Substitute 45 ° for θ in Equation (X). ω ˙ y = α cos 45 ° = α 2 Write the expression of angular acceleration along z ′ axis. ω ˙ z = α cos θ ........ (XI) Substitute 45 ° for θ in Equation (XI)

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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Chapter 18.2, Problem 18.63P

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The rate of change H˙A of the angular momentum HA of the plate assembly.

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Chapter 18.2, Problem 18.62P

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A homogeneous disk of weight W= 6 lb rotates at the constant rate W1= 16 rad/s with respect to arm ABC,which is welded to a shaft DCE rotating at the constant rate w 2= 8 rad/s. Determine the angular momentum HA of the disk about its center A.

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Chapter 18 Solutions

Vector Mechanics For Engineers

Ch. 18.1 - Prob. 18.1P Ch. 18.1 - Prob. 18.2P Ch. 18.1 - Prob. 18.3P Ch. 18.1 - A homogeneous disk of weight W=6 lb rotates at the...Ch. 18.1 - Prob. 18.5P Ch. 18.1 - A solid rectangular parallelepiped of mass m has a...Ch. 18.1 - Solve Prob. 18.6, assuming that the solid...Ch. 18.1 - Prob. 18.8P Ch. 18.1 - Determine the angular momentum HD of the disk of...Ch. 18.1 - Prob. 18.10P

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The rate of change H ˙ A of the angular momentum H A of the plate assembly.

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The rate of change H˙A of the angular momentum HA of the plate assembly.

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Chapter 18 Solutions