Explanation Given information: The expression for the constant rate ( n ) is n = I ′ − I I ′ ω 2 . The transverse moment of inertia ( I ′ ) is 0.9967 I . Calculation: Write the expression for the expression for the velocity of the body as follows: ω = − ϕ ˙ sin θ i + ( ϕ ˙ cos θ + ψ ˙ ) k Consider the reference frame G x y z be rotating with angular velocity Ω . Write the expression for the expression for the angular velocity ( Ω ) as follows: Ω = − ϕ ˙ sin θ i + ϕ ˙ cos θ k Calculate the expression for the angular acceleration of the body ( α ) using the relation: α = ω ˙ G xyz + Ω × ω Substitute 0 for ω ˙ G xyz , ( − ϕ ˙ sin θ i + ϕ ˙ cos θ k ) for Ω , and − ϕ ˙ sin θ i + ( ϕ ˙ cos θ + ψ ˙ ) k for ω . α = 0 + | i j k − ϕ ˙ sin θ 0 ϕ ˙ cos θ − ϕ ˙ sin θ 0 ϕ ˙ cos θ + ψ ˙ | = i ( 0 − 0 ) − j ( − ϕ ˙ sin θ ( ϕ ˙ cos θ + ψ ˙ ) + ϕ ˙ 2 sin θ cos θ ) + k ( 0 − 0 ) = − j ( − ϕ ˙ 2 sin θ cos θ − ψ ˙ ϕ ˙ sin θ + ϕ ˙ 2 sin θ cos θ ) = ψ ˙ ϕ ˙ sin θ j (1) The rate of change of angular velocity as observed from the body is − α . Consider that − α may be represented as the angular velocity vector rotating with angular velocity ( l i + m j + n k ) . Write the expression for the term ( − α ) as follows: − α = ( l i + m j + n k ) × ( ω x i + ω z k ) = | i j k l m n ω x 0 ω z | = i ( m ω z − 0 ) − j ( l ω z − n ω x ) + k ( 0 − m ω x ) = m ω z i + ( n ω x − l ω z ) j − m ω x k (2) Compare the Equation (1) and (2). ψ ˙ ϕ ˙ sin θ j = m ω z i + ( n ω x − l ω z ) j − m ω x k Compare i component

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 18.3, Problem 18.123P

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Chapter 18.3, Problem 18.122P

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Vector Mechanics for Engineers: Statics and Dynamics

Ch. 18.1 - A thin, homogeneous disk of mass m and radius r...Ch. 18.1 - Prob. 18.2P Ch. 18.1 - 18.3 Two uniform rods AB and CE, each of weight 3...Ch. 18.1 - A homogeneous disk of weight W = 6 lb rotates at...Ch. 18.1 - Prob. 18.5P Ch. 18.1 - A solid rectangular parallelepiped of mass m has a...Ch. 18.1 - Prob. 18.8P Ch. 18.1 - Determine the angular momentum HD of the disk of...Ch. 18.1 - Prob. 18.10P Ch. 18.1 - Prob. 18.11P

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