Explanation Given information: The expression for the constant rate ( n ) is n = I ′ − I I ′ ω 2 . Calculation: Write the expression for the velocity of the body. 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 angular velocity Ω . Ω = − ϕ ˙ sin θ i + ϕ ˙ cos θ k Determine the expression for the angular acceleration of the body. 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 . − 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. m ω z = 0 m = 0 Compare j component. ψ ˙ ϕ ˙ sin θ = ( n ω x − l ω z ) Substitute − ϕ ˙ sin θ for ω x and ( ϕ ˙ cos θ + ψ ˙ ) for ω z

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Chapter 18.3, Problem 18.123P

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The period of precession of the north pole of the earth about the axis of symmetry of the earth.

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Chapter 18.3, Problem 18.122P

Chapter 18.3, Problem 18.124P

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Chapter 18 Solutions

VECTOR MECH. FOR EGR: STATS & DYNAM (LL

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The period of precession of the north pole of the earth about the axis of symmetry of the earth.

Solution Summary: The author explains the period of precession of the earth about the axis of symmetry. The expression for the constant rate (n) is n=Iprime

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The period of precession of the north pole of the earth about the axis of symmetry of the earth.

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