Explanation Given information: The radius of aluminum sphere ( r ) is 4 inch. The speed at which joint A is rotated ( N ) is 60 rpm. The speed ( N ′ ) is 600 rpm. The length of the rod ( l ) is 10 in. Calculation: Calculate the angular velocity of the precision of joint A ( φ ˙ ) using the formula: φ ˙ = 2 π N 60 Substitute 60 rpm for N . φ ˙ = 2 π ( 60 rpm ) 60 = 6.2832 rad / s Calculate the rate of spin of the sphere ( N ′ ) corresponding to the speed using the relation: ψ ˙ = 2 π N ′ 60 Substitute 600 rpm for N ′ . ψ ˙ = 2 π N ′ 60 = 2 π ( 600 rpm ) 60 = 62.832 rad / s Take point A as origin. Consider the coordinates system with z -axis along rod AB . Consider the unit vectors along x , y , and z directions as i , j , and k , respectively. Show the coordinates system with z -axis taken along rod AB as in Figure (1). The angular velocity of the precision of joint A is φ ˙ , which is at an angle 90 − β with x -axis. Resolve the angular velocity of the precision of joint A is φ ˙ along x and z directions. φ ˙ = ( φ ˙ ) at an angle 90 − β = ( φ ˙ cos ( 90 − β ) ) along positive x -direction + ( φ ˙ sin ( 90 − β ) ) along negative z -direction = φ ˙ cos ( 90 − β ) i + φ ˙ sin ( 90 − β ) ( − k ) = φ ˙ sin β i − φ ˙ cos β k The rate of spin is about z -axis. Thus, take the rate of spin in vector form as − ψ ˙ k . Calculate the angular velocity of the solid sphere ( ω ) using the relation: ω = φ ˙ + ψ ˙ Here, ψ ˙ is the rate of spin of the solid sphere. Consider the coordinates of the angular velocity of the solid sphere along x , y , and z directions as ω x , ω y , and ω z . Thus, the angular velocity of the solid sphere in vector form is ω x i + ω y j + ω z k . Substitute ω x i + ω y j + ω z k for ω , φ ˙ sin β i − φ ˙ cos β k for φ ˙ , and − ψ ˙ k for ψ ˙ . ω x i + ω y j + ω z k = ( φ ˙ sin β i − φ ˙ cos β k ) + ( − ψ ˙ k ) = φ ˙ sin β i + ( − ψ ˙ − φ ˙ cos β ) k (1) Compare the terms of equation (1) and equate the x -components or i terms. ω x = φ ˙ sin β Compare the terms of equation (1) and equate the x -components or i terms. ω y = 0 Compare the terms of equation (1) and equate the x -components or i terms. ω z = − ( ψ ˙ + φ ˙ cos β ) Calculate the angular momentum of the sphere about A ( H A ) using the formula: H A = I x ω x i + I y ω y j + I z ω z k Here, I x , I y and I z are the mass moment of inertia of the sphere along the coordinates axes. Substitute φ ˙ sin β for ω x , zero for ω y , and − ( ψ ˙ + φ ˙ cos β ) for ω z . H A = I x ( φ ˙ sin β ) i + I y ( 0 ) + I z ( − ( ψ ˙ + φ ˙ cos β ) ) k = I x φ ˙ sin β i − I z ( ψ ˙ + φ ˙ cos β ) k Consider the reference frame with the angular velocity as that of joint A . Thus, the angular velocity of the reference frame is φ ˙ sin β i − φ ˙ cos β k . Calculate the rate of change of angular momentum ( H ˙ A ) using the formula: H ˙ A = Ω × H A Here, Ω is the angular velocity of the reference frame. Substitute φ ˙ sin β i − φ ˙ cos β k for Ω and I x φ ˙ sin β i − I z ( ψ ˙ + φ ˙ cos β ) k for H A

Connect 1 Semester Access Card for Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9781259639272

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 18.112P

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The angle (β) for which the sphere will process about a vertical axis at the constant rate of 60 rpm.

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Chapter 18.3, Problem 18.111P

Chapter 18.3, Problem 18.113P

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Chapter 18 Solutions

Connect 1 Semester Access Card for 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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The angle ( β ) for which the sphere will process about a vertical axis at the constant rate of 60 rpm.

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The angle (β) for which the sphere will process about a vertical axis at the constant rate of 60 rpm.

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Chapter 18 Solutions