Explanation Given information: The mass of the disk is m . The radius of the disk is r . The angle ( θ ) formed by the plane of disk is 20 ° . The angular velocity ( ω ) is ω i . The angular acceleration ( α ) is α i . Calculation: Consider the principle axes x ′ , y ′ and z ′ . Show the components of vector with principal axis as in Figure (1). Write the expression for the components of vector along x ′ axis as follows: i ′ = cos β i + sin β j Write the expression for the components of vector along y ′ axis as follows: j ′ = − sin β i + cos β j Write the expression for the components of vector along x axis as follows: i = cos β i ′ − sin β j ′ Write the expression for the components of vector along y axis as follows: j = sin β i ′ + cos β j ′ Calculate the angular velocity ( ω ) of system using the relation: ω = ω i Substitute cos β i ′ − sin β j ′ for i . ω = ω ( cos β i ′ − sin β j ′ ) = ω cos β i ′ − ω sin β j ′ Calculate the angular acceleration ( α ) of system using the relation: α = α i Substitute cos β i ′ − sin β j ′ for i . α = α ( cos β i ′ − sin β j ′ ) = α cos β i ′ − α sin β j ′ Write the expression for the equation of mass moment of inertia ( I y ′ ) of the system along y ′ -axis as follows: I y ′ = 1 4 m r 2 Write the expression for the equation of mass moment of inertia ( I z ′ ) of the disk along z ′ -axis as follows: I z ′ = 1 4 m r 2 Calculate the equation of mass moment of inertia ( I x ′ ) of the disk along x ′ -axis using the relation: I x ′ = I x ′ + I z ′ Substitute 1 4 m r 2 for I y ′ and 1 4 m r 2 for I z ′ . I x ′ = 1 4 m r 2 + 1 4 m r 2 = 1 2 m r 2 Calculate the angular momentum H G of the system about point G using the formula: H G = I x ′ ω x ′ i ′ + I y ′ ω y ′ j ′ + I z ′ ω z ′ k ′ Substitute 1 2 m r 2 for I ¯ x ′ , ω cos β for ω x ′ , 1 4 m r 2 for I ¯ y ′ , − ω sin β for ω y ′ , 1 4 m r 2 for I ¯ z ′ , and 0 for ω z ′ . H G = ( 1 2 m r 2 ) ( ω cos β ) i ′ + ( 1 4 m r 2 ) ( − ω sin β ) j ′ + ( 1 4 m r 2 ) ( 0 ) k ′ = 1 2 m r 2 ω cos β i ′ − 1 4 m r 2 ω sin β j ′ + 0 = 1 2 m r 2 ω cos β i ′ − 1 4 m r 2 ω sin β j ′ Write the expression for the angular velocity of the disk ( Ω ) as follows: Ω = ω j Write the expression for angular velocity with respect to rotating frame Gxyz ( H ˙ G ) G x y z as follows: ( H ˙ G ) G x y z = I x ′ α x ′ i ′ + I y ′ α y ′ j ′ + I z ′ α z ′ k ′ Calculate the rate of change of angular momentum ( H ˙ G ) using the formula: H ˙ G = ( H ˙ G ) G x y z + Ω ( H G ) Substitute I x ′ α x ′ i ′ + I y ′ α y ′ j ′ + I z ′ α z ′ k ′ for ( H ˙ G ) A x y z , ω j for Ω and 1 2 m r 2 ω cos β i ′ − 1 4 m r 2 ω sin β j ′ for H G

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

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

Publisher: McGraw-Hill Education

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Chapter 18.2, Problem 18.64P

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The rate of change (H˙G) of the angular momentum (HG) of the disk.

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Chapter 18.2, Problem 18.63P

Chapter 18.2, Problem 18.65P

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Chapter 18 Solutions

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 - Prob. 18.3P Ch. 18.1 - A homogeneous disk of weight W = 6 lb rotates at...Ch. 18.1 - A homogeneous disk of mass m = 8 kg rotates at the...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 - Determine the angular momentum HO of the disk of...

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The rate of change ( H ˙ G ) of the angular momentum ( H G ) of the disk.

Solution Summary: The author shows the components of vector with principal axis as in Figure (1).

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The rate of change (H˙G) of the angular momentum (HG) of the disk.

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Chapter 18 Solutions