Explanation Given information: The weight ( W ) of the disk is 6 lb. The angular velocity ( ω 1 ) of the disk is 16 rad/s. The angular velocity ( ω 2 ) of the shaft is 8 rad/s. The radius ( r ) of the disk is 8 inch. Calculation: Calculate the mass ( m ) of the disk using the formula: m = W g Here, g is the acceleration due to gravity. Substitute 6 lb for W and 32.2 ft/s 2 for g . m = 6 32.2 = 0.186335 lb ⋅ s 2 /ft Calculate the angular velocity of disk using the relation: ω = ω 2 j+ ω 1 i Substitute 8 rad/s for ω 2 and 16 rad/s for ω 1 . ω = ( 8 rad/s ) j + ( 16 rad/s ) i Consider the axes of disk x ′ , y ′ , and z ′ parallel to the axes x , y , and z respectively with origin at A . Calculate the mass moment of inertia ( I ¯ x ′ ) of the disk along x ′ -axis using the formula: I ¯ x ′ = 1 4 m r 2 Substitute 0.186335 lb ⋅ s 2 /ft for m and 8 in. for r . I ¯ x ′ = 1 4 ( 0.186335 ) ( 8 in . × 1 ft 12 in . ) 2 = 0.020704 lb ⋅ s 2 ⋅ ft Calculate the mass moment of inertia ( I ¯ z ′ ) of the disk along z ′ -axis using the formula: I ¯ z ′ = 1 4 m r 2 Substitute 0.186335 lb ⋅ s 2 /ft for m and 8 in. for r . I ¯ z ′ = 1 4 ( 0.186335 ) ( 8 in . × 1 ft 12 in . ) 2 = 0.020704 lb ⋅ s 2 ⋅ ft Calculate the mass moment of inertia ( I ¯ y ′ ) of the disk along y ′ -axis using the relation: I ¯ y ′ = I ¯ x ′ + I ¯ z ′ Substitute 0.020704 lb ⋅ s 2 ⋅ ft for I ¯ x ′ and 0.020704 lb ⋅ s 2 ⋅ ft for I ¯ z ′ . I ¯ y ′ = 0.020704 + 0.020704 = 0.041408 lb ⋅ s 2 ⋅ ft Calculate the angular momentum H A of the disk about its center A using the formula: H A = I ¯ x ′ ω x ′ i ′ + I ¯ y ′ ω y ′ j ′ + I ¯ z ′ ω z ′ k ′ Substitute 0

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 18.2, Problem 18.58P

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

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Chapter 18.2, Problem 18.57P

Chapter 18.2, Problem 18.59P

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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 - 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 rate of change ( H ˙ A ) of the angular momentum ( H A ) of the disk.

Solution Summary: The author calculates the angular velocity of the disk using the formula: m=Wg

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

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Chapter 18 Solutions