(a) Explanation Given information: The radius of a homogeneous disk ( a ) is 9 in.. The length of the rod AG ( l ) is 18 in.. The angle that the rod AG lies ( θ 0 ) is 90 ° . The angular velocity of the shaft AB ϕ ˙ m is twice its initial value ϕ ˙ 0 . Calculation: Consider Z axis be vertical. The principal moment of inertia for principal axes xyz with origin at A are I ′ = I x = I y = m [ 1 4 a 2 + ( 2 a ) 2 ] and I = I z = 1 2 m a 2 . Draw the Free body diagram of homogeneous disk and its vectors position as in Figure (1). Draw the FBD of dimensions of the disk as in Figure (2). Write the expression for the unit vector K. K = − i sin θ + k cos θ Write the expression for the angular momentum about point A . H A = I x ω x i + I y ω y j + I z ω z k Substitute I ′ for I x , ϕ ˙ sin θ for ω x , I ′ for I y , − θ ˙ for ω y , and I for I z . H A = I ′ ( ϕ ˙ sin θ ) i − I ′ θ ˙ j + I ω z k Determine the value of kinetic energy T . T = 1 2 I x ω x 2 + 1 2 I y ω y 2 + 1 2 I z ω z 2 Substitute I ′ for I x , ϕ ˙ sin θ for ω x , I ′ for I y , − θ ˙ for ω y , and I for I z . T = 1 2 I x ω x 2 + 1 2 I y ω y 2 + 1 2 I z ω z 2 = 1 2 I ′ ( ϕ ˙ sin θ ) 2 + 1 2 I ′ θ ˙ 2 + 1 2 I ω z 2 = 1 2 I ′ ( ϕ ˙ 2 sin 2 θ + θ ˙ 2 ) + 1 2 I ω z 2 Determine the angular velocity along z axis ω z using the relation. ω z = ψ ˙ + ϕ ˙ cos θ For constraint of clevis ψ ˙ = 0 . Substitute 0 for ψ ˙ . ω z = ϕ ˙ cos θ Determine the conservation of angular momentum about fixed Z axis H A ⋅ K . Substitute I ′ ( ϕ ˙ sin θ ) i − I ′ θ ˙ j + I ω z k for H A and ( i sin θ + k cos θ ) for K. H A ⋅ K = ( I ′ ( ϕ ˙ sin θ ) i − I ′ θ ˙ j + I ω z k ) . ( i sin θ + k cos θ ) = I ′ ϕ ˙ sin 2 θ + I ω z cos θ (1) Substitute α for H A ⋅ K , m [ 1 4 a 2 + ( 2 a ) 2 ] for I ′ , 1 2 m a 2 for I , and ϕ ˙ cos θ for ω z in Equation (1). α = m [ 1 4 a 2 + ( 2 a ) 2 ] ϕ ˙ sin 2 θ + 1 2 m a 2 ( ϕ ˙ cos θ ) cos θ = 17 4 m a 2 ϕ ˙ sin 2 θ + 1 2 m a 2 ϕ ˙ cos 2 θ Determine the value of conservation of energy using the relation (b) To determine The initial angular velocity ϕ ˙ 0 of shaft AB .

12th Edition

ISBN: 9781260265453

Author: BEER

Publisher: MCG

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Question

Chapter 18.3, Problem 18.135P

(a)

To determine

The minimum value of θm.

(b)

To determine

The initial angular velocity ϕ˙0 of shaft AB.

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Chapter 18.3, Problem 18.134P

Chapter 18.3, Problem 18.136P

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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 minimum value of θ m .

Solution Summary: The author calculates the principal moment of inertia for principal axes xyz with origin at A.