(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 .

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ISBN: 9781260804638

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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The minimum value of θ m .

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