(a) Explanation Given information: The side ( a ) of the square is 300 mm. The constant angular velocity ( ω ) of the clevis is 12 rad/s. Calculation: Let point O be the midpoint of axle AB and consider principal axes x ′ , y ′ , and z ′ with origin at point O . Sketch the axes of the system as shown in Figure (1). Refer Figure (1), Write the equation of the mass moment of inertia about x ′ -axis. I x ′ = 1 12 m a 2 Write the equation of the mass moment of inertia about z ′ -axis. I z ′ = 1 3 m a 2 Find the principal moment of inertia about y ′ -axis using the equation: I y ′ = I x ′ + I z ′ Substitute 1 12 m a 2 for I x ′ and 1 3 m a 2 for I z ′ . I y ′ = 1 12 m a 2 + 1 3 m a 2 = 5 12 m a 2 Write the vector form of angular velocity ( ω ) of the system. ω = − ω sin β i ′ + ω cos β j ′ Write the equation of angular momentum ( H O ) about point O . H O = I x ′ ω x ′ i + I y ′ ω y ′ j + I z ω z k (1) Substitute − ω sin β i ′ for ω x ′ i , ω cos β j ′ for ω y ′ j , and 0 for ω z . H O = I x ′ ( − ω sin β i ′ ) + I y ′ ( ω cos β j ′ ) + I z ( 0 ) k = − I x ′ ω sin β i ′ + I y ′ ω cos β j ′ Write the angular velocity of reference frame ( Ω ) . Ω = ω Substitute − ω sin β i ′ + ω cos β j ′ for ω . Ω = − ω sin β i ′ + ω cos β j ′ The rate of change of angular momentum ( H ˙ O ) O x y z about the reference frame is 0. Because the reference frame rotating about fixed point O . Write the equation of rate of change of angular momentum ( H ˙ O ) of the system. H ˙ O = ( H ˙ O ) O x y z + Ω × H O Substitute 0 for ( H ˙ O ) O x y z , − ω sin β i ′ + ω cos β j ′ for Ω , and − I x ′ ω sin β i ′ + I y ′ ω cos β j ′ for H O (b) To determine The largest value of angular velocity ( ω ) for which the plate remains vertical.

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 18.87P

(a)

To determine

The value of angular velocity (ω) for which the plate forms a constant angle β=60° with the horizontal x-axis.

(b)

To determine

The largest value of angular velocity (ω) for which the plate remains vertical.

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Chapter 18.2, Problem 18.86P

Chapter 18.2, Problem 18.88P

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Vector Mechanics for Engineers: Statics and Dynamics

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The value of angular velocity ( ω ) for which the plate forms a constant angle β = 60 ° with the horizontal x -axis.

Solution Summary: The author calculates the angular velocity and the mass moment of inertia using the following equations.