Explanation Given information: The weight of the triangular plate ( W ) is 10 lb. The angular velocity ( ω ) is 8 rad/s. The length of AB ( h ) is 24 in. The width of the plate ( b ) is 12 inch. Calculation: Show the diagram of the system as in Figure (1). Write the expression for the angular momentum about point A as follows: H A = − I x y ω i + I y ω j − I y z ω k (1) Here, H A is the angular momentum of about point A , I x y is the angular momentum about A in the xy direction, I y is the angular momentum about A in the y direction and I y z is the angular momentum about A in the yz direction. Write the expression for the product of inertia of area for plate ABC with respect to xy axis ( I x y ) a r e a as follows: ( I x y ) a r e a = 1 24 b 2 h 2 Write the expression for the area of the triangle ABC ( A ) as follows: A = 1 2 b h Calculate the mass moment of inertia in the xy direction ( I x y ) using the formula: I x y = m A ( I x y ) a r e a Substitute 1 24 b 2 h 2 for ( I x y ) a r e a and 1 2 b h for A . I x y = m ( 1 24 b 2 h 2 ) 1 2 b h = 1 12 m b h Write the expression or the product of inertia of area for plate ABC with respect to y axis ( I y ) a r e a as follows: ( I y ) a r e a = 1 12 b 3 h Write the expression for the mass moment of inertia in the y direction I y as follows: I y = m A ( I y ) a r e a Substitute 1 12 b 3 h for ( I y ) a r e a and 1 2 b h for A . I y = 1 12 b 3 h m 1 2 b h = 1 6 b 2 m The z coordinate is negligible. Thus, I y z = 0 Find the angular momentum about point A : Substitute 0 for I y z , 1 6 b 2 m for I y , and 1 12 m b h for I x y in Equation (1). H A = − ( 1 12 m b h ) ω i + ( 1 6 b 2 m ) ω j − ( 0 ) ω k = − 1 12 m b h ω i + 1 6 m b 2 ω j Write the expression for moment about A ( M A ) as follows: M A = h j × ( B x i + B z k ) Here, B x is the force at B in the x direction and B z is the force at B in the z direction. The matrix multiplication for vector product is done. M A = h B z i − h B x k (2) Write the expression for the angular velocity of the reference frame ( Ω ) as follows: Ω = ω j Write the expression for the moment about A as follows: M A = ( H ˙ A ) x y z + Ω × H A Here, ( H ˙ A ) x y z is the angular momentum of the reference frame Axyz . The angular velocity is constant, so the value of ( H ˙ A ) x y z is zero. Substitute 0 for ( H ˙ A ) x y z , ω j for Ω , and − 1 12 m b h ω i + 1 6 m b 2 ω j for H A

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 18.66P

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Vector Mechanics for Engineers: Statics and Dynamics

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The dynamic reactions at A ( A ) and B ( B ).

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