Explanation Given information: The mass of the uniform rod is m , the length of the shaft is 2 b and the angular velocity is ω Write the expression of moment of inertia of the uniform rod about y ′ axes. I y ′ = 1 3 m b 2 Here, the mass of the rod is m and the half length of the rod is b . Write the expression of moment of inertia of the uniform rod about z ′ axes. I z ′ = 1 3 m b 2 Write the expression of moment of inertia of the uniform rod about x ′ axes. I x ′ = 0 Write the expression of angular velocity of the shaft about x ′ axes. ω x ′ = − ω sin β Here, the angular velocity of the shaft is ω and the angle between x ′ and x axes is β . Write the expression of angular velocity of the shaft about y ′ axes. ω y ′ = ω cos β Write the expression of angular velocity of the shaft about z ′ axes. ω z ′ = 0 Write the expression of angular momentum of the system about point G . H G = I x ′ ω x ′ i ′ + I y ′ ω y ′ j ′ + I z ′ ω z ′ k ′ ......... (I) Here, the unit vector along the x ′ axis is i ′ , the unit vector along the y ′ axis is j ′ and the unit vector along the z ′ axis is k ′ . Substitute 0 for I x ′ , 1 3 m b 2 for I y ′ , 1 3 m b 2 for I z ′ , − ω sin β for ω x ′ , ω cos β for ω y ′ and 0 for ω z ′ in Equation (I). H G = ( 0 ) ( − ω sin β ) i ′ + ( 1 3 m b 2 ) ( ω cos β ) j ′ + ( 1 3 m b 2 ) ( 0 ) k ′ = 0 + ( 1 3 m b 2 ) ( ω cos β ) j ′ + 0 = ( 1 3 m b 2 ) ( ω cos β ) j ′ ........ (II) Write the expression of unit vector in y ′ axis. j ′ = i sin β + j cos β Substitute i sin β + j cos β for j ′ in Equation (II). H G = ( 1 3 m b 2 ) ( ω cos β ) ( i sin β + j cos β ) ......... (III) Write the expression of rate of change of angular momentum of the system. H ˙ G = ( H ˙ G ) G x y z + ( Ω × H G ) ........ (IV) Here, the angular acceleration is Ω . Write the expression of angular acceleration. Ω = ω = 0 Substitute 0 for Ω and ( 1 3 m b 2 ) ( ω cos β ) ( i sin β + j cos β ) for H G in Equation (IV). H ˙ G = ( H ˙ G ) G x y z + ( ( 0 ) ( ( 1 3 m b 2 ) ( ω cos β ) ( i sin β + j cos β ) ) ) = ( H ˙ G ) G x y z + 0 = ( H ˙ G ) G x y z To determine (b) The dynamic reactions at C . The dynamic reactions at D .

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

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Chapter 18.2, Problem 18.71P

To determine

(a)

The angular acceleration of the assembly.

To determine

(b)

The dynamic reactions at C.

The dynamic reactions at D.

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Chapter 18.2, Problem 18.70P

Chapter 18.2, Problem 18.72P

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(a) The angular acceleration of the assembly.

Solution Summary: The author explains the angular acceleration of the assembly.