Explanation Given information: The figure shown below shows the free body diagram of the system Figure-(1) Write the expression for the rate change of momentum. .........(I) Here, the rate of change of angular momentum is H ˙ G , angular momentum with respect to the frame of reference and angular velocity of the xyz frame is Ω . Write the expression for the momentum about the plane x y z . H G x y z = I x α i − I x y α j − I x z α k Write the expression for the angular momentum about the centre of mass. H G = I x ω i − I x y ω j − I x z ω k ......... (II) Here, momentum of inertia about x direction is I x , momentum of inertia about x y plane is I x y , momentum of inertia about x z plane is I x z and the angular velocity is x z . The shaft lies between x z plane hence the value of I x y is 0 . Substitute I x ω i − I x y ω j − I x z ω k for H G , ω i for Ω and I x α i − I x y α j − I x z α k for H G x y z in Equation (I). H ˙ G = I x α i − I x y α j − I x z α k + ω i × ( I x ω i + I x y ω j + I x z ω k ) = I x α i − I x z α k ......(III) The magnitude of A and B is Equal and opposite in direction because the center of mass lies on the line A B as it is symmetrical A and B forms a couple. A = − ( B ) .........(IV) Write the expression for the force at A . F A = M 0 i + 4 a i × ( B y j + B z k ) = M 0 i + 4 a B y k − 4 a B z k .........(V) The magnitude of A in the z direction is 0 . Substitute 0 for B z in Equation (V). F A = M 0 i + 4 a B y k − ( 0 ) = M 0 i + 4 a B y k .........(VI) Here, the magnitude of couple at B is F B the distance between a A to B is 4 a .the magnitude of B in z direction is B z . The magnitude of the couple and the rate of change of angular momentum are Equal. Equate Equation (VII) and Equation (IV). M 0 i + 4 a B y k = I x α i − I x z α k .........(VII) Compare the vector k in Equation (VII). I x z α k = 4 a B y k I x z α = 4 a B y B y = I x z α 4 a .........(VIII) Compare the vector i in Equation (VII). M 0 i = I x α i M 0 = I x α To determine (b) The dynamic reactions at A . The dynamic reactions at B .

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 18.74P

To determine

(a)

The couple M0.

To determine

(b)

The dynamic reactions at A.

The dynamic reactions at B.

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Chapter 18.2, Problem 18.73P

Chapter 18.2, Problem 18.75P

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