Explanation Given information : The radius of the curved rod is 5 ft. The moment at point C ( M C ) is 80 lb ⋅ ft . Show the free body diagram of the forces acting on the rope as in Figure (1). Determine the position vector for point AB . r A B = r B − r A = ( x B i + y B j + z B k ) − ( x A i + y A j + z A k ) = ( x B − x A ) i + ( y B − y A ) j + ( z B − z A ) k (I) Here, the position vector for point B is r B , the position vector for point A is r A , the distance of point B from the origin along x direction is x B , the distance of point B from the origin along y direction is y B , the distance of point B from the origin along z direction is z B , the distance of point A from the origin along x direction is x A , the distance of point A from the origin along y direction is y A , the distance of point A from the origin along z direction is z A , the vector component along x direction is i, the vector component along y direction is j, and the vector component along z direction is k. Determine the modulus of position vector for point AB . | r A B | = ( x B − x A ) 2 + ( y B − y A ) 2 + ( z B − z A ) 2 (II) Determine the position vector for point CA . r C A = r A − r C = ( x A i + y A j + z A k ) − ( x C i + y C j + z C k ) = ( x A − x C ) i + ( y A − y C ) j + ( z A − z C ) k (III) Here, the position vector for point C is r C , the distance of point C from the origin along x direction is x C , the distance of point C from the origin along y direction is y C , and the distance of point C from the origin along z direction is z C . Force in rope AB : F A B = F [ r A B | r A B | ] (IV) Write the equation for moment of force F A B about point C . M C = r C A × F A B (V) Conclusion : Substitute 6 ft for x B , 0 ft for x A , ( 7 − 5 sin 60 ° ) ft for y B , 0 ft for y A , 0 ft for z B , and ( 5 cos 60 ° ) ft for z A in Equation (I). r A B = ( 6 − 0 ) i + ( ( 7 − 5 sin 60 ° ) − 0 ) j + ( 0 − 5 cos 60 ° ) k = ( 6 i + 2.669 j − 2.5 k ) ft Substitute 6 ft for x B , 0 ft for x A , ( 7 − 5 sin 60 ° ) ft for y B , 0 ft for y A , 0 ft for z B , and ( 5 cos 60 ° ) ft for z A in Equation (II)

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 4.4, Problem 41P

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The smallest force F that must be applied along the rope.

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Chapter 4.4, Problem 40P

Chapter 4.4, Problem 42P

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