Explanation Draw the magnitude of force in the direction of AC with coordinates of O , A , and C as in Figure (1). Establish a position vector for points A , C , and O . The coordinates are: point A ( 0 , 6 , 2 ) ft , point O ( 0 , 0 , 0 ) ft , and point C ( 5 cos 60 ° , 0 , 5 sin 60 ° ) ft . Express the position vector r in the direction of AC . r A C = ( x C − x A ) i + ( y C − y A ) j + ( z C − z A ) k (I) Here, the coordinates of A and C are x A , x C , y A , y C , z A , and z C , respectively. Express the position vector r in the direction of AO . r A O = ( x O − x A ) i + ( y O − y A ) j + ( z O − z A ) k (II) Here, the coordinates of A and O are x A , x O , y A , y O , z A , and z O , respectively. Express the magnitude of the direction of AC . r A C = ( x C − x A ) 2 + ( y C − y A ) 2 + ( z C − z A ) 2 (III) Express the magnitude of the direction of AO . r A O = ( x O − x A ) 2 + ( y O − y A ) 2 + ( z O − z A ) 2 (IV) Determine the unit vector of AC . u A C = ( r A C r A C ) (V) Here, the position vector in the direction of AC is r A C , and the magnitude of the position vector in the direction of AC is r A C . Determine the force vector, F A C . F A C = F A C u A C (VI) Here, the force in the direction of AC is F A C . Determine the unit vector of AO . u A O = ( r A O r A O ) (VII) Here, the position vector in the direction of AO is r A O , and the magnitude of the position vector in the direction of AO is r A O . Use the vector dot product to determine the magnitude of the projected component of F A B along the strut AO . ( F A C ) A O = F A C ⋅ u A O (VIII) Determine the projected component of force F A C along the axis of strut AO . ( F A C ) A O = ( F A C ) A O u A O (IX) Conclusion: Substitute 0 ft for x A , 6 ft for y A , 2 ft for z A , ( 5 cos 60 ° ) ft for x C , 0 ft for y C , and ( 5 sin 60 ° ) ft for z C in Equation (I). r A C = ( ( 5 cos 60 ° ) ft − 0 ) i + ( 0 − 6 ) j + ( ( 5 sin 60 ° ) ft − 2 ) k = ( 2.5 i − 6 j + 2.33 k ) ft Substitute 0 ft for x O , 0 ft for y O , 0 ft for z O , 0 ft for x A , 6 ft for y A , and 2 ft for z A in Equation (II). r A O = ( 0 − 0 ) i + ( 0 − 6 ) j + ( 0 − 2 ) k = ( 0 i − 6 j − 2 k ) ft Substitute 0 ft for x A , 6 ft for y A , 2 ft for z A , ( 5 cos 60 ° ) ft for x C , 0 ft for y C , and ( 5 sin 60 ° ) ft for z C in Equation (III)

Engineering Mechanics: Statics and Study Pack (13th Edition)

13th Edition

ISBN: 9780133027990

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 2.9, Problem 125P

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The projected component of force FAC along the axis of strut AO.

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Chapter 2.9, Problem 124P

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Engineering Mechanics: Statics and Study Pack (13th Edition)

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The projected component of force F A C along the axis of strut AO .

The projected component of force FAC along the axis of strut AO.

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