Explanation Given: The magnitude of force in the direction of Dc , F D C = 600 N . The magnitude of force in the direction of DO , F D O = 400 N . The magnitude of force in the direction of DB , F D B = 800 N . Establish a position vector from D to O . The coordinates of points D and O are D ( 0 , 0 , 0 ) m and O ( 20 , 15 , − 24 ) m . Express the position vector r in the direction of DO . r D O = ( x O − x D ) i + ( y O − y D ) j + ( z O − z D ) k (I) Here, the coordinates of O and D are x O , x D , y O , y D , z O , and z D , respectively. Express the force F D O as a Cartesian vector form and position vector r directed from point D to point O . F D O = F D O u D O = F D O ( r D O r D O ) (II) Here, the unit vector in the direction of DO is u D O ; the position vector in the direction of DO is r D O ; and the magnitude of the position vector in the direction of DO is r D O . Write the magnitude of the position vector in the direction of DO . r D O = ( x O − x D ) 2 + ( y O − y D ) 2 + ( z O − z D ) 2 (III) Establish a position vector from D to B . The coordinates of points D and B are D ( 0 , 0 , 0 ) m and B ( − 6 , 4 , − 24 ) m . Express the position vector r in the direction of DB . r D B = ( x B − x D ) i + ( y B − y D ) j + ( z B − z D ) k (IV) Here, the coordinates of D and B are x D , x B , y D , y B , z D , and z B , respectively. Express the force F D B as a Cartesian vector form and position vector r directed from point D to point B . F D B = F D B u D B = F D B ( r D B r D B ) (V) Here, the unit vector in the direction of DB is u D B ; the position vector in the direction of DB is r D B ; and the magnitude of the position vector in the direction of DB is r D B . Write the magnitude of the position vector in the direction of DB . r D B = ( x B − x D ) 2 + ( y B − y D ) 2 + ( z B − z D ) 2 (VI) Establish a position vector from D to C . The coordinates of point D and C are D ( 0 , 0 , 0 ) m and C ( 16 , − 18 , − 24 ) m . Express the position vector r in the direction of DC using Figure (1). r D C = ( x C − x D ) i + ( y C − y D ) j + ( z C − z D ) k (VII) Here, the coordinates of D and C are x D , x C , y D , y C , z D , and z C , respectively. Express the force F D C as a Cartesian vector form and position vector r directed from point D to point C . F D C = F D C u D C = F D C ( r D C r D C ) (VIII) Here, the unit vector in the direction of DC is u D C ; the position vector in the direction of DC is r D C ; and the magnitude of the position vector in the direction of DC is r D C . Write the magnitude of the position vector in the direction of DC . r D C = ( x C − x D ) 2 + ( y C − y D ) 2 + ( z C − z D ) 2 (IX) Express the resultant force as a Cartesian vector. F R = F D O + F D B + F D C (X) Express the magnitude of F R . F R = ( F R ) x 2 + ( F R ) y 2 + ( F R ) z 2 (XI) Write the cosines of the coordinate direction angles. cos α = ( F R ) x F R (XII) cos β = ( F R ) y F R (XIII) cos γ = ( F R ) z F R (XIV) Conclusion: Substitute 0 m for x D , 20 m for x O , 0 m for y D , 15 m for y O , 0 for z D , and 24 m for z O in Equation (I). r D O = { ( 20 − 0 ) i + ( 15 − 0 ) j + ( − 24 − 0 ) k } m = { 20 i + 15 j − 24 k } m Substitute 0 m for x D , 20 m for x O , 0 m for y D , 15 m for y O , 0 for z D , and 24 m for z O in Equation (III). r D O = ( 20 − 0 ) 2 + ( 15 − 0 ) 2 + ( − 24 − 0 ) 2 = 400 + 225 + 576 = 34.66 m Substitute –6 m for x B , 0 m for x D , –4 m for y B , 0 m for y D , –24 m for z B , and 0 m for y D in Equation (IV)

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 2.8, Problem 94P

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The magnitude of resultant force FR and its coordinate direction angles (α,β,γ).

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Chapter 2 Solutions

Engineering Mechanics: Statics

Ch. 2.3 - Determine the magnitude of the resultant force...Ch. 2.3 - Determine the magnitude of the resultant force....Ch. 2.3 - Determine the magnitude of the resultant force and...Ch. 2.3 - Resolve the 30-lb force into components along the...Ch. 2.3 - Resolve this force into components acting along...Ch. 2.3 - along the v axis. Prob. F2-6 Ch. 2.3 - Determine the magnitude of the resultant force FR...Ch. 2.3 - If = 60 and F = 450 N, determine the magnitude of...Ch. 2.3 - If the magnitude of the resultant force is to be...Ch. 2.3 - Determine the magnitude of the resultant force FR...

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The magnitude of resultant force F R and its coordinate direction angles ( α , β , γ ) .

The magnitude of resultant force FR and its coordinate direction angles (α,β,γ).

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