Explanation Given: The magnitude of force in the direction of AB , F A B = 250 N . The magnitude of force in the direction of AC , F A C = 400 N . The angle between x axis and point B is 40 ° . Show the magnitude of forces F A B and F A C as in Figure (1). Show the coordinate direction angles with magnitude of the resultant force as in Figure (2). Establish a position vector from A to C . The coordinates of the tail A ( 0 , − 0.75 , 3 ) m are subtracted from the coordinates of the head C ( 2 , − 1 , 0 ) m . Express the position vector r in the direction of AC using Figure (1). 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 force F A C in Cartesian vector form and determine the position vector r which is directed from point A to point C . F A C = F A C u A C = F A C ( r A C r A C ) (II) Here, the unit vector in the direction of AC is u A C , 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 . Write the magnitude of the position vector in the direction of AC . r A C = ( x C − x A ) 2 + ( y C − y A ) 2 + ( z C − z A ) 2 (III) Establish a position vector from A to B . The coordinates of the tail A ( 0 , − 0.75 , 3 ) m are subtracted from the coordinates of the head B ( 2 cos 40 ° , 2 s i n 40 ° , 0 ) . Express the position vector r in the direction of AB using Figure (1). r A B = ( x B − x A ) i + ( y B − y A ) j + ( z B − z A ) k (IV) Here, the coordinates of A and B are x A , x B , y A , y B , z A , and z B respectively. Express the force F A B in Cartesian vector form and determine the position vector r which is directed from point A to point B . F A B = F A B u A B = F A B ( r A B r A B ) (V) Here, the unit vector in the direction of AB is u A B , the position vector in the direction of AB is r A B , and the magnitude of the position vector in the direction of AB is r A B . Write the magnitude of the position vector in the direction of AB . r A B = ( x B − x A ) 2 + ( y B − y A ) 2 + ( z B − z A ) 2 (VI) Express the resultant force in Cartesian vector form. F R = F A C + F A B (VII) Express the magnitude of F R . F R = ( F R ) x 2 + ( F R ) y 2 + ( F R ) z 2 (VIII) Write the coordinate direction angles. cos α = ( F R ) x F R (IX) cos β = ( F R ) y F R (X) cos γ = ( F R ) z F R (XI) Conclusion: Substitute 2 m for x C , 0 for x A , 1 m for y C , 0.75 m for y A , 0 for z C , and 3 m for z A in Equation (I). r A C = { ( 2 − 0 ) i + ( − 1 − ( − 0.75 ) ) j + ( 0 − 3 ) k } m = { 2 i − 0.25 j − 3 k } m Substitute 2 m for ( x C − x A ) , 0.25 m for ( y C − y A ) , and –3 m for ( z C − z A ) in Equation (III). r A C = ( 2 ) 2 + ( − 0.25 ) 2 + ( − 3 ) 2 = 4 + 0.0625 + 9 = 3.61 m Substitute 2 cos 40 ° for x B , 0 for x A , 2 sin 40 ° for y B , 0

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 2.8, Problem 92P

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

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Chapter 2.8, Problem 91P

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Chapter 2 Solutions

Engineering Mechanics: Statics & Dynamics (14th Edition)

Ch. 2.3 - In each case, construct the parallelogram law to...Ch. 2.3 - In each case, show how to resolve the force F into...Ch. 2.3 - Determine the magnitude of the resultant force...Ch. 2.3 - Two forces act on the hook. Determine the...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 - The force F = 450 lb acts on the frame. Resolve...Ch. 2.3 - If force F is to have a component along the u axis...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...

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

Solution Summary: The author shows the magnitude of the resultant force F_R and its coordinate direction angles as in Figure (1).

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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