Explanation Given: The magnitude of the vector force at A , B , and C point, F A , B , C = 60 lb . Establish a position vector from O to A , O to B , and O to C . The coordinates of points are: O ( 0 , 0 , 6 ) ft , A ( 4 sin 120 ° , 4 cos 120 ° , 0 ) ft , B ( − 4 sin 120 ° , 4 cos 120 ° , 0 ) , and C ( 0 , 4 , 0 ) ft. Express the position vector r in the direction of OA . r O A = ( x A − x O ) i + ( y A − y O ) j + ( z A − z O ) k (I) Here, the coordinates of A and O are x A , x O , y A , y O , z A , and z O , respectively. Express the force F A as a Cartesian vector form and the position vector r is directed from point O to point A . F A = F A u O A = F A ( r O A r O A ) (II) Here, the unit vector in the direction of OA is u E A and the magnitude of the position vector in the direction of OA is r O A . Write the magnitude of the position vector in the direction of OA . r O A = ( x A − x O ) 2 + ( y A − y O ) 2 + ( z A − z O ) 2 (III) Express the position vector r in the direction of OB . r O B = ( x B − x O ) i + ( y B − y O ) j + ( z B − z O ) k (IV) Here, the coordinates of O and B are x O , x B , y O , y B , z O , and z B , respectively. Express the force F B as a Cartesian vector form and the position vector r is directed from point O to point B . F B = F B u O B = F B ( r O B r O B ) (V) Here, the unit vector in the direction of OB is u O B and the magnitude of the position vector in the direction of OB is r O B . Write the magnitude of the position vector in the direction of OB . r O B = ( x B − x O ) 2 + ( y B − y O ) 2 + ( z B − z O ) 2 (VI) Express the position vector r in the direction of OC . r O C = ( x C − x O ) i + ( y C − y O ) j + ( z C − z O ) k (VII) Here, the coordinates of O and C are x O , x C , y O , y C , z O , and z C , respectively. Express the force F C as a Cartesian vector form and the position vector r is directed from point O to point C . F C = F C u O C = F C ( r O C r O C ) (VIII) Here, the unit vector in the direction of OC is u O C and the magnitude of the position vector in the direction of OC is r O C . Write the magnitude of the position vector in the direction of OC . r O C = ( x C − x O ) 2 + ( y C − y O ) 2 + ( z C − z O ) 2 (IX) Determine the resultant force. F R = F A + F B + F C (X) Determine the magnitude of the resultant force. F R = ( F R x ) 2 + ( F R y ) 2 + ( F R z ) 2 (XI) Here, the magnitude of resultant force in the x , y , and z coordinate axes are F R x , F R y , and F R z , respectively. Determine the coordinate direction angles in the x , y , and z direction. α = cos − 1 ( F R x F R ) (XII) β = cos − 1 ( F R y F R ) (XIII) γ = cos − 1 ( F R z F R ) (XIV) Conclusion: Substitute 0 for x O , ( 4 sin 120 ° ) ft for x A , 0 for y O , ( 4 cos 120 ° ) ft for y A , 6 ft for z O , and 0 for z A in Equation (I). r O A = ( ( 4 sin 120 ° ) ft − 0 ) i + ( ( 4 cos 120 ° ) ft − 0 ) j + ( 0 − 6 ) k = ( 3.464 i − 2 j − 6 k ) ft Substitute 0 for x O , ( 4 sin 120 ° ) ft for x A , 0 for y O , ( 4 cos 120 ° ) ft for y A , 6 ft for z O , and 0 for z A in Equation (III). r O A = ( ( 4 sin 120 ° ) ft − 0 ) 2 + ( ( 4 cos 120 ° ) ft − 0 ) 2 + ( 0 − 6 ) 2 = ( 3.434 ) 2 + ( − 2 2 ) + ( − 6 ) 2 = 7.211 ft Substitute 7.211 ft for r O A , ( 3.464 i − 2 j − 6 k ) ft for r O A , and 60 lb for F A in Equation (II). F A = 60 lb ( ( 3.464 i − 2 j − 6 k ) ft 7

Engineering Mechanics: Statics & Dynamics, Student Value Edition Plus Modified Mastering Engineering Revision with Pearson eText -- Access Card Package

14th Edition

ISBN: 9780135841532

Author: HIBBELER, Russell , HIBBLER

Publisher: PEARSON

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Chapter 2.9, Problem 108P

To determine

The vector force at point A, B, and C in Cartesian form, (FA,FB, and FC).

The magnitude of the resultant force, FR

The coordinate direction angles, (α,β, and γ).

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Chapter 2.9, Problem 107P

Chapter 2.9, Problem 109P

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Chapter 2 Solutions

Engineering Mechanics: Statics & Dynamics, Student Value Edition Plus Modified Mastering Engineering Revision with Pearson eText -- Access Card Package

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The vector force at point A , B , and C in Cartesian form, ( F A , F B , and F C ) . The magnitude of the resultant force, F R The coordinate direction angles, ( α , β , and γ ) .

Solution Summary: The author explains the vector force at points A, B, and C in Cartesian form. The magnitude of the resultant force is F_R.

Videos

The vector force at point A, B, and C in Cartesian form, (FA,FB, and FC).

The magnitude of the resultant force, FR

The coordinate direction angles, (α,β, and γ).

Want to see the full answer?

Chapter 2 Solutions

The vector force at point A , B , and C in Cartesian form, ( F A , F B , and F C ) . The magnitude of the resultant force, F R The coordinate direction angles, ( α , β , and γ ) .

Solution Summary: The author explains the vector force at points A, B, and C in Cartesian form. The magnitude of the resultant force is F_R.

Videos

The vector force at point A, B, and C in Cartesian form, (FA,FB, and FC). The magnitude of the resultant force, FR The coordinate direction angles, (α,β, and γ).

Want to see the full answer?

Chapter 2 Solutions

The vector force at point A, B, and C in Cartesian form, (FA,FB, and FC).

The magnitude of the resultant force, FR

The coordinate direction angles, (α,β, and γ).