Explanation Given: The magnitude of force, F 1 = 80 lb . The magnitude of resultant force, F R = 150 lb . The angle between F R and the z axis is γ The angle between F R and the x axis is 120 ° The angle between F 2 and the y axis is 130 ° . Show the magnitude of component forces F 1 and F 2 into x , y , and z direction as in Figure (1). Show the coordinate direction angles of the magnitude of F 2 as in Figure (2). Write the sum of coordinate direction angles equal to 1. cos 2 α + cos 2 β + cos 2 γ = 1 (I) Express the resultant force F R as a Cartesian vector using Figure (1). F R = F R ( u F ) = F R ( cos α i + cos β j + cos γ k ) (II) Here, the unit vectors are i , j , and k , the magnitude of the unit vector in the direction of F R is u F . Express the resultant force as a Cartesian vector. F R = F 1 + F 2 (III) Express the magnitude of F 2 . F 2 = ( F 2 ) x 2 + ( F 2 ) y 2 + ( F 2 ) z 2 (IV) Write the coordinate direction angles. cos α 2 = ( F 2 ) x F 2 (V) cos β 2 = ( F 2 ) y F 2 (VI) cos γ 2 = ( F 2 ) z F 2 (VII) Conclusion: Calculate the coordinate direction angle β from Figure (1). β = 180 ° − 130 ° = 50 ° Substitute 120 ° for α and 50 ° for β in Equation (I). cos 2 120 ° + cos 2 50 ° + cos 2 γ = 1 ( − 0.5 ) 2 + ( 0.642 ) 2 + cos 2 γ = 1 cos 2 γ = 0.3378 cos γ = 0.3378 γ = cos − 1 0.581 γ = 54.47 ° Refer to Figure (1), the range of γ < 90 ° . Thus, the value of γ is 54.47 ° . Substitute 150 lb for F R , 120 ° for α , 50 ° for β , and 54.47 ° for γ in Equation (II). F R = 150 lb ( cos 120 ° i + cos 50 ° j + cos 54.43 ° k ) = 150 lb ( ( − 0.5 ) i + ( 0.642 ) j + ( 0

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 2.6, Problem 72P

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The magnitude of F2 and its coordinate direction angles (α2,β2,γ2).

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Chapter 2.6, Problem 71P

Chapter 2.6, Problem 73P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The magnitude of F 2 and its coordinate direction angles ( α 2 , β 2 , γ 2 ) .

Solution Summary: The author illustrates the magnitude of component forces F_1 and f2 into x, y, and z direction.

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The magnitude of F2 and its coordinate direction angles (α2,β2,γ2).

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