Explanation Given information : The moment M 4 is 150 lb ⋅ ft . The resultant of the couple moment ( M C ) R is zero. Show the free body diagram of the assembly as in Figure (1). Use Figure (1). Express moment M 1 in vector form as follows: M 1 = ( M 1 j ) lb ⋅ ft (I) Here, the vector component along the y direction is j. Express moment M 2 in vector form as follows: M 2 = ( − M 2 i ) lb ⋅ ft (II) Here, the vector component along the x direction is i. Express position vector r 3 for moment M 3 as follows: r 3 = x i + y j + z k (III) Here, the vector component along the z direction is k. Determine the magnitude of the position vector of M 3 as follows: | r 3 | = ( x i ) 2 + ( y j ) 2 + ( z k ) 2 (IV) Express moment M 3 in vector form as follows: M 3 = M 3 ( r 3 | r 3 | ) (V) Express moment M 4 in vector form along the x , y , and z directions. M 4 = ( M 4 cos θ sin θ 1 ) i − ( M 4 cos θ cos θ 1 ) j − ( M 4 sin θ ) k (VI) Here, the resolving angles of M 4 in the x, y, and z directions are θ & θ 1 . Determine the resultant moment using the formula; ( M C ) R = ∑ M = M 1 + M 2 + M 3 + M 4 (VII) Show the calculation of moments as follows: Conclusion : Substitute 2.0 ft for x , –2.0 ft for y , and 1.0 ft for z in Equation (III). r 3 = ( 2.0 i − 2.0 j + 1 k ) ft Substitute 2.0 ft for x i, –2.0 ft for y j, and 1.0 ft for z k in Equation (IV). | r 3 | = ( 2.0 ) 2 + ( − 2.0 ) 2 + ( 1.0 ) 2 = 3.0 ft Substitute ( 2.0 i − 2.0 j + 1 k ) ft for r 3 and 3.0 ft for | r 3 | in Equation (V). M 3 = M 3 ( 2.0 i − 2.0 j + 1 k 3.0 ) = 2.0 M 3 i − 2.0 M 3 j + 1 M 3 k 3.0 = ( 2.0 M 3 i 3 − 2.0 M 3 j 3 + M 3 k 3 ) lb ⋅ f t Substitute 150 lb ⋅ ft for M 4 , 45 ° for θ , and 45 ° for θ 1 in Equation (VI). M 4 = ( 150 cos 45 ° sin 45 ° ) i − ( 150 cos 45 ° cos 45 ° ) j − ( 150 sin 45 ° ) k = ( 75 i − 75 j − 106

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 4.6, Problem 84P

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The magnitude of couple moments M1 , M2 , and M3 .

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Chapter 4.6, Problem 83P

Chapter 4.6, Problem 85P

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The magnitude of couple moments M 1 , M 2 , and M 3 .

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The magnitude of couple moments M1 , M2 , and M3 .

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