Explanation Given information: The inclined force acting at point B ( F B ) is 50 N. The inclined force acting at point C ( F C ) is 50 N. The force acting at the point E ( F E ) is 12.5 N. The force acting at the point D ( F D ) is 12.5 N. The height between point C and F ( CF ) is 120 mm. The height between point F and A ( FA ) is 120 mm. The horizontal distance between point E and D ( ED ) is 192 mm. The height between point E and B ( EB ) is 144 mm. The distance between point C and D ( CD ) is 160 mm. Calculation: Replace the couple in the ABCD plane with two couple’s P and Q . Draw the couple ABCD by replacing with two couple’s P and Q as in Figure (1). Calculate the distance CG using the Pythagoras theorem: C G = C D 2 + C F 2 Substitute 160 mm for CD and 120 mm for CF . C G = ( 160 ) 2 + ( 120 ) 2 = 200 mm Calculate the force ( P ) acting at the point B using the formula: P = F B C D C G Substitute 50 N for F B , 160 mm for CD , and 200 mm for CG . P = 50 ( 160 200 ) = 40 N Calculate the force ( Q ) acting at the point C using the formula: Q = F C C F C G Substitute 50 N for F C , 120 mm for CF , and 200 mm for CG . Q = 50 ( 120 200 ) = 30 N Draw the axis by showing the moments and angle acting as in Figure (2). Calculate the angle ( θ ) using the relation: tan θ = E B E D Substitute 144 mm for EB and 192 mm for ED . tan θ = 144 192 θ = tan − 1 ( 0.75 ) θ = 36.87 ° Calculate the couple vector ( M 1 ) perpendicular to plane ABCD using the relation: M 1 = [ P × ( C F + F A ) ] − [ Q ( C D ) ] Substitute 40 N for P , 120 mm for CF , 120 mm for FA , 30 N for Q , and 160 mm for CD . M 1 = [ 40 × ( 120 + 120 ) ] − [ 30 ( 160 ) ] = 4800 N ⋅ mm × 1 1000 m mm = 4.80 N ⋅ m Write the couple vector ( M 1 ) perpendicular to plane ABCD in vector form: M 1 = ( M 1 cos θ ) j + ( M 1 sin θ ) k Substitute 4.80 N ⋅ m for M 1 and 36

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 3.3, Problem 3.78P

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The magnitude of the couple (M) and direction of its axis after replacing the two couples in single equivalent couple.

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Chapter 3.3, Problem 3.77P

Chapter 3.3, Problem 3.79P

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The magnitude of the couple ( M ) and direction of its axis after replacing the two couples in single equivalent couple.

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