Vector Mechanics for Engineers: Statics, 11th Edition
Vector Mechanics for Engineers: Statics, 11th Edition
11th Edition
ISBN: 9780077687304
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek
Publisher: McGraw-Hill Education
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Chapter 3.2, Problem 3.55P
To determine

Find the moment about AB (MAB).

Expert Solution & Answer
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Answer to Problem 3.55P

The moment about AB (MAB) is 207lbft_.

Explanation of Solution

Given information:

The length of the vertical rod CD (lCD) is 23 in.

The length of the rod AB (lAB) is 50 in.

The force (P) is 235 lb.

The distance between the points O to B (x) is 32 in.

The distance between the points O to A (y) is 24 in.

The vertical height of OA (hOA) is 30 in.

The vertical height of OH (hOH) is 17 in.

The horizontal distance between point O and midpoint C (dOC) is 16 in.

The horizontal distance between midpoint C and point G over x axis (dCG) is 21 in.

The horizontal distance between point B and midpoint C (dBC) is 12 in.

The horizontal distance between midpoint C and point G over z axis (h) is 18 in.

Calculation:

Calculate the position vector of AB using the relation:

AB=xihOAjyk

Substitute 32 in. for x, 30 in. for hOA and 24 in. for y.

AB=32i30j24kAB=(32)2+(30)2+(24)2=50in.

Calculate the unit vector of AB using the formula:

λAB=ABAB

Substitute 32i30j24k for AB and 50 in. for AB.

λAB=32i30j24k50=0.64i0.6j0.48k

Apply the force P at the point G.

Calculate the position vector of from point G to point B (rG/B) using the relation:

rG/B=[x(dOC+dCG)]i+(dBC+h)k

Substitute 32 in. for x, 16 in. for dOC, 21 in. for dCG, 12 in. for dBC and 18 in. for h.

rG/B=[32(16+21)]i+(12+18)k=5i+30k

Calculate the position vector of DG using the relation:

DG=dCGi(hOA2+lCD)j+hk

Substitute 21 in. for dCG, 30 in. for hOA, 23 in. for lCD and 18 in. for h.

DG=21i(302+23)j+18k=21i38j+18kDG=(21)2+(38)2+(18)2=47in.

Calculate the force (P) using the formula:

P=PDGDG

Substitute 235 lb for P, 21i38j+18k for DG and 47 in. for DG.

P=235×21i38j+18k47=235×(0.447i0.808j+0.383k)=105i189.9j+90k105i190j+90k

Calculate the moment about AB (MAB) using the relation:

MAB=λAB(rG/B×P)

Substitute 0.64i0.6j0.48k for λAB, 5i+30k for rG/B and 105i190j+90k for P.

MAB=|0.640.60.48503010519090|=[0.64(0(5700))][(0.6)(4503150)][0.48(9500)]=36481620+456

=2484lbin.×112ftin.=207lbft

Therefore, the moment about AB (MAB) is 207lbft_.

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Chapter 3 Solutions

Vector Mechanics for Engineers: Statics, 11th Edition

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