Vector Mechanics for Engineers: Statics
Vector Mechanics for Engineers: Statics
12th Edition
ISBN: 9781259977268
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek
Publisher: McGraw-Hill Education
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Chapter 2.4, Problem 2.96P

For the plate of Prob. 2.89; determine the tensions in cables AB and AD knowing that the tension in cable AC is 54 N and that the resultant of the forces exerted by the three cables at A must be vertical.

Expert Solution & Answer
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To determine

The tensions in cables AB and AD.

Answer to Problem 2.96P

The tensions in cables AB and AD are 490N_ and 515N_ respectively.

Explanation of Solution

Write the equation to find the magnitude of the vector AB.

|AB|=ABx2+ABy2+ABz2 (I)

Write the equation to find the unit vector along AB.

λ=AB|AB| (II)

Write the equation of TAB in terms of its rectangular components.

TAB=TABλ (III)

Here, TAB is the tension in the cable AB, and λ is the unit vector along AB.

Write the equation to find the magnitude of the vector AC.

|AC|=ACx2+ACy2+ACz2 (IV)

Write the equation to find the unit vector along AC.

λ=AC|AC| (V)

Write the equation of TAC in terms of its rectangular components.

TAC=TACλ (VI)

Here, TAC is the tension in the cable AC, and λ is the unit vector along AC.

Write the equation to find the magnitude of the vector AD.

|AD|=ADx2+ADy2+ADz2 (VII)

Write the equation to find the unit vector along AD.

λ=AD|AD| (VIII)

Write the equation of TAD in terms of its rectangular components.

TAD=TADλ (IX)

Here, TAD is the tension in the cable AD, and λ is the unit vector along AD.

Write the equation to find the resultant of the forces exerted by the three cables at A.

R=TAB+TAC+TAD (X)

Conclusion:

Refer Fig.P2.89 and calculate the vector coordinates of the vector AB.

AB=(320mm)i(480mm)j+(360mm)k

Substitute 320mm for ABx, 480mm for ABy, and 360mm for ABz in the equation (I).

AB=(320mm)2+(480mm)2+(360mm)2=680mm

Substitute (320mm)i(480mm)j+(360mm)k for AB, and 680mm for AB in equation (II).

λ=(320mm)i(480mm)j+(360mm)k680mm=320680i480680j+360680k

Substitute 408N for TAB, and 320680i480680j+360680k for λ in equation (III).

TAB=TAB(320680i480680j+360680k)

Refer Fig.P2.89 and calculate the vector coordinates of the vector AC.

AC=+(450mm)i(480mm)j+(360mm)k

Substitute 450mm for ACx, 480mm for ACy, and 360mm for ACz in the equation (IV).

AC=(450mm)2+(480mm)2+(360mm)2=750mm

Substitute +(450mm)i(480mm)j+(360mm)k for AC, and 750mm for AC in equation (V).

λ=+(450mm)i(480mm)j+(360mm)k750mm=450750i480750j+360750k

Substitute 54N for TAC and 450750i480750j+360750k for λ in the above equation (VI).

TAC=(54N)(450750i480750j+360750k)=(32.4N)i(34.560)j+(25.920)k

Refer Fig.P2.89 and calculate the vector coordinates of the vector AD.

AD=(250mm)i(480mm)j(360mm)k

Substitute 250mm for ADx, 480mm for ADy, and 360mm for ADz in the equation (VII).

AD=(250mm)2+(480mm)2+(360mm)2=650mm

Substitute (250mm)i(480mm)j(360mm)k for AD, and 650mm for AD in equation (VIII).

λ=(250mm)i(480mm)j(360mm)k650mm=250650i480650j360650k

Substitute 250650i480650j360650k for λ in the above equation (IX).

TAD=TAD(250650i480650j360650k)

Substitute TAB(320680i480680j+360680k) for TAB, (32.4N)i(34.560)j+(25.920)k for TAC and TAD(250650i480650j360650k) for TAD in equation (X).

R=TAB(320680i480680j+360680k)+(32.4)i(34.56)j+(25.920)k+TAD(250650i480650j360650k)=(320680TAB+32.4+250650TAD)i+(480680TAB(34.56)480650TAD)j+(360680TAB+(25.920)360650TAD)k

Substitute 0 for the coefficients of i and k from the above equation.

320680TAB+32.4+250650TAD=0 (XI)

360680TAB+(25.920)360650TAD=0 (XII)

Multiply the equation (XI) by 3.6.

14485TAB+116.64+1813TAD=0 (XIII)

Multiply the equation (XI1) by 2.5.

4534TAB+64.81813TAD=0 (IX)

Add the equations (XIII) and (IX).

14485TAB+116.64+1813TAD+4534TAB+64.81813TAD=0181.4463170TAB=0TAB=(181.44)(17063)=489.60N (X)

Substitute 489.60N for TAB in equation (X) and solve for TAD.

360680(489.60N)+(25.920)360650TAD=0TAD=(285.12)650360=514.8N

Therefore, the tensions in cables AB and AD are 490N_ and 515N_ respectively.

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Chapter 2 Solutions

Vector Mechanics for Engineers: Statics

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