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 3.4, Problem 3.139P

(a)

To determine

The resultant force.

(a)

Expert Solution
Check Mark

Answer to Problem 3.139P

The resultant force is (2300N)i^+(500N)j^+(1850N)k^_.

Explanation of Solution

Write the equation of the distance between AC.

dAC=r12+r22+r32 (I)

Here, the distance between AC is dAC, the coordinates are r1,r2,r3

Write the equation of the distance between BD.

dBD=r42+r52+r62 (II)

Here, the distance between BD is dBD, the coordinates are r4,r5,r6

Write the equation of the momentum about AC.

TAC=(dACdAC)F1 (III)

Here, the momentum about AC is TAC, the force is F1.

Write the equation of the momentum about BD.

TBD=(dBDdBD)F2 (IV)

Here, the momentum about BD is TBD, the force is F2.

Write the equation of resultant force.

R=TAC+TBD (V)

Here, the resultant force is R.

Conclusion:

Substitute, 6m for r1, 2m for r2, 9m for r3 in equation (I)

dAC=(6m)2+(2m)2+(9m)2=11m

Substitute, 14m for r4, 2m for r5, 5m for r5 in equation (II)

dBD=(14m)2+(2m)2+(5m)2=15m

Substitute, (6m)i^+(2m)j^+(9m)k^ for dAC, (11m) for dAC, 1650N for F1 in equation (III).

TAC=([(6m)i^+(2m)j^+(9m)k^](11m))(1650N)=[(900N)i^+(300N)j^+(1350N)k^]

Substitute, (14m)i^+(2m)j^+(5m)k^ for dBD, (15m) for dBD, 1500N for F2 in equation (IV).

TBD=([(14m)i^+(2m)j^+(5m)k^](15m))(1500N)=[(1400N)i^+(200N)j^+(500N)k^]

Substitute, [(1400N)i^+(200N)j^+(500N)k^] for TBD, [(900N)i^+(300N)j^+(1350N)k^] for TAC in equation (V).

R=[(1400N)i^+(200N)j^+(500N)k^]+[(900N)i^+(300N)j^+(1350N)k^]=[(2300N)i^+(500N)j^+(1850N)k^]

The magnitude of the resultant force,

|R|=(2300N)2+(500N)2+(1850N)2=2993.7N

Thus, the resultant force is (2300N)i^+(500N)j^+(1850N)k^_.

(b)

To determine

The pitch of the wrench.

(b)

Expert Solution
Check Mark

Answer to Problem 3.139P

The pitch of the wrench is 0.201m_.

Explanation of Solution

Write the equation of pitch of the wrench.

p=M1R (VI)

Here, the pitch of the wrench is p, the momentum along wrench is M1

Since, the R and MoR are not in same direction, so the form of the wrench is,

M1=λaxisMoR (VII)

Here, the constant is λaxis.

Write the expression for the constant is,

λaxis=RR

Write the equation of momentum.

MoR=[(rA×TAC)+(rB×TBD)] (VIII)

Here, the momentum is MoR, the distance is rA,rB.

Rewrite the expression for the momentum of the wrench is,

M1=(RR)[(rA×TAC)+(rB×TBD)] (IX)

Conclusion:

Substitute, [(2300N)i^+(500N)j^+(1850N)k^] for R, 2993.7N for R, (12m)k^ for rA, [(900N)i^+(300N)j^+(1350N)k^] for TAC, (9m)i^ for rB, [(1400N)i^+(200N)j^+(500N)k^] for TBD in equation (IX).

M1=([(2300N)i^+(500N)j^+(1850N)k^](2993.7N))[((12m)k^×[(900N)i^+(300N)j^+(1350N)k^])+((9m)i^×[(1400N)i^+(200N)j^+(500N)k^])]=[(461.93N-m)i^+(100.42N-m)j^+(371.56N-m)k^]=601.26N-m

Substitute, 601.26N-m for M1, (2993.7N) for R in equation (VI)

p=(601.26N-m)(2993.7N)=0.201m

Thus, the pitch of the wrench is 0.201m_.

(c)

To determine

The point at which the axis of wrench intersects the yz-plane.

(c)

Expert Solution
Check Mark

Answer to Problem 3.139P

The axis of wrench intersects the yz-plane at y=0.944m,z=2.78m_.

Explanation of Solution

The diagram for the force-couple system is given below:

<x-custom-btb-me data-me-id='1725' class='microExplainerHighlight'>Vector</x-custom-btb-me> Mechanics for Engineers: Statics, Chapter 3.4, Problem 3.139P

Refer fig 1,

Write the equation for the force couple system for the wrench.

MoR=M1+M2 (X)

Here, the momentum is M2.

Write the expression for the momentum at which the wrench intersects the xz-plane.

M2=rQ/O×R (XI)

Here, the position vector is rQ/O.

Write the expression for the position vector is,

rQ/O=(yj^+zk^)

Here, the coordinates are y,z.

Conclusion:

Substitute, (12m)k^ for rA, [(900N)i^+(300N)j^+(1350N)k^] for TAC, (9m)i^ for rB, [(1400N)i^+(200N)j^+(500N)k^] for TBD in equation (VIII).

MoR=[((12m)k^×[(900N)i^+(300N)j^+(1350N)k^])+((9m)i^×[(1400N)i^+(200N)j^+(500N)k^])]=[(3600N-m)i^+(6300N-m)j^+(1800N-m)k^]

Substitute, [(3600N-m)i^+(6300N-m)j^+(1800N-m)k^] for MoR, [(461.93N-m)i^+(100.42N-m)j^+(371.56N-m)k^] for M1 in equation (X).

[(3600N-m)i^+(6300N-m)j^+(1800N-m)k^]=[(461.93N-m)i^+(100.42N-m)j^+(371.56N-m)k^]+M2M2=[(3138.1N-m)i^+(6400.4N-m)j^+(2171.6N-m)k^]

Substitute, [(3138.1N-m)i^+(6400.4N-m)j^+(2171.6N-m)k^] for M2, [(2300N)i^+(500N)j^+(1850N)k^] for R, (xi^+zk^) for rQ/O in equation (XI).

[(3138.1N-m)i^+(6400.4N-m)j^+(2171.6N-m)k^]=(yj^+zk^)×[(2300N)i^+(500N)j^+(1850N)k^]

Comparing the coefficients of the y and z components both sides,

y=0.944m,z=2.78m

Therefore, the axis of wrench intersects the yz-plane at y=0.944m,z=2.78m_.

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Chapter 3 Solutions

Vector Mechanics for Engineers: Statics

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