VECTOR MECHANICS FOR ENGINEERS: STATICS
VECTOR MECHANICS FOR ENGINEERS: STATICS
12th Edition
ISBN: 9781260912814
Author: BEER
Publisher: MCG
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Chapter 3, Problem 3.156RP

A 77-N force F1 and a 31-N·m couple M1 are applied to corner E of the bent plate shown. If F1 and M1 are to be replaced with an equivalent force-couple system (F2, M2) at corner B and if (M2)z = 0, determine (a) the distance d, (b) F2 and M2.

Chapter 3, Problem 3.156RP, A 77-N force F1 and a 31-Nm couple M1 are applied to corner E of the bent plate shown. If F1 and M1

(a)

Expert Solution
Check Mark
To determine

The magnitude of d.

Answer to Problem 3.156RP

The value of d is 135mm_.

Explanation of Solution

The z-component of the replaced equivalent couple M2z is zero, magnitude of F1 is 77N and moment M1 is 31Nm.

Write the expression for the vector from B to H.

rBH=310mm(1m1000mm)i^23.3mm(1m1000mm)j^=0.31i^0.0233j^ (I)

Here rBH is the vector from B to H.

Write the expression for the vector from E to H.

rEH=60mm(1m1000mm)i^+60mm(1m1000mm)j^70mm(1m1000mm)k^

Here rEH is the vector from E to H.

Write the expression for the force F1 in the direction of E to H.

F1=(77N)60mm(1m1000mm)i^+60mm(1m1000mm)j^70mm(1m1000mm)k^(60mm(1m1000mm))2+(60mm(1m1000mm))2+(70mm(1m1000mm))2=(42i^+42j^49k^)N (III)

Write the expression for the vector from E to J.

rEJ=di^+30mm(1m1000mm)j^70mm(1m1000mm)k^

Write the expression for the force F1 in the direction of E to J.

M1=(31Nm)di^+30mm(1m1000mm)j^70mm(1m1000mm)k^(d)2+(30mm(1m1000mm))2+(70mm(1m1000mm))2 (II)

Write the expression for the z-component of the moment.

k^(rBH×F1)+M1z=0

Conclusion:

Substitute 0.31i^0.0233j^ for rBH, (42i^+42j^49k^)N for F1 and the z-component of M1 in the above equation to calculate d.

k^|i^j^k^0.310.02330424249|+70mm(1m1000mm)(31Nm)(d)2+(30mm(1m1000mm))2+(70mm(1m1000mm))2=0(13.02+0.9786)Nm+(0.07m)(31Nm)d2+0.0058=013.9986Nm=2.17Nm2d2+0.0058(d2+0.0058)m2=(2.17Nm213.9986Nm)2(d2+0.0058)m2=0.024m2d2=0.0182m2d=0.135m(1000mm1m)=135mm

Thus, the value of d is 135mm_.

(b)

Expert Solution
Check Mark
To determine

The vectors F2 and M2.

Answer to Problem 3.156RP

The vectors F2 and M2 are respectively (42i^+42j^49k^)N_ and 25.858i^+21.19j^_.

Explanation of Solution

Since the force F1 and F2 is same the vector F2 is (42i^+42j^49k^)N.

Write the expression to calculate M2.

M2=(rBH×F1)+M1

Conclusion:

Substitute (I), (II), (III) and 135mm for d in the above equation to calculate M2.

M2=|i^j^k^0.310.02330424249|+(135mm(1m1000mm)i^+30mm(1m1000mm)j^70mm(1m1000mm)k^)(31Nm)(135mm)2+(30mm(1m1000mm))2+(70mm(1m1000mm))2=(1.1417i^+15.19j^+14k^)Nm+(0.135i^+0.03j^0.07k^)m(31Nm)0.155m=(1.1417i^+15.19j^+14k^)Nm+(27i^+6j^14k^)Nm=25.858i^+21.19j^0k^=25.858i^+21.19j^

Thus, the vectors F2 and M2 are respectively (42i^+42j^49k^)N_ and 25.858i^+21.19j^_.

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Chapter 3 Solutions

VECTOR MECHANICS FOR ENGINEERS: STATICS

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