PROBLEM 8.6 SOLUTION. FBD block (Impending motion up) = N. = tan (0.25) (a) (Note: For minimum P, P^ Then. = ( N)sin β = 14.

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1 PROBLEM 8.6 Knowing that the coefficient of friction between the 25-kg block and the incline i μ =.25, determine (a) the mallet value of P required to tart the block moving up the incline, (b) the correponding value of β. FBD block (Impending motion up) W = mg = (25 kg)(9.8 m/ ) = N φ = tan μ = tan (.25) = 4.4 (a) (Note: For minimum P, P^ R o β = φ.) Then P= Win (3 + φ ) = ( N)in P = 7.5 N (b) We have β = φ β = 4.4 min PROPRIETARY MATERIAL. 23 The McGraw-Hill ompanie, Inc. All right reerved. No part of thi Manual ma be diplaed, reproduced or ditributed in an form or b an mean, without the prior written permiion of the publiher, or ued beond the limited ditribution to teacher and educator permitted b McGraw-Hill for their individual coure preparation. If ou are a tudent uing thi Manual, ou are uing it without permiion. 227

2 PROBLEM 8. The 2-lb block A and the 3-lb block B are upported b an incline that i held in the poition hown. Knowing that the coefficient of tatic friction i.5 between the two block and zero between block B and the incline, determine the value of θ for which motion i impending. Since motion impend, F = μ N between A + B Free bod: Block A Impending motion: Σ F = : N = 2coθ Σ F = : T 2inθ μ N = T = 2inθ +.5(2co θ) Free bod: Block B T = 2inθ + 3coθ () Impending motion: Σ F = : T 3inθ + μn = T = 3inθ μ N = 3inθ.5(2 co θ) T = 3inθ 3coθ (2) Eq. ()-Eq. (2): 2inθ + 3coθ 3inθ + 3coθ = 6 6coθ = in θ : tan θ = ; θ = 3.96 θ = 3. PROPRIETARY MATERIAL. 23 The McGraw-Hill ompanie, Inc. All right reerved. No part of thi Manual ma be diplaed, reproduced or ditributed in an form or b an mean, without the prior written permiion of the publiher, or ued beond the limited ditribution to teacher and educator permitted b McGraw-Hill for their individual coure preparation. If ou are a tudent uing thi Manual, ou are uing it without permiion. 232

3 PROBLEM 8.6 A 4-kg packing crate i pulled b a rope a hown. The coefficient of tatic friction between the crate and the floor i.35. If α = 4, determine (a) the magnitude of the force P required to move the crate, (b) whether the crate will lide or tip. Force P for which liding i impending (We aume that crate doe not tip) 2 W = (4 kg)(9.8 m/ ) = N Σ F = : N W + Pin4 = Subtitute for N from Eq. ():.35( W Pin 4 ) Pco 4 = N = W Pin 4 () Σ F = :.35 N Pco4 = Force P for which crate rotate about (We aume that crate doe not lide).35w P =.35in 4 + co 4 P=.3532W Σ M = : ( Pin 4 )(.8 m) + ( Pco 4 )(.5 m) W (.4 m) =.4W P = =.4458W.8in 4 +.5co 4 rate will firt lide P =.3532(392.4 N) P = 38.6 N PROPRIETARY MATERIAL. 23 The McGraw-Hill ompanie, Inc. All right reerved. No part of thi Manual ma be diplaed, reproduced or ditributed in an form or b an mean, without the prior written permiion of the publiher, or ued beond the limited ditribution to teacher and educator permitted b McGraw-Hill for their individual coure preparation. If ou are a tudent uing thi Manual, ou are uing it without permiion. 237

4 PROBLEM 8.49 Solve Problem 8.48 auming that the force P i directed to the right. PROBLEM 8.48 The machine part AB i upported b a frictionle hinge at B and a wedge at. Knowing that the coefficient of tatic friction at both urface of the wedge i.2, determine (a) the force P required to move the wedge, (b) the component of the correponding reaction at B. φ = tan.2 =.3 Free bod: Part AB Σ M = : (8 N)(.35 m) R co.3 (.6 m) = B Free bod: Wedge R = 5.3 N Force triangle: (a) Law of ine: γ = 9.3 = P 5.3 N = in ( ) in P = 234 N P = 234 N (b) Return to Part AB: Σ F = : B + 8 N + R in.3 = B + 8 N + (5.3 N)in.3 = B = 824 N B 824 N = Σ F = : B + R co.3 = B + (5.3 N)co.3 = B = 5 N B 5 N = PROPRIETARY MATERIAL. 23 The McGraw-Hill ompanie, Inc. All right reerved. No part of thi Manual ma be diplaed, reproduced or ditributed in an form or b an mean, without the prior written permiion of the publiher, or ued beond the limited ditribution to teacher and educator permitted b McGraw-Hill for their individual coure preparation. If ou are a tudent uing thi Manual, ou are uing it without permiion. 28

5 PROBLEM 8.5 The elevation of the end of the teel beam upported b a concrete floor i adjuted b mean of the teel wedge E and F. The bae plate D ha been welded to the lower flange of the beam, and the end reaction of the beam i known to be kn. The coefficient of tatic friction i.3 between two teel urface and.6 between teel and concrete. If the horizontal motion of the beam i prevented b the force Q, determine (a) the force P required to raie the beam, (b) the correponding force Q. Free bod: Wedge F φ = tan.3 = 6.7 (a) P = ( kn) tan ( kn) tanφ P = 5.29kN+ 3kN P = 8.29 kn P = 8.3 kn ( kn) R = =.94 kn co 26.7 Free bod: Beam, plate, and wedge E (b) Q= W tan 26.7 = ( kn) tan 26.7 Q = 5.29 kn Q = 5.3 kn PROPRIETARY MATERIAL. 23 The McGraw-Hill ompanie, Inc. All right reerved. No part of thi Manual ma be diplaed, reproduced or ditributed in an form or b an mean, without the prior written permiion of the publiher, or ued beond the limited ditribution to teacher and educator permitted b McGraw-Hill for their individual coure preparation. If ou are a tudent uing thi Manual, ou are uing it without permiion. 282