is at Point B. PROBLEM 12.7 ‘The acceleration of a package sliding at Point A is 3 mvs’ Assuming that the coefficient of kinetic friction is the same ~ for each section, determine the acceleration of the package SOLUTION For any angle 0 Use x and y coordinates as shown, a,=0 4/ EP, =ma,: N=mg cos 0=0 N=mg cos 0 AER, =ma,: mg sin O- 4,N = ma, a, = g(sin 6-1, cos 6) At Point A 8 1g 608 30° _9.81sin 30°-3 9.81.08 30° = 0.22423 At Point B. O=15%, yy, =0.22423 9.81(sin 15° 0.22423 ¢0s 15°) = 0414 mis a=0414 m/s? 15° PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision 0s- PROBLEM 12.13 ‘The two blocks shown are originally at rest. Neglecting the ‘masses of the pulleys and the effect of friction in the pulleys and between block A and the incline, determine (a) the acceleration of each block, (b) the tension in the cable, mk, LB ‘sol SOLUTION (@) Wenote that a5 Blocka | 200 AF, may: TCO01n30°= 200, o Bloke 22 a 3508 meh, + vp - 350 (1 |EF,=mgay: 350 b-2r = 7%) ® (@) Multiply Bq, (1) by 2 and add Eq. (2) in order to eliminate 7 -20200)in30 43502220, + 250/14, saa" 3a0l2") 515 3224 a, =8.40 fis? 2230" Fea fs’), 20 1us'| (6) FromEg. (1), T= (200)sin30°= 2298 40) T=15221 PROPRIETARY MATERIAL. © 2013 The MeGraw-Hill Companies, fn, All ights reserved. No pert of hie Manual may be diplayed reproduced of dizrbuted im any form or by any mean, without the prior writen permision of the publisher, of used Beyond the lied ‘erowton to teachers and educators permed by MeGraw-Bl for ther individual course preparation. Ifyou area student wsng tis Mara, Sow are using i thou! permission ms— PROBLEM 12.17 ge ‘A 5000-1b truck is being used to lift 1000 Ib boulder B that is ona 200 Ib pallet A. Knowing the acceleration of the truck gs th oud, (te fre Been th boulder and the pallet. SOLUTION Kinematics: ap =i m/s? — ay =a, =05 ms? Masses: amy = 0 = 155.28 sags 200 = 6211 slugs 3 i 1000 sm, = 12% 31,056 slugs 32.2 es Let 7’be the tension in the cable, Apply Newton's second law to the lower pulley, pallet and boulder. TOT t Cm + mg) Ay @ A A (rt Me) F Vertical components +{ 2 (0m, 41mg) 8 = (mg +5), 27 — (37.267)(32.2) = (37.267)(0.5) 7 =609,32 Ib Apply Newton’s second law to the truck. A _ may mo Fite PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision PryPROBLEM 12.17 (Continued) Horizontal components > @ F-T= may Horizontal force between lines and ground. T+ ma, = 609.32 + (155.28)(1.0) F=765 1b 4 Apply Newton’s second law to the boulder. o gS Ge Fae Vertical components +|; F, mpg = mya, Fag =Mmy(g +a) 1,056(32.2 +0.5) (@) Contact force: 016 1b PROPRIETARY MATERIAL. © 2013 The MeGraw-Hill Companies, fn, All ights reserved. No pert of hie Manual may be diplayed reproduced of dizrbuted im any form or by any mean, without the prior writen permision of the publisher, of used Beyond the lied ‘erowton to teachers and educators permed by MeGraw-Bl for ther individual course preparation. Ifyou area student wsng tis Mara, Sow are using i thou! permission mPROBLEM 12.27 A spring AB of constant kis attached to a support at A and to a collar of mass m. The unstretched length of the spring is Knowing that the collar is released from rest at x=. and neglecting friction between the collar and the horizontal rod, determine the magnitude of the velocity of the collar as it passes through Point C. SOLUTION ‘Choose the origin at Point C and let x be positive to the right, Then x is a position coordinate of the slider B and is its initial value, Let L be the siretched length of the spring. Then, from the right triangle ‘The elongation of the spring is £,and the magnitude of the force exerted by the spring is WEE -6 7 By geometry, ma answers ve fF (Pr -1) PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision a0aN ay (i PROBLEM 12.28 Block A has a mass of 10 kg, and blocks B and C have masses of 5 kg each, ‘Knowing that the blocks are initially at rest and that B moves through 3 m in 2s, determine (a) the magnitude of the force P, (b) the tension in the cord. AD. Neglect the masses of the pulleys and axle friction, SOLUTION Let the position coordinate y be positive downward. Constraint of cord AD: y+ yp = constant 0, Mm +¥D ay tay =0 Constraint of cord BC: (4 ~ yp) +(e ~ Yp) = constant 2p =0, dy +ac—2ay =0 Eliminate a, 2ay tay +a, =0 o We have uniformly accelerated motion because all of the forces are constant Je=OvoHOnattagl, (Vy) =0 og = AZO _ 8) say? v (2° PulleyD: ——44BF) <0: ye —Tap =0 Block A or @) PROPRIETARY MATERIAL. © 2013 The MeGraw-Hill Companies, fn, All ights reserved. No pert of hie Manual may be diplayed reproduced of dizrbuted im any form or by any mean, without the prior writen permision of the publisher, of used Beyond the lied ‘erowton to teachers and educators permed by MeGraw-Bl for ther individual course preparation. Ifyou area student wsng tis Mara, Sow are using i thou! permissionPROBLEM 12.28 (Continued) he Bock: HER, =ma,: We~Tae= mote @ L>} " Substituting the value for ay and Egs. 2) and (3) into ig, (1), and solving Wet P MBI for Tae. Block B: (a) Magnitude of P. P=Tye -Wy +505 =51.55-5(9.81)+5(1.5) P=10.00N 4 (b) — Tension in cord AD. Pap = Myo = (2151.55) hp 103.1 N PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision anP ) PROBLEM 12.30 ‘The coefficients of friction between blocks A and C and the horizontal surfaces are /1, =0.24 and 44, = 0.20. Knowing that m,=5kg, my =10kg, and m,=10kg, determine (@) the tension in the cord, (b) the acceleration of each 6 block, SOLUTION We first check that static equilibrium is not maintained: Em + Fede = Hila + Me B = 0.245 +10)¢ =3.68 Since Wy 10g > 3.6, equilbsium is nor maintained. |) HS oes Block A EE: Nyame Fy=H,Nq=0.2m,¢ LEE, =myay: T-0.2mg8 = mate ® ms lock Ne =e: apy ms Block C: BF: Ne=meg Fe=M,No=02meg AEF, =meag: T-0.2meg = meade ey at Block 8: +|EF, = myay + = QI mpg — 23 8) ute, ed dy 1 From kinematics: ay =—(a, +a.) @ 2 (a) Tension in cord. Given data: Eq.(): T-0.2(5)g =Sa, © Eq.(2): T-0.210)g =10a, © Hq.(3): 10g -2F =10a5 o PROPRIETARY MATERIAL. © 2013 The MeGraw-Hill Companies, fn, All ights reserved. No pert of hie Manual may be diplayed reproduced of dizrbuted im any form or by any mean, without the prior writen permision of the publisher, of used Beyond the lied ‘erowton to teachers and educators permed by MeGraw-Bl for ther individual course preparation. Ifyou area student wsng tis Mara, Sow are using i thou! permissionPROBLEM 12.30 (Continued) Substitute into (4 e-0ar S021 -0.2¢ +0.17 -0.2g) 12 24 2 9.81 mi 7 O81 mis) (®) Substitute for Tinto (5), (7), and (6): (24) 2 a4 =02| 2 )-02¢ = 048570981 mis? =2-02[ Fs |= 0.31439.81 mis?) (24 =u ae )-0.2¢ =0.14286(9.81 m/s") (74) 3.6N 4 a= 4.76 mis'— a, =3.08 m/s*| € =1401 m/s’ PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision MsPROBLEM 12.31 A 10-lb block B rests as shown on a 20-Ib bracket A. The coefficients of friction are 1, =0.30 and 44, =0.25 between block B and bracket A, and there is no friction in the pulley or between the bracket and the horizontal surface. (a) Determine the maximum weight of block C if block B is not to slide on bracket A. (b) If the weight of block C is 10% larger than the answer found in a determine the accelerations of A, B and C. SOLUTION Kinematics, Let x, and x, be horizontal coordinates of A and B measured from a fixed vertical line to the left of A and B. Let yc be the distance that block C is below the pulley. Note that yc increases when C moves downward. See figure. le ‘The cable length L is fixed. Ls (xp — 4) + Op —14) + Ye teonstant Differentiating and noting that fp Vp—20y +¥e =O ay tay tae a Here, a, and ay are positive to the right, and a, is positive downward. Kinetics. Let Tbe the tension in the cable and Fy, be the friction force between blocks A and B. The free "Ls ree fk Ny Bracket A: PROPRIETARY MATERIAL. © 2013 The MeGraw-Hill Companies, fn, All ights reserved. No pert of hie Manual may be diplayed reproduced of dizrbuted im any form or by any mean, without the prior writen permision of the publisher, of used Beyond the lied ‘erowton to teachers and educators permed by MeGraw-Bl for ther individual course preparation. Ifyou area student wsng tis Mara, Sow are using i thou! permissionPROBLEM 12.31 (Continued) Block 8 mg. Yada pal = 1, Tuo Block € - mae W, Bracket A na, 2 Fy = "ay @ 8 Block B: Q) o: N Block C: +[3F, =ma,: me -T= "2a, o 8 Adding Eas. (2), (@) and (4), and transposing, Mag + Mea, Meas =We © ge te Subtacting Bg. (4) from Bg, 3) and wansposing, 6) (a) _Noslip between A and 8. From Bq. (1) From Eq. (5), For impending slip, PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permisionPROBLEM 12,34 ~ Xx fy” Be Sse ec is surpored bythe 2.4 Mok Aandi attached to a cord to which a 225-N horizontal force is applied as shown, Neglecting friction, determine (a) the acceleration of block A, (b) the acceleration of block B relative to A. SOLUTION (@) First wenote a, =a, +a,,, where a,,, is directed along the inclined surface of A. B Nae, P-W, sin25°= mya, cos25°+mydy,—B my ea w] or 225 -15g sin 25° —15(a, c0525° ath Pee or 15—gsin25° o lay Wy 008 25° Nag =15(g c0825°— a sin25' A mag: P= Poos25* +N sin 25° A or Nag =[25a, ~225(1~cos25°)]/sin25° pat Equating the two expressions for Ny 25a, ~ 225(1—cos 25" sin 25" 15¢g.c0s25° ~ a, sin 25 3(0.81) c08 25° sin 25° + 45(0 — cos 25 54+3sin? 25° =2.7979 mvs? a, =2.80 mis*— () FromEq.() Aya =15~(9.81)sin 25° ~2.7979 cos 25° or aig =8:32mst 25° € PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision as: PROBLEM 12.35 Block B of mass 10-kg rests as shown on the upper surface of a 22-kg 20 ‘6 wedge A. Knowing that the system is released from rest and neglecting friction, determine (a) the acceleration of B, (b) the velocity of B relative to A at 1=0.5 s SOLUTION @) 4/BF,=mya,: W,sin30°4N, Now we note: ay Wy FApig, Where ay, is directed along the (op surface of A. ’ pene, ay Nhe NN cs Yap ~Ws cos 20° =—ma, sin 50° 8, |B, =may: or Nap =10(g.c0820°~ asin 0") Equating the two expressions for Ny (e-4e) af a,-1¢ \ 2%) 10(g 605 20° —a, sin 50°) cosa 8 " (@.81)(.1+c0520*c0s 40°) 2.2 c0s40°sin 50" or 4, = 6.4061 mis* NEF emyay! Wy sin 20°= mya ~mga 0350" or dng = #in20" +a, 00550" = (9.81sin 20° + 6.4061cos 50°) m/s* = 7.4730 mis? PROPRIETARY MATERIAL. © 2013 The MeGraw-Hill Companies, fn, All ights reserved. No pert of hie Manual may be diplayed reproduced of dizrbuted im any form or by any mean, without the prior writen permision of the publisher, of used Beyond the lied ‘erowton to teachers and educators permed by MeGraw-Bl for ther individual course preparation. Ifyou area student wsng tis Mara, Sow are using i thou! permissionPROBLEM 12.35 (Continued) Finally ay =a, tang a; = 64061? + 7.4730" ~2(6.40617.4730)c0s50" or a, =5.9447 mis? T4730 _ 5.9447 sina sins" or ane a, =594 mis? 75.6 (®) Note: We have uniformly accelerated motion, so that O+ar Now Vp—V4q=apt—ayt 05s 7.4730 mis? x0.5 s or 14 mis 20" PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision 38a PROBLEM 12.45 > Vea reaches the crest A of a hill, (a) Determine the radius of y traveling at a constant speed of 50 mi/h, passes through A. SOLUTION (a) Note: 100 mi/h = 146.667 fUs 2 Mone mae g eo ne? ge men or (146.667 fs)" ? 32.2 ftls* or p= 608 € (©) Note: vis constant = a, = 0; 50 mish =73.333 Us +|2K, =m, W Ea &- b . alee or N= (160 1b)| 1-73 333 Us) _ (62.2 fvs*(668.05 £0) =120.0 b| 4 PROPRIETARY MATERIAL. © 2013 Tae McGraw-Hill Companies, Ine, All rights reserved. No part of thi Mansa! may be diplayed, reproduced op diated in any Jorm or By any means, without the prior wren permission ofthe publisher. of wed beyond the lined Adsrbaton fo teachers and educators permed by Me Craw Hil for ther indiidal course preparation Ifyou area student using this Manual Sow reusing i without permision 369