Introduction to and Andy Ruina and Rudra Pratap

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Problems for Chapter 2 723Problems for Chapter 2Vector skills for mechanics2.1 Vector notation and vectoraddition2.1 Draw the vector r ⇀ = (5m)î + (5m) j ˆ.2.2 A vector ⇀ a is 2 m long and points northwestat an angle 60 o from the north. Draw thevector.2.3 The position vector of a point B measures3 m and is directed at 40 o from the negativex-axis towards the negative y-axis. Show theposition vector vector.2.4 Draw a force vector that is given as ⇀ F =2Nî + 2Nˆ j + 1Nuk.2.5 Represent the vector r ⇀ = 5mî − 2mˆ j inthree different ways.2.6 Which one of the following representationsof the same vector F ⇀ is wrong and why?a) b)2Nĵ3N -3 Nî + 2Nĵîc) d)√ √13 N 13 Nproblem 2.6:2323(Filename:pfigure2.vec1.2)2.7 There are exactly two representations thatdescribe the same vector in the following pictures.Match the correct pictures into pairs.a) b)4N4Nĵ30 o 30 oîc) d)2N2 √ 3N 2N(-î + √ 3ĵ)e) f)3Nî + 1Nĵ 3N( 1 3î + ĵ)problem 2.7:(Filename:pfigure2.vec1.3)2.8 Find the sum of forces F ⇀ 1 = 20 Nî −2Nˆ j , F ⇀ 2 = 30 N( √ 1 î + √ 1 j ˆ), and F ⇀ 2 2 3 =−20 N(−î + √ 3j ˆ).2.9 The forces acting on a block of massm = 5 kg are shown in the figure, whereF 1 = 20 N, F 2 = 50 N, and W = mg. Findthe sum F ⇀ (= F ⇀ 1 + F ⇀ 2 + W ⇀ )?4problem 2.9:⇀F 1⇀F 23⇀W43(Filename:pfigure2.vec1.6)2.10 Given that the sum of four vectors F ⇀ i , i =1 to 4, is zero, where F ⇀ 1 = 20 Nî, F ⇀ 2 =−50 Nj ˆ, F ⇀ 3 = 10 N(−î + j ˆ), find F ⇀ 4 .2.11 Three forces F ⇀ = 2Nî − 5Nˆ j , R ⇀ =10 N(cos θî+sin θj ˆ) and W ⇀ =−20 Nj ˆ, sumup to zero. Determine the angle θ and draw theforce vector R ⇀ clearly showing its direction.2.12 Given thatR ⇀ 1 = 1Nî+1.5Nˆ j and R ⇀ 2 =3.2Nî − 0.4Nˆ j , find 2R ⇀ 1 + 5R ⇀ 2 .2.13 Find the magnitudes of the forces F ⇀ 1 =30 Nî − 40 Njând F ⇀ 2 = 30 Nî + 40 Nj ˆ.Draw the two forces, representing them withtheir magnitudes.⇀2.14 Two forces R = 2N(0.16î +0.80j ˆ) and W ⇀ =−36 Njâct on a particle.Find the magnitude of the net force. What isthe direction of this force?2.15 In the figure shown, F 1 = 100 N andF 2 = 300 N. Find the magnitude and directionof F ⇀ 2 − F ⇀ 1 .yF 1F 230 o45 o ⇀F 2 - ⇀ F 1problem 2.15:x(Filename:pfigure2.vec1.17)2.16 Two points A and B are located in the xyplane. The coordinates of A and B are (4 mm,8 mm) and (90 mm, 6 mm), respectively.(a) Draw position vectors r ⇀ A and r ⇀ B .(b) Find the magnitude of r ⇀ A and r ⇀ B .(c) How far is A from B?2.17 Three position vectors are shown in thefigure below. Given that r ⇀ B/A = 3m( 1 2î +√32j ˆ) and r ⇀ C/B = 1mî − 2mˆ j , find r ⇀ A/C .BĵîAproblem 2.17:C(Filename:pfigure2.vec1.7)2.18 In the figure shown below, the positionvectors are r ⇀ AB = 3ftˆk, r ⇀ BC = 2ftˆ j , and⇀r CD = 2ft( j ˆ + ˆk). Find the position vector⇀r AD .BACr ⇀ BCr ⇀ CD⇀r ABproblem 2.18:ˆkĵD(Filename:pfigure2.vec1.5)2.19 In the figure shown, a ball is suspendedwith a 0.8 mlong cord from a 2 mlong hoist OA.(a) Find the position vector r ⇀ B of the ball.(b) Find the distance of the ball from theorigin.yA2mOproblem 2.19:0.8 mB45 ox(Filename:pfigure2.vec1.20)2.20 A cube of side 6 inis shown in the figure.
724 APPENDIX 2. Contact: friction and collisions(a) Find the position vector of point F, r ⇀ F ,from the vector sum r ⇀ F = r ⇀ D + r ⇀ C/D +⇀r F/c .(b) Calculate | r ⇀ F |.(c) Find r ⇀ G using r ⇀ F .zEyFyh30 olBCT 2y60 o 45 o T 1xDHCGOproblem 2.23:x(Filename:pfigure2.vec1.27)Wproblem 2.29:(Filename:pfigure2.vec1.12)Aproblem 2.20:Bx(Filename:pfigure2.vec1.23)2.21 Find the unit vector ˆλ AB , directed frompoint A to point B shown in the figure.y3mB1mA1mproblem 2.21:2mx(Filename:pfigure2.vec1.25)2.22 Find a unit vector along string BA andexpress the position vector of A with respect toB, r ⇀ A/B , in terms of the unit vector.yA2.5 m1.5 m3mzproblem 2.22:B1mx(Filename:pfigure2.vec1.26)2.23 In the structure shown in the figure,l = 2ft, h = 1.5 ft. The force in the springis F ⇀ = kr ⇀ AB , where k = 100 lbf/ ft. Finda unit vector ˆλ AB along AB and calculate thespring force F ⇀ = F ˆλ AB .2.24 Express the vector r ⇀ A = 2mî − 3mˆ j +5mˆk in terms of its magnitude and a unit vectorindicating its direction.⇀2.25 Let F = 10 lbfî + 30 lbfjând W ⇀ =−20 lbfj ˆ. Find a unit vector in the direction ofthe net force F ⇀ + W ⇀ , and express the the netforce in terms of the unit vector.2.26 Let ˆλ 1 = 0.80î + 0.60jând ˆλ 2 = 0.5î +0.866j ˆ.(a) Show that ˆλ 1 and ˆλ 2 are unit vectors.(b) Is the sum of these two unit vectors alsoa unit vector? If not, find a unit vectoralong the sum of ˆλ 1 and ˆλ 2 .2.27 For the unit vectors ˆλ 1 and ˆλ 2 shownbelow, find the scalars α and β such that α ˆλ 2 −3ˆλ 2 = βj ˆ.y1ˆλ 260 o1problem 2.27:ˆλ 1x(Filename:pfigure2.vec1.11)2.28 If a mass slides from point A towards pointB along a straight path and the coordinates ofpoints A and B are (0 in, 5 in, 0 in) and (10 in,0 in, 10 in), respectively, find the unit vectorˆλ AB directed from A to B along the path.2.29 In the figure shown, T 1 = 20 √ 2N, T 2 =40 N, and W is such that the sum of the threeforces equals zero. If W is doubled, find α andβ such that α ⇀ T 1 ,β ⇀ T 2 , and 2 ⇀ W still sum up tozero.2.30 In the figure shown, rods AB and BC areeach 4 cm long and lie along y and x axes,respectively. Rod CD is in the xz plane andmakes an angle θ = 30 o with the x-axis.(a) Find r ⇀ AD in terms of the variable lengthl.(b) Find l and α such that⇀r AD = r ⇀ AB − r ⇀ BC + α ˆk.zA4cm30 ox Cproblem 2.30:lDB4cmy(Filename:pfigure2.vec1.13)2.31 In Problem 2.30, find l such that the lengthof the position vector ⇀ r AD is 6 cm.2.32 Let two forces P ⇀ and Q ⇀ act in the directionshown in the figure. You are allowed tochange the direction of the forces by changingthe angles α and θ while keeping the magnitudesfixed. What should be the values of αand θ if the magnitude of P ⇀ + Q ⇀ has to be themaximum?yPαproblem 2.32:θQx(Filename:pfigure2.vec1.18)2.33 A1m× 1 m square board is supportedby two strings AE and BF. The tension in thestring BF is 20 N. Express this tension as avector.
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802 APPENDIX 9. Kinematics using ti
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804 APPENDIX 9. Kinematics using ti
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806 APPENDIX 10. Contact: friction
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832 ANSWERSf) Can’t do in one sho
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834 ANSWERSc) T = 4mπ 2 r.τ 2d) T
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836 ANSWERSb) φ = sin −1 [1√(
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