TAM203_spring00_Prelim1_solutions

TAM203_spring00_Prelim1_solutions - ~ . 9g ouramezflnmw //...

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Unformatted text preview: ~ . 9g ouramezflnmw // 0Iu+mm§H Y N /——————‘ Your TA: M— // T&AM 203 Prelim 1 Tuesday February 29, 2000 7:30 — 9:00+ PM Draft February 26, 2000 3 problems, 100 points, and 90+ minutes. it. Please follow these directions to ease grading and to maximize your score. a) No calculators, books or notes allowed. Two pages of formulas, from the front of the text, and a blank page for tentative scrap work are provided at the back. Ask for extra scrap paper if you need it. b) Full credit if x / . . . o -—->free body d1agrams<— are drawn whenever llnear or angular momentum balance 1s used; correct vector notation is used, when appropriate; T—> any dimensions, coordinates, variables and base vectors that you add are clearly defined; :l: all signs and directions are well defined with sketches and/ or words; —)| reasonable justification, enough to distinguish an informed answer from a guess, is given; you ciearly state any reasonable assumptions if a problem seems Wily] defined; 0 work is ) neat, II. ) clear, and III.) well organized; - your answers are TIDILY REDUCED (Don’t leave simplifiable algebraic expressions); El your answers are boxed in; and >> unless otherwise stated, you will get full credit for, instead of doing a calculation, presenting Matlab code that would generate the desired answer. To ease grading and save space, your Matlab code can use shortcut notation like “07 = 18” instead of, say, “theta'ldot = 18”. c) Substantial partial credit if your answer is in terms of well defined variables and you have not substi- tuted in the numerical values. Substantial partial credit if you reduce the problem to a clearly defined set of equations to solve. I )ié'TLe fuzfl‘d‘i‘l'd’li ave becausc I hope. N Jim my "folu‘l'imqj" laid/e Problem 1; 30430 Somc rehaihihg Error}, Problem 2; 35135 (em-or. fig 1C (mural-((1 have.) Problem 3: 35'135 TOTAL: I d0 [100 V ‘ 1)(30 pts) The problems below are independent. MATLAB commands are not allowed except in part 1a) (10 pts) A mass m is connected to a spring k and released from rest with the spring stretched a distance d from its static equilibrium position. It then oscillates back and forth repeatedly crossing the equilibrium. How much time passes from release until the mass moves through the equilibrium position for the second time? (Answer in terms of some or all of m, k, and d.) [Neglect gravity and friction] 9 X = + Serahd W I A=W («2155515 3% : 311/2 fi—N‘ ' flow—9 M4, Xklw93’0 =7£= 311’“ 1b) (10 pts) Assuming equal masses and equal forces in the two cases, what is the ratio of the : M acceleration of point A to that of point B? [assume massless ideal pulleys etc] : p l / I M I " : X3 F/‘m =6‘fi‘x‘lfléo‘d ’70 : Lunatic/(3:th ((03+):(ififl'x'cl1‘ v(0hS+:/OGE+ 29m! I WWW 1c) (10 pts) In three-dimensional space with no gravity a particle with m = 3 kg at A is pulled by three strings which pass through points B, C7 and D respectively. The acceleration is known to be 9; = (1i + 2 j + 3k) m/s2. The position vectors of B, C, and D relative to A are given in the first few lines of code below. Complete the code to find the three tensions. The last line should read T = . . . with T being assigned to be a 3-e1ement column vector With the three tensions in Newtons. [ Hint: If x, y, and z are three column vectors then A= [x y z] is a matrix with x, y, and 2 as columns] 0 ' K =ma El). [C "8 13/1173 Tfilnach3m+To_nD . 6th a! um‘l 990‘ch in (alum n] W W mm in 9 be «GHr [T] Z a MATLAB script file to find 3 tensions a = [1 2 3]]; I‘AB = [2 3 5]]; rAC = [—3 4 21'; I'AD = [1 1 11/; uAB = rAB/norm(rAB); Z norm gives vector magnitude Z You write the code below (4 to 5 lines). % Don’t copy any of the numbers above. X Don’t do any arithmetic on the side. MAC: mC/horméFAC '/ o/a Th‘e oHeIo-é—am UHD= FAD/WWW( VAC); “[0 Uml'vec “I H = [0‘93 “AC 0119013 ‘/0 affemlrle F) T: AQn-yaq 0/0 Solve with bukflaul: 2)(35 pts) A particle of mass m moves in a viscous fluid which resists motion with a force of magnitude F = c|1|, where 1 is the velocity. Do not neglect gravity. a) (10 ptss) In terms of some or all of g, m, and 0, what is the particle’s terminal (steady-state) falling speed? b) (15 pts) Starting with a free body diagram and linear momentum balance7 find two second order scalar differential equations that describe the two—dimensional motion of the particle. c) (10 pts) (challenge, do last, long calculation) Assume the particle is thrown from g = Q with X = um i + vyo j at a vertical wall a distance d away. Find the height h along the wall where the particle hits. (Answer in terms of some or all of 1710, v90, m, g, c, and d.) [Hint i) find 33(t) and y(t) like in the homework, ii) eliminate t, iii) substitute :1: = d. The answer is not tidy. In the limit d —> 0 the answer reduces to a sensible dependence on d (The limit (2 —» 0 is also sensible). If you use Matlab, start your code by assigning any non-trivial values to all constants] £51)" 0 . :-cv(-:y~l’"m3i ’ 31Lde sealer; vx=o, (9:0 :7 v —» szzncj, :3 Sff’adr 5+3“: tailing JPCEJ =l(¢) 82 S I C1 m {/x + .74; Vx: Vx@)=vxo fizvx [4) 7(6)): 0 -»' x M WM c... J __ : [‘51: if; = 1—60”? ‘5 = "5%) {CHE 11“ "'V’“ X0 :7 rte {'Lij (hen-510$: ‘6': ‘2':- EL ) i: p/OSSI'Ue becatde ' WV” fad-Elle alwayj M006! 5 if: . I Problem 2 continued. 'So\V€ 40V T — an (7+3 vet>= :(v,o+.rg2)e*> .. gait +64 (8*? 0 = ‘€vao+m)~+<q =7 (Fig—manta”) ‘7 V = T's-(w £1>(l~e-*'~’t) .. £217: (2) (like homewavk J [Bertha : Chub. {Mei tc“: "/c V 514(«17 Sfik/ Mg: (Ha t=‘tc M I — X/xh.) {3) (2):? tc(vh+vs)(l_6—f/£¢)’ VS 7: §uL5+£4He {4) iH’oé) Y = éc(\/yo+v)(I—(l—></xh)) + V, ta £04m) .3. r o tile: 5:1", “0"” X l x“ = In W /C V0 d __ If -_—_ 31314 (which makeJ senje/ SPQSW) :7 (yr?)_;hfi Zvyo/fivxo/fl we - a 112 i..— 7‘ C% O 7’? J's-u (1‘ d/Xh) IV —d/Xh‘é(d/XI~)L g h" -£(C)(c ) vxo“) _ WOVW l )7— Pambd 1K ) w "‘ “VT; " "i on trayth 6! I 3) 35 pts) Car accelerating. A car (mass 2m) With a big motor, front- / ________ "M y! wheel drive and a stiff suspension accelerates to the right with the //’”M‘l w“ 7 ’4__/ : front wheels over-powered and skidding (friction coefficient = p) f x and back wheels turning freely. L a) (5 pts) Assuming the car starts from rest and has constant acceleration a, how far has it travelled in time t? (Answer in terms of a, and 15.) [Not a trick7 just easy]. v66): ([email protected]+=§444=4++<l v(o)=o——9 vm=a+ u/fl x55), Svgm- : §4H+ == gig/312+Q MPG :5 CL: 0 3) x65 )3 12: 4‘6 b) (30 pts) Find a in terms of any or all of €T,€f,h,m,g and y p. [Hint: all the directions on the cover page apply. Your O . . . n c H (I ) answer should reduce to a = ZTg/h 1n the limit [.1 —> 00.] I F B D M 9 fr; Mane {66+ AM 1249040) )1 ELY/E: [Ecrxémfli'l T = Q; 4— (154+ 5) ("53 ) I Problem 3 continued. ...
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This note was uploaded on 07/09/2009 for the course ENGRD 2030 taught by Professor Ruina during the Spring '08 term at Cornell University (Engineering School).

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TAM203_spring00_Prelim1_solutions - ~ . 9g ouramezflnmw //...

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