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Unformatted text preview: Your Name: Age/.19—
Your TA: mama—— T&AM 203 FINAL EXAM
Wednesday May 17, 2000 Draft May 9. 2000 4 problems, 100 points, and 150 minutes. Please follow these directions to ease grading and to maximize your score. a) N0 calculators, books or notes allowed. Six pages of formulas from the front and back of the text are
provided. The back of the test can be used for tentative scrap work. Ask for extra scrap paper if you
need it. b) Full credit if '\ /' . . .
o —>free body diagramse— are drawn whenever linear or angular momentum balance is used; correct vector notation is used, when appropriate; , T—> any dimensions, coordinates; variables and base vectors that you add are clearly deﬁned;
2!: 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 clearly state any reasonable assumptions if a problem seems W W; 0 work is I. ) neat,
II. ) clear, and
III.) well organized; . your answers are TIDILY REDUCED (Don’t leave simpliﬁable algebraic expressions);
1:] 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 “67 = 18” instead of, say, “theta7d0t = 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. Problem 1: A Problem 2 _[25_
Problem 3 (25
Problem 3 [25
TOTAL: 100 1)(25 pts) Yoyo. A yo—yo of mass 2m is made of two identical disks (mass m, radius R,
thickness D) glued on either side of a massless spindle (radius 7', thickness d). A
string is wrapped around the spindle and unwinds without friction, The string has
total length L and is inﬁnitesimally thin and massless. G is at the yo—yo’s center of
mass. // /// a) (3 pts) Does G move in the mdirection as the yoyo falls
and unwinds? Why or why not? i a b) (6 pts) Find G’s vertical acceleration. Comment on the
two cases:
i) R<<r and ii) R>>r , c) (4 pts) Find the tension in the string.
d) (5 pts) Write an expression for the total kinetic energy of
the yo—yo when G’s speed is v. e) (5 pts) If 7" << L and the yo—yo starts from rest, ﬁnd 1)
when the string is fully unwound. I
l

f) (2 pts) Under what circumstances will the yoyo rewind ’ i H
i
l completely? 2.1"
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Earn? 93 5mm :7 0' I. Asia as lbW(UﬂHJ~l+WWINL€ iiAMsgsﬁmbﬁw ' 2 “(WW4
) 3 M1 [069 in mg‘ﬁlp i, ‘Ibow'laxrbolbm  9‘ jMEEﬂﬁ‘ 2)(25 pts) Particle on a springy leash. A particle with mass m slides on a rigid
horizontal frictionless plane. It is held by a string which is in turn connect
ed to a linear elastic spring with constant k. The string length is such that
the spring is relaxed when the mass is on top of the hole in the plane. The
position of the particle is F = xi + yj. For each of the statements below,
state the circumstances in which the statement is true (assuming the particle stays
on the plane). Justify your answer with convincing explanation and/or calculation. a) (2 pts) The force of the plane on the particle is 771913.
b) (2pts) E+ﬁz=0 c) (2pts) fj+ﬁy=0 d) (3 pts) 1“ + ﬁr = 0, where 7' = IF] e) (2 pts) r = constant 1') (3 pts) bl = constant g) (3 pts) r29 = constant h) (2 pts) m(si:2 + :02) + 1672 = constant i) (3 pts) The trajectory is a straight line segment. j) (3 pts) The trajectory is a circle. M3 . A ' [t7 :7 N = m
0;) emiti9 (9&4;th [M4 no utr‘l'laﬂ (2)5vxoi‘10m ) 2F: — 0 ’l‘ N Al 5 L
b.) th‘hﬁ N45 m Mtge11102 P 1m; 0.594 Mwﬁ Cat/+93% cmrcis, A A)
c.) (WY. ifws => ~kr=~(¥3+53)="‘(’““ﬁ~‘ y ‘ ll ﬂ,
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~kr£r= rre ér'(LM6) :7 —kr =m ‘9') gambit when mCréIji=4¢V (P416 rd'&*) 9 4]: Wei (5 coms'l' P) ngdewxab 9))9w}lilae cons’iwr'€ {"5 5
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(u r(ré)) or "10“?) ‘2 E MA ('5' (anéi’wi— 3,) rlé Is Mgdwmowm pawl‘i'rwua C" re, + r; é , n u
Ham * N553 ﬂee QCOn/ponfwl'g— if} 29): _2r9+r9 sbnce ’hxere cs no 99 atone, Wneuf dab4; «s alga—41:, hula/l :1 fume h.) Wisl'at‘detwm z(¢m¢+;kr). Thaw «Corn/4 commt1, 1.) As Md) SLMMBUW 4") A6 (AUti BlindC) men‘boneji Ctrcuﬂumo‘hoia lé 4005956 Wm é z}? 3)(25 pts) The problems below (a—d) are independent. a) (5 pts) A falling wire. A Ushaped wire (of the given dimensions) falls from
a height h without rotation and strikes a tabletop at T completely inelastically.
Brieﬂy defend your answers to the following questions. _.—=====— I 3 During impact: L/ ‘ k . .
i) (1 pt) Is the Wire’s linear momentum conserved? T— II “H
ii) (1 pt) Is the Wire’s angular momentum about G constant? in i
iii) (1 pt) Is the Wire’s angular momentum about T constant?
iv) (1 pt) Is the wire’s angular velocity conserved? v) (1 pt) Is the Wire’s total mechanical energy (kinetic + po
tential) constant? L.) LMB :> E ,. 53‘; mg , ZX+CIHA( EDVECD J LL!) AME :? M6: E/q BMVMWJMQ
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M) H WMn’+ while? 59'ng MMWWWJ (Jug‘bﬂﬁ ). g (40+ Conjemcci. 0'.) @1533 e (057“ Cougiwﬂerw‘ﬂj mime (N1 M (stain . mum :uébm‘u
(“fabrﬁ , 50M) _ “(9M . . ’nOi'cavwimf E b) (5 pts) Mars Polar Lander. Last December when the Mars Polar Lander was
lost, some blamed its simple thrusters (devices that eject gas and thus are capable
of providing an impulse in a single direction). Argue using dynamical principles, the
minimum number of thrusters required to have complete threedimensional control. LMB Int’éli djnamliAc'xaQBe" Jet JVIPh/ld 0pCLbD% agave.“ by‘ (7‘; 3'9 5.33:6 ﬁt CM
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e¢ ice ‘3)(379 #1"le c) (5 pts) A falling tower. Frequently parents will build a tower of blocks for their ‘ children. Just as frequently, kids knock them down. In falling (even when they start to topple aligned), these towers invariably break in two (or more) pieces at 9 some point along their length. Why does this occur? What condition is satisﬁed at 9) the point of the break? Will the stack bend towards or away from the ﬂoor after
/ the break? F mﬁwaQmﬁh m Asoﬁlg plea} ml; ’9'
"59/7 M {0:5 ‘0 mérces mmmém ﬂocks ‘ Mammy: Md €441; gastg‘wx +0 W A, _ \ slilw'v]. amt: wisest) magma , 0 ”TH \ SIN}: umjlmd blocks mnofW’fmpn. 72,0 Mulching M‘l'accel‘nﬁ»: # (“if
Mme 4W6 Thom +09 beat; Gee ﬁrst. mama: at“ (paw: Us; 42ng
mvhohottesotl shots. gmg M =14; :2 we"; 24.2 by PM macaw sinks all mac: shame) : Hemteﬁgbrark—Ws WW (5 (”Luca 5r leWM‘l’o lach “”15 d) (10 pts) A pea shooter. A pea of mass m is being blown out of a tube at constant gram 1‘); 4,00”
speed '0. The tube itself is at a constant angle 0 to the vertical and spins at constant
angular velocity wok (1.6., it sweeps out a cone). At the instant shown, the tube is in the yzplane and the pea is at a distance R along the tube. i) (7 pts) What is the pea’s acceleration?
ii) (3 pts) What force acts on it? l7: mos/.21 swgiuct mi .13me;
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jw‘m Maw {21%qu ’ 43(25 pts) Double pendulum. Two identical homogeneous slender bars (weight W, length
’ L, frictionless hinges) hang vertically in a gravity ﬁeld. They are initially at rest
when a horizontal force Pi is suddenly applied at the center of the top bar. a) (10 pts) Write out expressions for the accelerations of the
centers of mass GT and G B in terms of the angular motions T A l 3 of the bars.
b) (10 pts) Write down sufﬁcient equations to solve for the reaction forces at A and C, and for the angular motions. L
c)( (5 pts) Describe how to use Matlab to solve these eua— L P L A A
a) BC“: 247$ EH) = drag L l
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 Spring '08
 RUINA

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