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Unformatted text preview: Fall 2011 AAE 340 — Dynamics and Vibrations Exam II Soluﬁ'on 5 Please read the problems carefully.
Write clearly and use diagrams when necessary. ( 30 points) 1. A dumbbell consists of two particles A and B ( of equal mass m) connected by a rigid
massless rod of length L. Particle A is constrained to move on a ﬁxed, frictionless circular track centered at point 0 and of radius R. The system moves on a smooth, HORIZONTAL surface. Both angles are free. A A u.
r, \ w n B
:96an A X ’8 A
/\ ‘ a \K a;
e: memol \( a
A . . xx A
u“ a in A em/”" in“ ra’ A A
a ’ w“/ "N ERR A
‘2 z m Pod Jig/f ; x\\:\ 81 :1 = Chm 1'33) Us.
€~DL ° /\ 5’ g ¢ ’31 = “5,2,0” +Cﬂ>ug
“— *' l“ ' A g m————’ g
()3 : (5 (L3 l“ I} A an A
8 — Y‘ o . A :K\ //// ¢ w :( + @)u5 I (b A
\ \xNiN fix] ’/ (a) Sketch the FBD. (b) Energy is constant. Justify this statement. Write an expression for E. (c) Total linear momentum is NOT constant. Why? (d) Total angular momentum about point 0 is a constant. Justify this statement. N unknown normal
clue +0 ‘h‘ctCK WA ) NA all, V73 QB 10 Pm 07C MD‘hOn (b) En ‘ Condan‘b
6.936% A ’ /\
mNv =Nw+R¢u&~—‘o VN=O
normaﬂ 40mg claw no leA (it) Any {owes normaﬂ +0 plane of: War/H
do no room (anﬂ We m ('23 dwwﬁm) (CM {minimal “Force in rod does no work smce,
du‘sw‘omae t)an parﬂ‘dcs is ﬁxed E=~7> 55m (avAfv’m imﬁv‘i We) 6.. O A VA s R¢ Ola .. A .. c . _ A r05: + LY“: "7 avg: R¢Q§U + L(¢ +gp)r~&
55 . A o o a A
v = chspn + [Wow L(¢+p):} Fa ( 45 points )
2. Use the force equation and the moment about A to derive the EOM for the system of
particles in Probleml. _. cm A
PC : Ralf/4‘ A 3V0“: twigs EGIHEWV €~Ocm av . co ,0 A ‘ ' a" A
A = R915 52— Rqﬂf, + 5w Hm.  5015+?) *7 2 [w R931“ " S Cé+ﬂ3>Kij (2
+ I R¢>+ itwgm — 5C0; myegﬁg a”; ’ ’\ = NIL: ORA EAcmx am 4 =59, >< am (Raul—Res my
: [mLRc'ﬁ’qb + mLRgéa‘sfbl {1‘3 a): so = mLQCéS'+E>S)+mLR<}5'C(5+ mLRgﬁ'Qsﬂ) \EOM ( 25 points )
3. In the ﬁgure, a circular disk D rotates at the end of an arm at the constant rate Q rad/s; the
arms passes through the disk center of mass. The arms rotates about the inertiallyﬁxed axis é2 at the arbitrary rate 7. (a) Deﬁne the appropriate sets of unit vectors. Write the unit vector relationships. (b) Determine the number of degrees of freedom. Use the disk center of mass as the
base point and justify your answer. ’Derh‘ne é: that‘ﬁO/Q 5t: thud in arm
0AM ﬁxed in dtsK 2191‘sk— MM EDP —‘~‘ (p
Trans 30F
choose am as béléé JOOWC ._. /\
ﬂooofm gm roam = ﬂail/4r L414 A 3 consMVYﬁe: ‘5» = 920 h 01)
L = Lo Role boll
0762M ch3~l< (Home am 1~9\“3
6 Oxbowt at 3 /\ 04 wow} d& /\ (9 WM 0L5 ConSNwhﬁ: o< = 0(0 ~ 3
ﬂ? 3 (50
e=ﬂ£+50 O Rot 30F ...
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 Fall '09
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