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Unformatted text preview: CWMcchaniCSml 3) c) 6 points; Th6 fr‘iationai force 1’ is; given by
f 3 MN, when: ,u is the coefﬁcient of sanding friction
and N is the: manual force. The normal fame N
is; the {My radial farce acting on the mags m.
Hanan fmm Nev/tank Second Law, N an ac :2" mﬁvii/R),
when: ac is the centripetal ameieration. Thus f ,um V3/R 4 puinm The frictional force f is {hit (miy tangential
farce: acting 0n the: masa m. Hencm fmm Nawmxm
Seasond Law, fathm where the minus; Sign accounts {or the directiwn
of the frictima} f0»er Thug m Wm ~~~~ n f5 paints The time {,3 mquired {my the bloak :0 come twmﬁféiﬁl‘"
ﬂan be: obtained from knowlcdgc: 0f the: tangential
accxaiaraéim in part (M (“iv a; LZI‘MMV’w/R a? :2: Swimming the variabms gives 29 Paims 2 points; 3 poimts 1 paint 3 points, 1 point
I paint 2 poinm 1 point ‘2 points CmMechanicsm’Z a) b) 3 points Tho two forces acting on the cylinder are
the tension T and the gravitational
force mg. T For the two vectors
For correct labels
mg ( w 1 ) loss of credit per extra irrelavant force 9 points The accoloration can be found by two mothods:
by calculating the torque or by considering the
energy. MQIEOVBI‘, each method hag two or three
variations. First mothod, Variation 1 : Torque about the center of mags
By wusidcring the rotation about the cottter of mass, the:
tmmitm must be calculated. *r 2: Ia TR gMRil ' (at/R)
T :2 (1/2)Ma Mg, AAAA v T :2: M21 Mg tr: (3/2}Ma ~ 2:: (mm Variation 2: Torqut: about the rim By considering the rotation about the point of contact
of (ht: tape, tho moment of incrtia about that point
must be caiculatod. 7:: 10¢ MgR 3:5 xram ' (ﬁx/R) 131m fr: (1/2)MR3 4» MR3
MgR (WED/{Rx * (ex/R)
a :2: (Ii/3);; (Parana! mos theorem) 30 Points 1 point
2 points I point
33 points;
i point
2 points
1 point
1 point, 1 point
4 points
2 points
1 point 1 point Samde Method: Energy
ng (1/2)(MR2/2)(v/R)2 + (1/2)Mv‘~’
v 11 a: (4/3 ) gy Variation 1: The: dﬁrivative with rcgpem to i yields
2V3. (4/3 )gv
a :2 (2/3)}; Variation 2: By comparing the, kinematic ralationship W3 3 v51 ~+ 2m; 01' v3 x 2ay
with the msuit
v?3 (4/3)gy. Again, a Z/Sg
a) 3 points; The (remix of the cyiinder moveg Straight dmwn
becausa than": am no horizonta} forcaa. 31 3 points
2 points 1 point 2 points I point 2 poinm I paint 3 points w”
W ,,
‘‘‘‘‘‘‘‘‘‘‘‘ H CmMechanicgmiS a) 5 points Tht‘: ﬁrm} Speed V 0f the biock M can be Obtainad
fmm aonservmmn of imam mamenmm. The initial momentum the ﬁnal momentum or
m m, 2:: m wax/3 w?“ MV 0r solving for V
V (2/3 ) m
m M... VO b) 3 pointa
me the daﬁnitmn 0f kinetic ermxjggfxa one may write
A K155 2:: (i/2)1nv§2 m (1/2)mvi‘¢ Subgtituting the: iniﬁal and fma} velocities gives; AKE :: (11/2)m (1/23mv03 ‘— (3/8)mv01~’
OF (3/4)(!/2 mvﬁ) 75% KEmmal c} 4 pwintr‘; F3er the deﬁnition of work, for a comatam farce F
applkid over a distance L, (me 1135 w {:L
The workkirxetic {energy thmmm States. W 2:: A KE
Therefwm for the case given FLU (1/2)}11‘1(my/3)fa W (1/2)”! (VOW
and in order to {amp the buliat PM (1/2)m(0)2 m (1/2}H3(V0)3
Smiving for L5 in terms 0f L0 yieids L3 :2 9/814, 32 Points 1 point 2 poims; 2 paints I paint 2 paints “I paint 1 1:30th .1 paint '3 paint W x . An aliematiw sainﬁon emplaying (ht: kinematic miationghip Vi? 21:12: Vail 238 I point can he, cmpiayﬁd whats a K F/m 1 paint
and
F140;: (1/2)?“ (VG/31W « {i/Z)m(v0)3~’ ipwint
Thus, if V0 I: 0, :3 3 LS and
L5 9/8 LU 1 paint (1) 3 points Wham the block is; free to move, the: congtant form: acts war :3 greater distance in the ﬁxed inertial System; hance from the wurk~kinetic energy theorem {hare Win be a gmater change? (negativa) in the bullct’s kinetic anargy, and it Wi“ emerge with a lower Spam, 3 paints;
Essaa‘tiaﬂy the 3mm argumem mm be mada by stating that a cairmin ammum (3? work is mguimd far the bullat m tunnel ‘mmugh [hC block“ This ammmt is the same if this block is mcwing or ﬁxed. In that farmer cam, howaven m: black aim carries off some: kimtia: energy. Thus, a5; Um allergy of the system is; a cmxmtam the kinatic anargy 0f the bulic‘t is 16:33. 33 CWE and Mimi vanm
a ) 4 points For Symmetry I pmint Far lines starting only
on inner, ending {mly (m outer 1 point For Qutside lines; directed away from Sphere K paint
For raiiu of 2:] (between: (miside) I paint
1‘3) 5 paints
The mu ﬂux (D is given by
cg) :5; A,ka mm“: L } paint
0 Since: the, ﬁcid is a constant for a givm value
at“ r, (busy Law yields h‘db 22:: EWer 2:2: 4Wk'20 3 gamma;
Salvimg for the field? we obtains a 21:0: V , ,, ‘ .
E1 2:212: 1W " ( R, *1, 3?“ <1. 3R) 1 pom: «) c) 4 paintg
TWO diﬁ'amm matbvdﬂ 0f attack can be: Empioyed. First Methmdz From the: definition of A V “Ma Ma
AV :2 «t/ Evedr '1 point
“ ‘sz
J; V :1» 1“ 73%. d1» 1 paint,
AV M 2%)] 3R } .
___ + r R pmnt , 2m M W 4 m A V 2‘ R 3R 3 R I pmm 36‘ Swarm Mammal: me hunk/1(3de of the: potential
aim: m a paint (xpilericai ciigtributiun) charge: V M 2m * ‘
m (mum)
22M) 4
\z m (mater) Justiﬁcatim for above
D 21(6) 23(0 _ 4 k0 “Mia ““““““ EWWM” d} 2 pminm
Na charge on the water spherch Q on Gutsidﬁ Sphﬁfl'ﬁ
(N0 axplammimn required) Fm ml); Mating Wayne at same pummial 37 I point I paint 1 mm: 1 paint 2 paints % paint ‘mw‘ guru in a! M ~ R a 13 £3» W mm 1m wwzt”; m Wm M: mum in magmmm:
mm: mem m dilﬂl‘ﬂi’iﬂl‘h M warm
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2%, a WMJHEMM fi'm’;_:a:n mm mmnm mg; ijﬁi w
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in mgiwm i mmi aﬁw ht: Wax/111% 1““:N'3;'ﬁ'w“ m»: {Em
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314%th my v Mama mm (d)? mum m 2: BMW/333%: 33?}? f”«,,3i2W‘ 'E wﬁm E pagxiygrts 1%.: f a:me ’i Wm 35 Wrsim: gt» a:&§x31% 11::mi ﬂ a WWW
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 Summer '07
 LECLAIR,A

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