This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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
9mm: m; 1w L“ 7‘ wig: i’mtw EM” m LEM:
2%, a WMJHEMM fi'm’;_:a:n mm mmnm mg; ijﬁi w
i Lu: W W H 91 d :3 i} $23M Ems; "Hm mataxiixmwiw Kama: Tm giwm m d L? 3% pmimM E Mum‘s Law mm: hm M V i mm M 2 wth '2? gmiaxﬁim Emisst E g1) 03H“; i; 32‘, an E 3;me %‘ warm; 2 WWW 2,1»; 3 puix‘i‘g TE: gﬁwmm MM WW3 :1 '3 a: :E a: )7 i p m 1, M mm" m mam: m a, “M w” grimh sf Hm hmmm: Hm m w m gracérpm h H " m M“: ‘,
mm m mum, :1 M72me Hm mum” Mi” H13 ﬂ {i 33% witﬁit‘m; Wmm‘mw Hit: par‘ﬁaxm marmm myswiacmmw m Wiwe‘mgi Mm: 2233:»; H 'Eirawm; E'iiégéim ﬂ {3:93:32}: m when Wm mm m; WWW Wm «rm it); than We: Exxzigrmiia: :fai‘xrm
in mgiwm i mmi aﬁw ht: Wax/111% 1““:N'3;'ﬁ'w“ m»: {Em
maxim: i5; mi: mam/mi in MEW" h " ‘ “ ‘ in
mgﬁm EL, Mm“ Hm mmwm {0mm a‘k‘um ﬁrm: ﬁmwwmm. M “3} AS the E31 5?in up” the: gthwgc mum, Em»: mgmiwm Hf Map!!in WNW? {32:3 I‘inm‘wm: fi.‘32"a1§fi§‘ ﬂew E M!“ 1m», “Mam, mm mic,” mm: Mmﬁm Hm: in mgkﬁgm M
ma ﬁnﬁiaé mmwm fm a pwﬁ’iw WHEN Wilmid ﬁrm deﬁ, "Wm, begmaw h Mifiamy dézi‘lmmﬂ my?» it mm W nwgmhw 1‘3 gimil‘im Wm ngmﬂém: “ﬁ’mmz Mb»; 2331:: mm}! 62m m:er himw ;,3.<.1r%17:§ QUWE‘L LE5 wéii‘rzm mm: m m mmimﬁamm in {Winn L Hm
mm; {mm mum km mmw w I‘m k, miggmtigz ‘ikmwgrm Wm m Hm aiming faimm ﬂufﬁszw WE (CW V 5 Wm 81mm firm»: Wmm uwmﬂuxwm a wzihwm ckmzmw
mmiatrmiwz m Ki'mgém‘a H maxim WM: e'mgnmiw:
{mm mm may wrim E77 meg:
Haw M: m WXR
Smmmg 35:31” W, mm wmmm
m :72: Mam/V
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
i minim E pw‘ij‘x‘if; ...
View
Full
Document
 Summer '07
 LECLAIR,A

Click to edit the document details