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# C1976 - CWMcchaniCSml 3 c 6 points Th6 fr‘iationai force...

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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 work-kirxetic {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 mar-mm 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; ...
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