Axis Thin spherical shell about any diameter


 Kellie Fisher
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1 Insucos: Fiel/che/Deweile PYSICS DEPATET PY 48 Finl Em Apil 5, 5 me pin, ls fis: Signue: On m hono, I he neihe gien no eceie unuhoie i on his eminion. YOU TEST UBE IS TE 5DIGIT UBE AT TE TOP OF EAC PAGE. Coe ou es nume on ou nswe shee use lines 76 8 on he nswe shee fo he 5igi nume. Coe ou nme on ou nswe shee. DAKE CICES COPETEY. Coe ou UFID nume on ou nswe shee. Pin ou nme on his shee n sign i lso. 3 Do ll scch wok nwhee on his em h ou like. Cicle ou nswes on he es fom. A he en of he es, his em pinou is o e une in. o cei will e gien wihou oh nswe shee n pinou. 4 Blcken he cicle of ou inene nswe compleel, using # pencil o lue o lck ink. Do no mke n s mks o some nswes m e coune s incoec. 5 The nswes e oune off. Choose he closes o ec. Thee is no penl fo guessing. If ou eliee h no lise nswe is coec, lee he fom lnk. 6 n in he nswe shee sepel. Use g 9.8 m/s Ais oop ou cenl is Ais Annul cline o ing ou cenl is Ais Soli cline o isk ou cenl is I I I Ais Soli cline o isk ou cenl imee Ais Thin o ou is hough cene pepenicul o lengh Ais Soli sphee ou n imee I 4 I I 5 Ais Thin spheicl shell ou n imee Ais oop ou n imee Ais Sl ou pepenicul is hough cene I 3 I I
2 PY48 Em Fomul Shee ecos k j i k j i ˆ ˆ ˆ ˆ ˆ ˆ gniues: Scl Pouc: gniue: cos ngle eween n eco Pouc: k j i ˆ ˆ ˆ gniue: sin smlles ngle eween n oion Displcemen: imension 3 imensions Aege eloci: e im e 3 im Aege Spee: s e ol isnce/ Insnneous eloci: im 3 im elie eloci: BC AB AC 3 im Aege Acceleion: e im e 3 im Insnneous Acceleion: im 3 im Equions of oion Consn Acceleion ewon s w n Weigh m F ne m mss Weigh ne he sufce of he Eh W mg use g 9.8 m/s gniue of he Ficionl Foce µ s sic coefficien of ficion, µ k kineic coefficien of ficion Sic: s s F f µ m Kineic: k k F f µ F is he mgniue of he noml foce Unifom Cicul oion ius, Tngenil Spee ω, Angul eloci ω Cenipel Acceleion & Foce: ω mω m F Peio: ω π π T Pojecile oion hoionl sufce ne Eh, iniil spee, iniil ngle wih hoionl nge: g sin eigh: g sin Time of fligh: g f sin Quic Fomul If: c Then: c 4 ±
3 PY48 Em Fomul Shee Wok W, echnicl Eneg E, Kineic Eneg KE, Poenil Eneg U W KE m Wok: W F F Powe: P F Cons n F Kineic Eneg: WokEneg Theoem: KE KE W Poenil Eneg: U F f i WokEneg: Weenl KE U Eheml Einenl Gi e he Sufce of he Eh is up: Sping Foce: F echnicl Eneg: ine omenum: k Cene of ss CO: e Foce: F omen of Inei: Angul omenum: U k F mg U mg U F Wok: W U E KE U Isole n Conseie Ssem: EKEU E f Ei ine omenum, Angul omenum, Toque f p p p m F Kineic Eneg: KE Impulse: J p m F P o m i i CO mi i o i o ne oco Po oco i I i m i i p iscee I Toque: m Conseion of ine omenum: if unifom p i CO Pllel Ais: pi o i I ICO h f τ F Wok: W τ i p hen p consn n p f pi τ hen consn n f i F ne Conseion of Angul omenum: if ne oionl les ω Angul Posiion: Angul eloci: ω Angul Acceleion: α Toque: τne Iα Angul omenum: Iω Kineic Eneg: E o Iω Powe: P τω I Ac engh: s Tngenil Spee: ω Tngenil Acceleion: α olling Wihou Slipping: CO CO ω CO α KE CO I CO ω oionl Equions of oion Consn Angul Acceleionα ω ω α ω ω ω α α i
4 gniue of Foce: Poenil Eneg: PY48 Em 3 Fomul Shee w of Giion mm mm G Escpe Spee: F g G G 6.67 m / kg U g G escpe Tension & Compession Y Young s oulus, B Bulk oulus F F ine: Y olume: P B A A Iel Fluis Pessue ile foce: P F A Pessue consn foce: F P Unis: P /m A Equion of Coninui: A consn olume flow e m A consn mss flow e Benoulli s Equion is up: P g P g consn Fluis es is up: P P g Buonc Foce: F g Simple monic oion S ngul fequenc cos m Iel Sping k sping consn: Phse: m m cos F sin k k m E m Buo flui f /T m m m m k consn Sinusoil Teling Wes fequenc f /T /, we nume k /, sin sin k  igh moing, lef moing m m k We Spee: we f We Spee igh sing: k T we n n,,, n n,,, n n,,, n n,,, Inefeence Consucie: Inefeence Desucie: Allowe Welenghs & Fequencies: Inensi W/m : Dopple Shif: f I os Sning Wes lengh, n hmonic nume we nwe n / n fn n,,3 P A f S Soun Wes P Powe P I 4 Isoopic Poin Souce: soun soun D S n souce Spee of Soun: soun f S fequenc of souce, S, D spee of souce, eeco Chnge D o D if he eeco is moing opposie he iecion of he popgion of he soun we. Chnge S o S if he souce is moing opposie he iecion of he popgion of he soun we. B
5 . Twouomoilessfomesonheissshowninhefigue. C is n c is. A he wo cs egin o moe in he posiie iecion consn cceleions. If he mgniue of he cceleion of c is wice he mgniue of he cceleion of c, wh poin long he is will he wo cs mee? C 3/ 3 4/ C is. Twouomoilessfomesonheissshowninhefigue. C is n c is. A he wo cs egin o moe in he posiie iecion consn cceleions. If he mgniue of he cceleion of c is hee imes he mgniue of he cceleion of c, wh poin long he is will he wo cs mee? C 3/ 3 4/ C is 3. Twouomoilessfomesonheissshowninhefigue. C is n c is. A he wo cs egin o moe in he posiie iecion consn cceleions. If he mgniue of he cceleion of c is fou imes he mgniue of he cceleion of c, wh poin long he is will he wo cs mee? C 4/3 3 3/ C is 4. On fl hoionl socce fiel ne he sufce of he Eh he gol keepe kicks he ll wih iniil spee m/s n ngle 45 wih he hoionl ows emme h is es isnce m w s shown in he figue. A he insn he ll is kicke, he emme egins unning consn spee w fom he gol keepe. Wh consn spee in m/s mus he emme un in oe o ech he ll he insn i his he goun? On fl hoionl socce fiel ne he sufce of he Eh he gol keepe kicks he ll wih iniil spee m/s n ngle 45 wih he hoionl ows emme h is es isnce m w s shown in he figue. A he insn he ll is kicke, he emme egins unning consn spee w fom he gol keepe. Wh consn spee in m/s mus he emme un in oe o ech he ll he insn i his he goun? On fl hoionl socce fiel ne he sufce of he Eh he gol keepe kicks he ll wih iniil spee m/s n ngle 45 wih he hoionl ows emme h is es isnce 3 m w s shown in he figue. A he insn he ll is kicke, he emme egins unning consn spee w fom he gol keepe. Wh consn spee in m/s mus he emme un in oe o ech he ll he insn i his he goun?
6 7. e he sufce of he Eh mn is siing in chi h is connece o he pulle ssem shown in he figue. The pulles e ficionless n ope is mssless. If he comine mss of he mn n he chi is 9 kg, wih wh foce in mus he mn pull on he ope if he is o ise consn eloci? chi ope 8. e he sufce of he Eh mn is siing in chi h is connece o he pulle ssem shown in he figue. The pulles e ficionless n ope is mssless. If he comine mss of he mn n he chi is kg, wih wh foce in mus he mn pull on he ope if he is o ise consn eloci? chi ope 9. e he sufce of he Eh mn is siing in chi h is connece o he pulle ssem shown in he figue. The pulles e ficionless n ope is mssless. If he comine mss of he mn n he chi is 5 kg, wih wh foce in mus he mn pull on he ope if he is o ise consn eloci? chi ope. A lock wih weigh W 735 is eing pushe up ficionless mp wih ngle 3 elie o he hoionl consn spee hoionl foce s shown in he figue. Wh is he mgniue of he foce in h he mp ees on he lock? F e W. A lock wih weigh W 735 is eing pushe up ficionless mp wih ngle 45 elie o he hoionl consn spee hoionl foce s shown in he figue. Wh is he mgniue of he foce in h he mp ees on he lock? F e W. A lock wih weigh W 735 is eing pushe up ficionless mp wih ngle 6 elie o he hoionl consn spee hoionl foce s shown in he figue. Wh is he mgniue of he foce in h he mp ees on he lock? F e W
7 3. e he sufce of he Eh, ll of mss hngs eicll on sing of lengh s shown in he figue. A lump of mu wih mss m n hoionl spee collies wih he ll. The mu sicks o he ll cusing i o swing on is sing. If m, n he mimum eicl heigh he mull ssem eches fe he collision is, wh is he iniil spee of he lump of mu? Befoe Afe 8g 8g 3 3g 4 g 5 g Bll 4. e he sufce of he Eh, ll of mss hngs eicll on sing of lengh s shown in he figue. A lump of mu wih mss m n hoionl spee collies wih he ll. The mu sicks o he ll cusing i o swing on is sing. If m /, n he mimum eicl heigh he mull ssem eches fe he collision is, wh is he iniil spee of he lump of mu? Befoe Afe 8g 8g 3 3g 4 g 5 g Bll 5. e he sufce of he Eh, ll of mss hngs eicll on sing of lengh s shown in he figue. A lump of mu wih mss m n hoionl spee collies wih he ll. The mu sicks o he ll cusing i o swing on is sing. If m /3, n he mimum eicl heigh he mull ssem eches fe he collision is, wh is he iniil spee of he lump of mu? Befoe Afe 3g 8g 3 8g 4 g 5 g Bll 6. Aunifom lehs lengh n mss. A winowwsheemps olen he le gins ficionless eicl wll s shown in he figue. e fins h he le slips on he goun when i is plce n ngle,, less hn 3 o he goun, u emins in plce when he ngle is gee hn 3. Wh is he coefficien of sic ficion eween he le n he goun? Aunifom lehs lengh n mss. A winowwsheemps olen he le gins ficionless eicl wll s shown in he figue. e fins h he le slips on he goun when i is plce n ngle,, less hn 4 o he goun, u emins in plce when he ngle is gee hn 4. Wh is he coefficien of sic ficion eween he le n he goun?
8 8. Aunifom lehs lengh n mss. A winowwsheemps olen he le gins ficionless eicl wll s shown in he figue. e fins h he le slips on he goun when i is plce n ngle,, less hn 5 o he goun, u emins in plce when he ngle is gee hn 5. Wh is he coefficien of sic ficion eween he le n he goun? e he sufce of he Eh soli ojec of mss, ius, n momen of inei I / olls smoohl long hoionl sufce wih spee n up mp h mkes n ngle of wih he hoionl sufce s shown in he figue. Wh mimum heigh oes he ojec ech efoe olling ck own he incline? 3 /4g /3g 3 5 /6g 4 /g 5 /g. e he sufce of he Eh soli ojec of mss, ius, n momen of inei I /3 olls smoohl long hoionl sufce wih spee n up mp h mkes n ngle of wih he hoionl sufce s shown in he figue. Wh mimum heigh oes he ojec ech efoe olling ck own he incline? /3g 3 /4g 3 5 /6g 4 /g 5 /g. e he sufce of he Eh soli ojec of mss, ius, n momen of inei I /3 olls smoohl long hoionl sufce wih spee n up mp h mkes n ngle of wih he hoionl sufce s shown in he figue. Wh mimum heigh oes he ojec ech efoe olling ck own he incline? 5 /6g 3 /4g 3 /3g 4 /g 5 /g. A poin picle wih mss n spee is moing in he plne s shown in he figue. A he momen i cosses he is, is eloci eco poins n ngle of 3 elie o he is n he mgniue of is ngul momenum ou he oigin, i.e., is. Wh is he mgniue of is ngul momenum ou poin P loce on he is isnce / 3 fom he oigin his sme momen? P O is / 3 o is 3. A poin picle wih mss n spee is moing in he plne s shown in he figue. A he momen i cosses he is, is eloci eco poins n ngle of 3 elie o he is n he mgniue of is ngul momenum ou he oigin, i.e., is. Wh is he mgniue of is ngul momenum ou poin P loce on he is isnce / 3 fom he oigin his sme momen? / P O is 3 o is
9 4. A poin picle wih mss n spee is moing in he plne s shown in he figue. A he momen i cosses he is, is eloci eco poins n ngle of 3 elie o he is n he mgniue of is ngul momenum ou he oigin, i.e., is. Wh is he mgniue of is ngul momenum ou poin P loce on he is isnce 3 fom he oigin his sme momen? P O is / 3 o is 5. An uomoile ss fom es on he is s shown in he figue. A i egins o ccelee wih n cceleion h epens on ime ccoing o he fomul, whee n e consns. If 4 m/s n m/s 3, wh ime in s oes he c eun o is sing poin? C is 6. An uomoile ss fom es on he is s shown in he figue. A i egins o ccelee wih n cceleion h epens on ime ccoing o he fomul, whee n e consns. If 6 m/s n m/s 3, wh ime in s oes he c eun o is sing poin? C is 7. An uomoile ss fom es on he is s shown in he figue. A i egins o ccelee wih n cceleion h epens on ime ccoing o he fomul, whee n e consns. If 8 m/s n m/s 3, wh ime in s oes he c eun o is sing poin? C is 8. Two lou spekes e loce 4 m p on n ouoo sge. A lisene is 8 m fom one n m fom he ohe. Duing he soun check, signl geneo ies he wo spekes in phse wih he sme mpliue n fequenc. The nsmie fequenc is swep hough he uile nge o k. Wh is he lowes fequenc in h gies minimum signl mimum esucie inefeence he lisene s locion? Tke he spee of soun o e 343 m/s Two lou spekes e loce 3 m p on n ouoo sge. A lisene is 9 m fom one n m fom he ohe. Duing he soun check, signl geneo ies he wo spekes in phse wih he sme mpliue n fequenc. The nsmie fequenc is swep hough he uile nge o k. Wh is he lowes fequenc in h gies minimum signl mimum esucie inefeence he lisene s locion? Tke he spee of soun o e 343 m/s
10 3. Two lou spekes e loce 3 m p on n ouoo sge. A lisene is m fom one n m fom he ohe. Duing he soun check, signl geneo ies he wo spekes in phse wih he sme mpliue n fequenc. The nsmie fequenc is swep hough he uile nge o k. Wh is he lowes fequenc in h gies minimum signl mimum esucie inefeence he lisene s locion? Tke he spee of soun o e 343 m/s e he sufce of he Eh, suen is using simple pulle o lif ce of mss 3 kg fom es o heigh pulling on he ope wih consn foce F s shown in he figue. If he eking sengh of he ope is 5, wh is he minimum ime in s equie fo he suen o hul he ce o heigh m? Assume h he pulle is mssless n ficionless ope F ce goun 3. e he sufce of he Eh, suen is using simple pulle o lif ce of mss 35 kg fom es o heigh pulling on he ope wih consn foce F s shown in he figue. If he eking sengh of he ope is 5, wh is he minimum ime in s equie fo he suen o hul he ce o heigh m? Assume h he pulle is mssless n ficionless ope F ce goun 33. e he sufce of he Eh, suen is using simple pulle o lif ce of mss 4 kg fom es o heigh pulling on he ope wih consn foce F s shown in he figue. If he eking sengh of he ope is 5, wh is he minimum ime in s equie fo he suen o hul he ce o heigh m? Assume h he pulle is mssless n ficionless ope F ce goun 34. Consie mss kg suspene e ligh sing fom he ceiling of ilwcne he sufceof he Eh. The c hs consn cceleion s shownin he figue, cusing he mss o hng n ngle wih he eicl. If he cceleion of he ilw c is m/s, wh is he ension in he sing in?
11 35. Consie mss 3 kg suspene e ligh sing fom he ceiling of ilwcne he sufceof he Eh. The c hs consn cceleion s shownin he figue, cusing he mss o hng n ngle wih he eicl. If he cceleion of he ilw c is m/s, wh is he ension in he sing in? Consie mss 4 kg suspene e ligh sing fom he ceiling of ilwcne he sufceof he Eh. The c hs consn cceleion s shownin he figue, cusing he mss o hng n ngle wih he eicl. If he cceleion of he ilw c is m/s, wh is he ension in he sing in? Jck wns o ow iecl coss ie fom he es shoe o poin on he wes shoe, s shown in he figue. The wih of he ie is 5 m n he cuen flows fom noh o souh.5 m/s. The ip kes Jck 4 minues. A wh spee in m/s wih espec o he sill we is Jck le o ow? We Cuen Jck W E Shoe Shoe S 38. Jck wns o ow iecl coss ie fom he es shoe o poin on he wes shoe, s shown in he figue. The wih of he ie is 5 m n he cuen flows fom noh o souh.5 m/s. The ip kes Jck 3 minues. A wh spee in m/s wih espec o he sill we is Jck le o ow? We Cuen Jck W E Shoe Shoe S 39. Jck wns o ow iecl coss ie fom he es shoe o poin on he wes shoe, s shown in he figue. The wih of he ie is 5 m n he cuen flows fom noh o souh.5 m/s. The ip kes Jck minues. A wh spee in m/s wih espec o he sill we is Jck le o ow? We Cuen Jck W E Shoe Shoe S 4. e he sufce of he Eh lock of mss is elese fom es heigh on ficionless incline s shown in he figue. The lock slies own he ficionless incline o ech fl hoionl sufce wih kineic coefficien of ficion µ k.. If he lock slies hoionl isnce 5 m long he sufce efoe coming o es, wh is he iniil heigh in m? h is
12 4. e he sufce of he Eh lock of mss is elese fom es heigh on ficionless incline s shown in he figue. The lock slies own he ficionless incline o ech fl hoionl sufce wih kineic coefficien of ficion µ k.4. If he lock slies hoionl isnce 5 m long he sufce efoe coming o es, wh is he iniil heigh in m? h is 4. e he sufce of he Eh lock of mss is elese fom es heigh on ficionless incline s shown in he figue. The lock slies own he ficionless incline o ech fl hoionl sufce wih kineic coefficien of ficion µ k.6. If he lock slies hoionl isnce 5 m long he sufce efoe coming o es, wh is he iniil heigh in m? h is 43. A wh wo isnces fom he cene of he Eh,, is he giionl foce on poin mss m equl o /4 of wh i is he sufce of he Eh? oe: E is he ius of he Eh. E n E /4 3 E n E /9 3 4 E n E /6 4 4 E n E /4 5 E n E / 44. A wh wo isnces fom he cene of he Eh,, is he giionl foce on poin mss m equl o /9 of wh i is he sufce of he Eh? oe: E is he ius of he Eh. 3 E n E /9 E n E /4 3 4 E n E /6 4 3 E n E /3 5 9 E n E /9 45. A wh wo isnces fom he cene of he Eh,, is he giionl foce on poin mss m equl o /6 of wh i is he sufce of he Eh? oe: E is he ius of he Eh. 4 E n E /6 E n E /4 3 3 E n E /9 4 4 E n E /4 5 6 E n E /6 46. Plne oon, wih mss of 7 4 kg n ius of,5 km, giionll cs meeoie h is iniill es elie o he plne, isnce ge enough o ke s infinie. The meeoie flls ow he plne. Assuming he plne is iless, wh is he spee in km/s of he meeoie elie o he plne when i eches he plne s sufce? Plne oon, wih mss of 7 4 kg n ius of 3, km, giionll cs meeoie h is iniill es elie o he plne, isnce ge enough o ke s infinie. The meeoie flls ow he plne. Assuming he plne is iless, wh is he spee in km/s of he meeoie elie o he plne when i eches he plne s sufce? Plne oon, wih mss of 7 4 kg n ius of 4,5 km, giionll cs meeoie h is iniill es elie o he plne, isnce ge enough o ke s infinie. The meeoie flls ow he plne. Assuming he plne is iless, wh is he spee in km/s of he meeoie elie o he plne when i eches he plne s sufce?
13 49. Wh is he mimum ol mss of he cgo incluing he mss of he emp lloon h spheicl helium lloon wih ius of 4. m cn lif off he goun? The ensi of helium n he i e ρ e.8 kg/m 3 n ρ i. kg/m 3, especiel. Assume h he ensi of he cgo is much lge hn he ensi of he i kg 534. kg kg kg 5. kg 5. Wh is he mimum ol mss of he cgo incluing he mss of he emp lloon h spheicl helium lloon wih ius of 5. m cn lif off he goun? The ensi of helium n he i e ρ e.8 kg/m 3 n ρ i. kg/m 3, especiel. Assume h he ensi of he cgo is much lge hn he ensi of he i kg 73.4 kg kg kg 5. kg 5. Wh is he mimum ol mss of he cgo incluing he mss of he emp lloon h spheicl helium lloon wih ius of 6. m cn lif off he goun? The ensi of helium n he i e ρ e.8 kg/m 3 n ρ i. kg/m 3, especiel. Assume h he ensi of he cgo is much lge hn he ensi of he i. 9.9 kg 73.4 kg kg kg 5. kg 5. e he sufce of he Eh, fl plfom is unegoing simple hmonic moion n is oscilling eicll, up n own ccoing o he fomul Asinω s shown in he figue. A lock of mss sis on plfom, u he lock is no che o he plfom. The ngul fequenc ω of he oscillions ss ou smll n inceses slowl. If he mpliue of he oscillions is A.45 m, wh lue of ω in /s oes he lock lose conc wih he plfom n s o ounce? is is e he sufce of he Eh, fl plfom is unegoing simple hmonic moion n is oscilling eicll, up n own ccoing o he fomul Asinω s shown in he figue. A lock of mss sis on plfom, u he lock is no che o he plfom. The ngul fequenc ω of he oscillions ss ou smll n inceses slowl. If he mpliue of he oscillions is A 9.8 m, wh lue of ω in /s oes he lock lose conc wih he plfom n s o ounce? is is e he sufce of he Eh, fl plfom is unegoing simple hmonic moion n is oscilling eicll, up n own ccoing o he fomul Asinω s shown in he figue. A lock of mss sis on plfom, u he lock is no che o he plfom. The ngul fequenc ω of he oscillions ss ou smll n inceses slowl. If he mpliue of he oscillions is A 39. m, wh lue of ω in /s oes he lock lose conc wih he plfom n s o ounce? is is
14 55. An iel spingnmss ssem is unegoing simple hmonic moion S wih peio of he oscillions T s. If he spee of he lock is. m/s when he isplcemen fom equiliium is. m, wh is he spee of he lock in m/s when he isplcemen fom equiliium is. m? An iel spingnmss ssem is unegoing simple hmonic moion S wih peio of he oscillions T 3 s. If he spee of he lock is. m/s when he isplcemen fom equiliium is. m, wh is he spee of he lock in m/s when he isplcemen fom equiliium is. m? An iel spingnmss ssem is unegoing simple hmonic moion S wih peio of he oscillions T 4 s. If he spee of he lock is. m/s when he isplcemen fom equiliium is. m, wh is he spee of he lock in m/s when he isplcemen fom equiliium is. m? A sion moion eeco on he is sens soun wes of fequenc 5, sshownin he figue. The wessen ou he eecoeeflece offuckelinglonghe isn hen eeceieckhe eeco. If he fequenc of he wes eceie ck he eeco is 4, wh is he componen of he eloci of he uck in m/s? Tke he spee of soun o e 343 m/s. oion Deeco Tuck is A sion moion eeco on he is sens soun wes of fequenc 5, sshownin he figue. The wessen ou he eecoeeflece offuckelinglonghe isn hen eeceieckhe eeco. If he fequenc of he wes eceie ck he eeco is 75, wh is he componen of he eloci of he uck in m/s? Tke he spee of soun o e 343 m/s. oion Deeco Tuck is A sion moion eeco on he is sens soun wes of fequenc 5, sshownin he figue. The wessen ou he eecoeeflece offuckelinglonghe isn hen eeceieckhe eeco. If he fequenc of he wes eceie ck he eeco is 35, wh is he componen of he eloci of he uck in m/s? Tke he spee of soun o e 343 m/s. oion Deeco Tuck is FOOWIG GOUPS OF QUESTIOS WI BE SEECTED AS OE GOUP FO EAC TYPE TYPE Q# S Q# S Q# S 3 TYPE Q# S 4 Q# S 5 Q# S 6 TYPE 3 Q# S 7 Q# S 8 Q# S 9 TYPE 4
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