Answer, Key Homework 6 David McIntyre Mar 25,

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1 Answe, Key Homewok 6 vid McInye 4513 M 5, This pin-ou should hve 0 quesions. Muliple-choice quesions my coninue on he nex column o pge find ll choices befoe mking you selecion. The due ime is Cenl ime. Chpes 5 nd 6 poblems. 001 (p 1 of ) 5 poins An musemen pk ide consiss of oing cicul plfom 11.4 m in dimee fom which 10 kg ses e suspended he end of 1.18 m mssless chins. When he sysem oes, he chins mke n ngle of 43.7 wih he veicl. The cceleion of gviy is 9.8 m/s. l Wh is he speed of ech se? Coec nswe: m/s. In he veicl diecion we hve θ d T cos θ m g, whee T is he ension in he chin. hoizonl diecion we hve Since T sin θ m v. In he 00 (p of ) 5 poins If child of mss 6.5 kg sis in se, wh is he ension in he chin (fo he sme ngle)? Coec nswe: N. M 6.5 kg om he fis p we hve T cos θ (m + M) g (m + M) g T cos θ (10 kg kg) (9.8 m/s ) cos N. 003 (p 1 of 1) 0 poins A figue of dnce on music box moves couneclockwise consn speed ound he ph shown below. The ph is such h he lenghs of is segmens, P Q, Q, S, nd SP, e equl. Acs Q nd SP e semicicles. S P Which of he following bes epesens he mgniude of he dnce s cceleion s funcion of ime duing one ip ound he ph, beginning poin? 1. Q we hve l sin θ + d (1.18 m) sin m m, v g n θ (9.8 m/s ) ( m) n m/s.. 3. Q Q Q S S S

2 Answe, Key Homewok 6 vid McInye 4513 M 5, Q S 3. v µ g g 4. v µ coec 5. v 1 µ g coec Q S uing S nd P Q, he cceleion is zeo becuse he velociy is consn boh in mgniude nd in diecion. uing Q nd SP, even hough he mgniude of he velociy is consn, i chnges is diecion, which will esul in n cceleion of v, whee v is he velociy nd is he dius of he c. So he cceleion is fixed non-zeo consn duing SP nd Q. 004 (p 1 of ) 0 poins An musemen pk ide consiss of lge veicl cylinde h spins bou is xis fs enough h ny peson inside is held up gins he wll when he floo dops wy (see figue). The coefficien of sic ficion beween he peson nd he wll is µ nd he dius of he cylinde is. 6. v g 7. v µ g 8. v µ π g 9. v µ g 10. v g Bsic Conceps: Cenipel foce: m v icionl foce: f s µ N fs mx Soluion: The mximum ficionl foce due o ficion is f mx µ N, whee N is he inwd dieced noml foce of he wll of he cylinde on he peson. To suppo he peson veiclly, his mximl ficion foce mus be lge hn he foce of gviy m g so h he cul foce, which is less hn µn, cn ke on he vlue m g in he posiive veicl diecion. Now, he noml foce supplies he cenipel cceleion v on he peson, so fom Newon s second lw, fs mx N m v. Since Wh is he minimum ngenil velociy needed o keep he peson fom slipping downwd? 1. v 1 g. v µ g ω f mx s µ N µ m v m g, he minimum speed equied o keep he peson suppoed is he limi of his inequliy, which is µ m vmin m g, o ( ) 1 g v min. µ

3 Answe, Key Homewok 6 vid McInye 4513 M 5, (p of ) 0 poins Suppose peson whose mss is m is being held up gins he wll wih consn ngenil velociy v gee hn he minimum necessy. ind he mgniude of he ficionl foce beween he peson nd he wll. 1. m v µ. m g coec 3. m v 4. µ m v 5. m g + µ m v 6. µ m g + m v 7. µ m v 8. m g µ 9. m v 10. µ m g m g µ m g The veicl ficion foce mus equl m g, in ode o blnce he foce of gviy nd no hve ny cceleion in he veicl diecion. 006 (p 1 of ) 5 poins A c of mss 69 kg vels ound fl, cicul ce ck of dius 59.1 m. The coefficien of sic ficion beween he wheels nd he ck is The cceleion of gviy is 9.8 m/s. Wh is he mximum speed v h he c cn go wihou flying off he ck? Coec nswe: m/s. When he ce c is going ound cuved ck, he mximum cenipel foce h cn be pplied by he ficionl foce is f s µ s mg. Noing h he mximum cenipel foce is equied when he ce c is moving he mximum llowed speed wihou slipping, we hve m vmx µ s m g v mx µ s g (0.93)(9.8 m/s )(59.1 m) m/s. 007 (p of ) 5 poins The sme c now vels on sigh ck nd goes ove hill wih dius 194 m he op. Wh is he mximum speed h he c cn go ove he hill wihou leving he od? Coec nswe: m/s. m v m g N whee N is he noml foce cing on he c fom he gound. The c will fly off he gound jus when N 0 so he mximum speed llowed will be v mx g (9.8 m/s )(194 m) m/s. 008 (p 1 of 4) 3 poins The following figue shows eis wheel h oes 5 imes ech minue nd hs dimee of 0 m. The cceleion of gviy is 9.8 m/s. Wh is he cenipel cceleion of ide? Coec nswe: m/s.

4 Answe, Key Homewok 6 vid McInye 4513 M 5, The peiod of he eis wheel is T 60 s/5 1 s. The speed of he wheel is v π T π (10 m) 1 s m/s, so he cenipel cceleion is v ( m/s) 10 m m/s. 009 (p of 4) 3 poins Wh foce does he se exe on 61 kg ide he lowes poin of he ide? Coec nswe: N. The foce exeed by he se blnces he gviy nd povides he cenipel foce, so l m [g + ] (61 kg) (9.8 m/s m/s ) N 010 (p 3 of 4) poins Wh foce does he se exe on 61 kg ide he highes poin of he ide? Coec nswe: N. The gviy is ply blnced by he foce exeed by he se nd his esuln povides he cenipel foce, so l m [g ] (61 kg) (9.8 m/s m/s ) N. 011 (p 4 of 4) poins Wh foce (mgniude) does he se exe on ide when he ide is hlfwy beween op nd boom? Coec nswe: N. In his cse, he foce exeed by he se hs wo componens: he veicl one blncing he gviy nd he hoizonl one poviding he cenipel foce. Thus we hve m m g + (61 kg) (9.8 m/s ) + ( m/s ) N. 01 (p 1 of 1) 0 poins A c veling on sigh od 7.9 m/s goes ove hump in he od. The hump my be egded s n c of cicle of dius 10 m. The cceleion of gviy is 9.8 m/s. Wh is he ppen weigh of (n) 381 N womn in he c s she ides ove he hump? Coec nswe: N. Thee e wo foces cing on he womn of mss m. ne is he gviy, W; he ohe is he suppoing foce fom he chi, which is equl o he ppen weigh W in mgniude. Thei esuln foce povides he cenipel foce so hen W W c m v W W c W W v g (381 N) (7.9 m/s) 381 N (9.8 m/s ) (10 m) N. 013 (p 1 of ) 0 poins A plne simil o he Eh hs dius m nd hs n cceleion of gviy of 10 m/s on he plne s sufce. The plne oes bou is xis wih peiod of

5 Answe, Key Homewok 6 vid McInye 4513 M 5, h. Imgine h he oionl speed cn be incesed. If n objec he equo is o hve zeo ppen weigh, wh is he new peiod? Coec nswe: h. When he ppen weigh is zeo, he gviy is equl o he cenipel foce ( ) π m g m ω m T f so his fse peiod, T f, is T f π g π 6 m 10 m/s s h. whee is he men dius of he plne. 014 (p of ) 0 poins By wh fco would he speed of he objec be incesed when he plne is oing he highe speed? Coec nswe: The velociy is v disnce ime π T. The peiod is T 5 h. We know he io of he coesponding velociies, v f, is he v invese of wo diffeen peiods, which is, fo his cse, v f v T T f 5 h h (p 1 of 1) 0 poins Given: G is he univesl gviionl consn. Two idenicl ss, fixed disnce p, evolve in cicle bou hei muul cene of mss, s shown below. Ech s hs mss M nd speed v. M v Which of he following is coec elionship mong hese quniies? 1. v G M. v 4 G M 3. v G M 4. v G M 5. v G M 6. v 4 G M 7. v G M coec 8. v M G om he bsic fomule fo he cicul obil movemen, he cenipel cceleion is v. v M Using he Newon s second lw of moion, we know he cceleion is M, whee is he foce beween wo ss nd is olly supplied by he univesl foce. So we obin v M G M

6 Answe, Key Homewok 6 vid McInye 4513 M 5, v G M. 016 (p 1 of ) 5 poins n he wy o he moon he Apollo sonus ech poin whee he Moon s gviionl pull is songe hn h of Eh s. eemine he disnce of his poin fom he cene of he Eh. The msses of he Eh nd he Moon e especively kg nd kg. The disnce fom he Eh o he Moon is m. Coec nswe: m. If e is he disnce fom his poin o he cene of he Eh nd m is he disnce fom his poin o he cene of he Moon, hen fom he fomul we obin G m M e e q m e n he ohe hnd, G m M m m Mm M e kg kg e + m. Elimining m fom he ls wo equliies, we obin e q m m. 017 (p of ) 5 poins Wh is he cceleion due o he Eh s gviy his poin? The univesl gviionl consn G is N m /kg. Coec nswe: m/s. om he elion m G M e e we obin h he cceleion due o he Eh s gviy his poin is N m /kg kg ( m) m/s. 018 (p 1 of 3) 0 poins Given: A sol sysem simil o ou Sun nd Eh, whee M eh kg eh m M sun kg sun eh m. eemine he mgniude of he chnge in gviionl foce h he Sun exes on 73.1 kg womn snding on he equo noon nd midnigh. Hin: Since is so smll, use diffeenils. Assume: The Sun nd he Eh e he only msses cing on he womn. Coec nswe: N. Bsic Conceps The gviionl foce exeed by m on m 1 is 1 G m 1 m 1 nd he mgniude of he foce is ˆ 1 (1) G m 1 m. () If we wn o find chnge due o chnge, we wie d d (3) since using diffeenils essenilly mouns o ppoximing d d (4)

7 Answe, Key Homewok 6 vid McInye 4513 M 5, fo smll chnges. Soluion: iffeeniing wih espec o, we find d d G m M s 3 (5) so, using diffeenil ppoximions, G m M s 3. (6) The chnge in disnce is equl o he wice he dius of he Eh (i.e., he dimee of he Eh), since he womn moves fom he closes poin (noon) o he fuhes poin (midnigh). E ( m) m. Theefoe G m M s 3. (6) ( N m /kg ) (73.1 kg) ( kg) ( m) 3 ( m) N N. 019 (p of 3) 0 poins eemine he fcionl pecen chnge in he Sun s gviionl foce % on he womn due o he oion of he Eh in he 1 hous beween noon nd midnigh. Assume: The Sun nd he Eh e he only msses cing on he womn. Coec nswe: %. Using Eqs. () nd (6), he fcionl chnge is m M s G 3 G m M (7) s ( m) m , so he pecenge decese is %. 00 (p 3 of 3) 0 poins This p is he sme s he pevious p excep we e concened wih he Moon s gviion foce insed of he Sun s gviionl foce on he womn. Given: A sysem simil o ou Eh nd Moon, whee M eh kg eh m M moon kg moon eh m. eemine he fcionl pecen chnge in he Moon s gviionl foce % on he womn due o he oion of he Eh in he 1 hous beween noon nd midnigh. Assume: The Eh nd he Moon e he only msses cing on he womn. Coec nswe: %. The fcionl chnge is m M moon G 3 G m M moon ( m) m , so he pecenge decese is %.