Physics 240: Worksheet 15 Name

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Lawrence Lang
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1 Physics 40: Woksht 15 Nam Each of ths poblms inol physics in an acclatd fam of fnc Althouh you mind wants to ty to foc you to wok ths poblms insid th acclatd fnc fam (i.. th so-calld "won way" by som popl), you should foc you mind to do ths th iht way by dict application of Nwton's laws and no intoduction of fictional focs! In ach of ths poblms, th moal to th stoy is do what you can to mak su you stay outsid th acclatd fnc fam and FORCE you mind to stay th! (1) Suppos a oll coast id has a loop of adius. What is th minimum spd that a oll coast can ha and still stay on th tack at th top? () Suppos a hollow sphical spac station of adius R is in spac and is otatin about on axis. With what anula locity must th spac station otat so that a pson locatd at th quato of th spac station xpincs an acclation qual to? How dos this acclation ay as th pson mos towads th pol alon th sph? (3) How lon would th day b on th Eath if an objct at th Eath's quato would float? How much is th appant wiht of a 100 k objct chand at th quato du to th otation of th Eath? (4) A tain ca has a mass hanin by som fishin lin fom its oof. How much acclation must th tain ca xpinc in od to poduc an anl of φ=5 0 btwn th stin and th tical. What happns if th tain is moin with a constant locity? (5) A cat is undoin an acclation as shown. A block is on th sid of th cat. If th is a static cofficint of fiction µ btwn th block and th cat, how much must th acclation b so that th block dos not slid? (6) Suppos you find youslf in a iant cntifu of adius R. Th walls of th cntifu ha a static cofficint of fiction of µ with you back. What must b th fquncy (f) with which th cntifu otats so that you will main stuck to th wall?

Physics 40: Woksht 15 Nam (1) Suppos a oll coast id has a loop of adius. What is th minimum spd that a oll coast of mass m can ha and still stay on th tack at th top?

2 Physics 40: Woksht 15 Nam (1) Suppos a oll coast id has a loop of adius. What is th minimum spd that a oll coast of mass m can ha and still stay on th tack at th top? Solution: Th a ways to look at this poblm... th iht way and th won way. W want to concntat on doin thins th iht way so I won't show th won way h. Th tack is xtin a nomal foc on th cat. Lt's look at a f body diaam of th systm. I commnd that if possibl you mak a mntal snapshot of thins and otat to poduc a positi acclation whn possibl. Accodin to Nwton's laws: F = ma and so h, w ha N+ m= ma c. Th cat will fall whn th nomal foc is qual to zo. Thus, this quis ac =. But th cntiptal acclation is in by a = so w can thn say that th dsid lation looks lik = =±. I ncoua you not to tansf youslf to th fam of fnc of th cat in this poblm sinc that intoducs focs which a ally not psnt. Somtims, it's ally had to foc you mind into thinkin about thins lik this. Also plas not, this poblm is not ally unifom cicula motion althouh it is indd closly latd (th spd is low on th top than it is on th bottom). c

3 Physics 40: Woksht 15 Nam () Suppos a hollow sphical spac station of adius R is in spac and is otatin about on axis. With what anula locity must th spac station otat so that a pson locatd at th quato of th spac station xpincs an acclation qual to? How dos this acclation ay as th pson mos towads th pol alon th sph? H is a pictu of th situation. Alon th quato, w ha th followin situation Th only foc psnt h is th Nomal foc and that is th foc which is poidin th cntiptal acclation. W thus apply Nwton's laws to this situation: F = ma. W want to simulat aity h so w ultimatly want this foc to b qual to m. Thus, w qui ma= m = m =±. But, th anula locity is latd to by =ω. So it is cla that w ha th quimnt: ± =ω =ω ω=± Th poblm of what happns away fom th quato is just a bit mo complicatd than this, how. Lt x psnt th ppndicula distanc fom this axis. This distanc is latd to by x= sin( θ ). Th tanntial locity, how, is oin to chan. If w kp th sam ω as in th fist pat, w can find th tanntial locity to b in by: =ω x= sin( θ ). Th cntiptal acclation at ach point is thn in by ( θ) a = = = = sin( θ) sin( ) sin ( θ) c x sin( θ) sin( θ) which coctly pdicts ac 0at th pols, and a c = at th quato..

4 Physics 40: Woksht 15 Nam (3) How lon would th day b on th Eath if an objct at th Eath's quato would float? How much is th appant wiht of a 100 k objct chand at th quato du to th otation of th Eath? Solution: Th situation is shown blow: So lon as th nomal foc is lss than th wiht of th objct, th acclation will b cntiptal and dictd towads th cnt of th Eath as shown in th scond fiu. Lt's apply Nwton's laws to this situation: F = ma. Application of Nwton s laws thn says m N= ma c. Th objct will float whn N=0. Thus w ham= ma, o ultimatly a c =. a c c Now sinc ω R = R = R =ω R w ha th sult in as: Rω = ω= = πf f= T= π 1 R R π R H f is fquncy and T is piod. W can cay this a littl bit futh to find out what how much th wiht is diffnt fom m at th quato. Pobably it's bst to imain a scal und a pson at th quato. Th scal is adin th nomal foc which it is xtin. Thus, w ha m mrω = N m( R ω ) = N W can thus ach th followin answs: usin R =6.37x10 6 w find that 6.37x10 6 T= π = s= min =. h Now, th anula fquncy of th Eath is π π "ad" ω= π f = T = =. x s. So to find out th appant 6 5 wiht of an objct, w ha N= m( R ω ) = 1009 ( x10 ( 7. 7x 10 ) )=977N This is compad to 980N fo a 100k objct at th noth pol if th Eath w pfctly sphical (i.. it's 3 N lss at th quato).

Physics 40: Woksht 15 Nam (4) A tain ca has a mass hanin by som fishin lin fom its oof. How much acclation must th tain ca xpinc in od to poduc an anl of φ=5 0 btwn th stin and th tical.

5 Physics 40: Woksht 15 Nam (4) A tain ca has a mass hanin by som fishin lin fom its oof. How much acclation must th tain ca xpinc in od to poduc an anl of φ=5 0 btwn th stin and th tical. What happns if th tain is moin with a constant locity? This is you fist poblm in which you must sol componnts of a foc...it is not difficult to do. Solution: Th situation is shown to th lft. In this cas, it is th tnsion in th stin which is poducin th acclation of th mass. Lt's daw in ach of th focs and analyz this poblm not fom th acclatd fnc fam (which som would call th "won" way ) but ath fom outsid this fam. W ha th focs which I ha shown h. Apply Nwton's laws to this situation: F = ma. Alon th y diction, w thn ha Ty m=0alon th x diction w ha: Tx = ma. Now w ha a connction btwn T x, T y and th anl which is in by = tan( θ ). W can find th Tx alu of T y by lookin at th y-quation of motion: m Ty= Tsin( θ ) = m T= sin( θ) which is th tnsion in th stin. W can thn find th quid acclation fom th x quation: T = Tcos( θ ) = cos( θ ) = mcot( θ ) = = ma. This thn is th m m x sin( θ) tan( θ) quid acclation as a= cot( θ ) If φ=5 0 thn θ=90-5=65 0. Thus, a = 9. 8cot( 65) = m/ s. You can ify in tms of φ that th quid acclation is a= tan( φ ). Ty

Claly, if th w no fictional foc, th block would fall off. Also, if th w no nomal foc, th block would not acclat. If th w no wiht, th block would not fall. If th w no acclation, th poblm wouldn't wok.

6 Physics 40: Woksht 15 Nam (5) A cat is undoin an acclation as shown. A block is on th sid of th cat. If th is a static cofficint of fiction µ btwn th block and th cat, how much must th acclation b so that th block dos not slid? Solution: You nd to analyz this poblm lookin at th focs inold. Claly, if th w no fictional foc, th block would fall off. Also, if th w no nomal foc, th block would not acclat. If th w no wiht, th block would not fall. If th w no acclation, th poblm wouldn't wok... tc. Lt's sktch th f body diaam fo this situation. A sktch of th focs inold looks lik this: Th f body diaam is shown to th iht. W apply Nwton's laws to this situation: F = ma.this is us th followin quations of motion: f m=0 and N= ma. Th fictional foc h is claly in th diction shown. You can st th cofficint to zo to s which way it slids... fiction is in th opposit diction. Th fictional foc is still in by f=µ N but you must sist th tmptation (by th il on phaps) to say that this is m. In fact, h th nomal foc is shown abo and is qual to ma. W thn ha µ ma m= 0 a = µ.

As usual, you want to daw a f body diaam fo this systm. A sktch of th situation (wh I ha otatd th systm to insu btt isualization of th focs) is shown to th iht.

7 Physics 40: Woksht 15 Nam (6) Suppos you find youslf in a iant cntifu of adius R. Th walls of th cntifu ha a static cofficint of fiction of µ with you back. What must b th fquncy (f) with which th cntifu otats so that you will main stuck to th wall? This situation is shown to th lft. You nd to analyz this poblm only in tms of th focs which a actually th. As usual, you want to daw a f body diaam fo this systm. A sktch of th situation (wh I ha otatd th systm to insu btt isualization of th focs) is shown to th iht. Th f body diaam fo this systm looks lik what is shown blow. W apply Nwton's laws to this systm: F = ma. Th fictional foc is still in by f=µn and aain you' ot to sist that tmptation to say this is m. Lt's wit out th componnts of Nwton's laws: N= ma fo th x-diction and f m=0 fo th y diction. W can sol fo th fictional foc now: f=µ N=µ ma c sinc th acclation psnt is, fo su, th cntiptal acclation. Us this in th y-quation of motion to obtain: ω R µ ma m= 0 µ a = a = = = =ω R. Lt's sol this fo ω: c c c µ R R ω R= ω = ω= = πf f= 1 µ µ R µ R π µ R

CHAPTER 5 CIRCULAR MOTION

CHAPTER 5 CIRCULAR MOTION and GRAVITATION 5.1 CENTRIPETAL FORCE It is known that if a paticl mos with constant spd in a cicula path of adius, it acquis a cntiptal acclation du to th chang in th diction