Physics 321 Solutions for Final Exam

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1 Page f 8 Physics 3 Slutins fr inal Exa ) A sall blb f clay with ass is drpped fr a height h abve a thin rd f length L and ass M which can pivt frictinlessly abut its center. The initial situatin is shwn belw. The blb strikes the rd at ne end, and is stuck t the rd thereafter. h M L a) (0 pts) ind the velcity f the blb just befre it strikes the rd. We slve this using cnservatin f energy: E E i f gh v v v gh gh b) (0 pts) ind the angular velcity f the rd+blb syste just after the blb strikes the rd. The angular entu just befre the cllisin is apprxiately the sae as that just after the cllisin (this is a sewhat subtle pint, since L 0 -- gravity supplies a trque n the syste. But if the cllisin is rapid enugh, L L t will be very sall.) Taking the rigin t be the center f the rd, we have: L gh Li v L L L Lf v + Iω ω + Iω

2 Page f 8 We need t calculate I abut the center f the rd: I L / 3 L / 3 3 y ρ L ρl ML ρy dy ρ 3 / L / 3 4 L With this, we have: L gh L ML ω + ω gh L L L M ω M ω ω gh 4 L M + 3 ) (0 pts) Tw identical particles f ass interact via a cnservative central frce n described by the ptential U( r ) kr. a) (0 pts) ind the radius f a circular rbit if the syste has an angular entu l. Circular rbits ccur at the iniu f the effective ptential: ( ) l ( ) l V r + U r + kr µ r µ r µ + dv l dr r l n+ nkr n+ l r nk n + nkr 3 l r nk / ( n+ ) A subtle pint: this r is the separatin between the particles, which in this case is the diaeter f the rbit. The radius is therefre 0 l nk / ( n+ ). n

3 Page 3 f 8 b) (0 pts) r the case n -, find the frequency f sall scillatins abut a circular rbit. l If n -, r (nte that this nly akes sense fr k < 0!). T find the frequency f k sall scillatins we need t evaluate the secnd derivative f V at this pint: l d kr 3 dv r 6l 3 + kr 4 dr dr r l l 6l 8l 8l k k l k k k k Since this is the effective spring cnstant fr the scillatin, we have: dv k k k ω dr l 6 3 µ / 4l l 3) (0 pts) Cnsider a rigid bdy with an axis f syetry, such that I I. The bdy is rtating abut its center f ass, but a frictinal trque acts t slw it dwn. The trque is given by: N bω If the initial angular velcity is ω, find the cpnent f angular velcity alng the syetry axis as a functin f tie. Yu shuld nt assue any special directin fr the initial angular velcity (i.e. it desn t have t be alng a principal axis). The syetry axis is the z axis in this case, s we use the Euler equatin fr the cpnent f angular entu alng this axis: ( ) Due t the syetry, this siplifies t: I ω I I ωω N bω

4 Page 4 f 8 I ω bω dω3 b dt ω I 3 3 bt lnω3 + C I ω Ce ω e 3 bt / I3 bt / I3 3 3,

5 Page 5 f 8 4) (5 pts) A siple harnic scillatr f ass and spring cnstant k is iersed in a fluid which prvides a daping frce with daping paraeter β. The syste initially scillates with aplitude A, and is bserved t scillate thrugh any cycles (with the aplitude f tin decreasing fr each cycle). a) (5 pts) ind the axiu pssible value fr β cnsistent with the described tin f the syste. The tin described is that f an underdaped scillatr, s β < ω k b) (5 pts) ind the tie it takes fr the scillatr t lse half f its initial kinetic energy. The energy f an scillatr is given by: We re lking fr the case where: r the equatin sheet, we knw that: E ka E ka E ka 4 A A At () Ae Ae βt βt ln ln t β c) (5 pts) What beces f the energy lst by the scillatr? e βt βt It ges int the fluid, as heat energy r wave tin. A

6 Page 6 f 8 5) (5 pts) A particle is slides withut frictin dwn a parablic rap (the surface is defined by the equatin y ax ). A cnstant gravitatinal field with agnitude g acts n the particle. a) (0 pts) ind the equatins f tin fr the particle using the Lagrangian technigue. In this case the Cartesian crdinates are a gd chice fr the generalized crdinates. We have: ( ) ( ( ) ) ( 4 ) T x + y x + axx x + a x x U gy gax L x ( + 4a x ) gax L d L 0 x dt x d 4a xx gax x ( + 4a x ) 0 dt 4a xx gax x ( + 4a x ) 8a xx 0 gax + x ( + 4a x ) + 4a xx 0 b) (0 pts) Deterine the frce exerted by the rap n the particle, in ters f the particle s psitin and velcity. Here we need t use Lagrange s undeterined ultipliers, which eans we start by nt assuing the cnstraint: The cnstraint enters as: L ( x + y ) gy f y ax 0 Thus the equatins f tin bece: L d L f + λ 0 x dt x x L d L f + λ 0 y dt y y 0 x + λ ( ax) 0 g y + λ 0

7 Page 7 f 8 Nw we apply the cnstraint, finding: y axx y ax + ax x Substituting this in t the equatins f tin gives: x + λ ( ax) 0 g ( ax + axx ) + λ 0 λ ( ax) x axx g λ + ax g λ + ax x ax ax λ ax g λ ax ax g λ ax ax ax 4 λ ax ax ax + g λ ax ax g Nw that we knw λ we can find the frces exerted by the rap in bth the x and y directins: ax + g x axλ xa ax + g y λ c) (5 pts) Shw that the answer btained in part (b) has the crrect liit as x 0 and x fr the case where the particle has zer velcity. If yu didn t cplete part (b), state what the expected liits are. r the zer-velcity case, the frce f cnstraint beces:

8 Page 8 f 8 x y g xa g We expect that when the particle is infinitely far fr the rigin, the rap is vertical, and thus exerts n nral frce n the particle. At x 0, the particle is sitting n a lcally hrizntal surface, s the nral frce is just g, pinting upwards. Our expressins in these liits bece: x 0: x y 0 g S we d find the expected liits. x : x y g 0 g 0

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