A2Physics物理出國英語4.3FurtherMechanicsMS

1.Aballofmass0.2kgismovingwithavelocityof5m/stowardsawall.Itreboundsfromthewallwithavelocityof3m/sintheoppositedirection.Whatisthechangeinmomentumoftheball?

Answer:

Theinitialmomentum(p_i=mv_i),where(m=0.2spacekg)and(v_i=5spacem/s),so(p_i=0.2times5=1spacekgcdotm/s)

Thefinalmomentum(p_f=mv_f),with(v_f=3spacem/s)(oppositedirection),so(p_f=0.2times(3)=0.6spacekgcdotm/s)

Thechangeinmomentum(Deltap=p_fp_i=0.61=1.6spacekgcdotm/s).Themagnitudeofthechangeinmomentumis(1.6spacekgcdotm/s).

2.Acarofmass1200kgismovingataconstantspeedof20m/s.Whatisthekineticenergyofthecar?

Answer:

Theformulaforkineticenergyis(E_k=frac{1}{2}mv^{2}).

Substituting(m=1200spacekg)and(v=20spacem/s)intotheformula,weget(E_k=frac{1}{2}times1200times20^{2}=frac{1}{2}times1200times400=240000spaceJ=240spacekJ).

3.Aforceof50Nactsonanobjectofmass10kgfor4s.Whatistheimpulseexertedontheobject?

Answer:

Theimpulse(I)isgivenbytheformula(I=FDeltat).

Here,(F=50spaceN)and(Deltat=4spaces),so(I=50times4=200spaceNcdots).

4.Apendulumbobofmass0.5kgisreleasedfromaheightof0.2maboveitslowestpoint.Whatisitsspeedatthelowestpoint?

Answer:

Usingtheconservationofmechanicalenergy.Theinitialpotentialenergy(E_p=mgh)isconvertedintokineticenergy(E_k=frac{1}{2}mv^{2})atthelowestpoint.

(mgh=frac{1}{2}mv^{2}),wecancanceloutthemass(m)onbothsides.

Given(h=0.2spacem)and(g=9.8spacem/s^{2}),then(v=sqrt{2gh}=sqrt{2times9.8times0.2}=sqrt{3.92}approx1.98spacem/s).

5.Aspringhasaspringconstant(k=200spaceN/m).Howmuchworkisdoneinstretchingthespringby0.1m?

Answer:

Theworkdoneonaspringisgivenby(W=frac{1}{2}kx^{2}).

Substituting(k=200spaceN/m)and(x=0.1spacem)intotheformula,weget(W=frac{1}{2}times200times(0.1)^{2}=1spaceJ).

6.Aplanethasamass(M=5times10^{24}spacekg)andaradius(R=6times10^{6}spacem).Whatisthegravitationalfieldstrength(g)atthesurfaceoftheplanet?

Answer:

Theformulaforgravitationalfieldstrengthis(g=frac{GM}{R^{2}}),where(G=6.67times10^{11}spaceNm^{2}/kg^{2}).

(g=frac{6.67times10^{11}times5times10^{24}}{(6times10^{6})^{2}}=frac{33.35times10^{13}}{36times10^{12}}approx9.26spaceN/kg).

7.Anobjectofmass2kgismovinginacircularpathofradius3mwithaconstantspeedof4m/s.Whatisthecentripetalforceactingontheobject?

Answer:

Thecentripetalforce(F_c=frac{mv^{2}}{r}).

Substituting(m=2spacekg),(v=4spacem/s)and(r=3spacem)intotheformula,weget(F_c=frac{2times4^{2}}{3}=frac{32}{3}approx10.67spaceN).

8.Abodyisinsimpleharmonicmotionwithanamplitude(A=0.1spacem)andaperiod(T=2spaces).Whatisitsmaximumvelocity?

Answer:

Theangularfrequency(omega=frac{2pi}{T}),with(T=2spaces),so(omega=pispacerad/s).

Themaximumvelocity(v_{max}=omegaA).

Substituting(omega=pispacerad/s)and(A=0.1spacem),weget(v_{max}=pitimes0.1approx0.314spacem/s).

9.Aparticleofmass(m)ismovingwithavelocity(v).Ifitsvelocityisdoubled,whathappenstoitskineticenergy?

Answer:

Theinitialkineticenergy(E_{k1}=frac{1}{2}mv^{2}).

Whenthevelocityisdoubled((v_2=2v)),thenewkineticenergy(E_{k2}=frac{1}{2}m(2v)^{2}=frac{1}{2}mtimes4v^{2}=4timesfrac{1}{2}mv^{2}).

Sothekineticenergybecomes4timestheinitialkineticenergy.

10.Aforce(F=(3hat{i}+4hat{j})spaceN)actsonanobjectwhichmovesfromtheorigin((0,0))tothepoint((2,3)spacem).Whatistheworkdonebytheforce?

Answer:

Thedisplacementvector(vec{s}=(2hat{i}+3hat{j})spacem).

Theworkdone(W=vec{F}cdotvec{s}=(3hat{i}+4hat{j})cdot(2hat{i}+3hat{j})=3times2+4times3=6+12=18spaceJ).

11.Arocketofmass(m)ejectsgasataconstantrate(frac{dm}{dt})witharelativevelocity(v_{rel})withrespecttotherocket.Whatisthethrustforceontherocket?

Answer:

Thethrustforce(F=v_{rel}frac{dm}{dt}).

12.AsatelliteisorbitingtheEarthataheight(h)abovetheEarth'ssurface.TheradiusoftheEarthis(R)andthemassoftheEarthis(M).Whatistheorbitalspeedofthesatellite?

Answer:

Thecentripetalforceforthesatellite'scircularmotionisprovidedbythegravitationalforce.(frac{mv^{2}}{R+h}=frac{GMm}{(R+h)^{2}}).

Wecancanceloutthemass(m)ofthesatelliteonbothsidesandsolvefor(v),getting(v=sqrt{frac{GM}{R+h}}).

13.Arodoflength(L)andmass(M)ispivotedatoneend.Whatisitsmomentofinertiaaboutthepivot?

Answer:

Themomentofinertiaofarodpivotedatoneendis(I=frac{1}{3}ML^{2}).

14.Awheelofradius(r)isrollingwithoutslipping.Ifitsangularvelocityis(omega),whatisthelinearvelocityofapointontherimofthewheel?

Answer:

Forawheelrollingwithoutslipping,therelationshipbetweenlinearvelocity(v)andangularvelocity(omega)is(v=romega).

15.Ablockofmass(m)isplacedonaninclinedplaneofangle(theta).Thecoefficientofstaticfrictionis(mu_s).Whatisthemaximumangle(theta)forwhichtheblockremainsatrest?

Answer:

Theforcesactingontheblockalongtheinclinedplanearethecomponentofgravitationalforce(mgsintheta)andthefrictionalforce(F_f=mu_sN),where(N=mgcostheta).

Fortheblocktoremainatrest,(mgsinthetaleqslantmu_smgcostheta).

(tanthetaleqslantmu_s),sothemaximumangle(theta=arctan(mu_s)).

16.Abulletofmass(m)isfiredintoablockofmass(M)whichisinitiallyatrestonafrictionlesssurface.Thebulletembedsitselfintheblock.Iftheinitialvelocityofthebulletis(v_0),whatisthevelocityofthecombinedmassafterthecollision?

Answer:

Usingtheprincipleofconservationofmomentum.Theinitialmomentumis(p_i=mv_0)andthefinalmomentumis(p_f=(m+M)v).

By(p_i=p_f),wehave(mv_0=(m+M)v),so(v=frac{mv_0}{m+M}).

17.Asystemisinequilibriumwhenthenettorqueaboutanypointiszero.Explainwhythisisthecase.

Answer:

Torquecausesrotationalmotion.Ifthenettorqueaboutanypointisnonzero,therewillbeanangularaccelerationaccordingtotheequation(tau=Ialpha)(where(tau)istorque,(I)ismomentofinertiaand(alpha)isangularacceleration).Inequilibrium,thesystemshouldhavenoangularacceleration,i.e.,itshouldeitherbeatrestorrotatingwithaconstantangularvelocity.Sothenettorqueaboutanypointmustbezero.

18.Aspringmasssystemisoscillatingvertically.Atwhatpositionisthevelocityofthemassmaximum?

Answer:

Thevelocityofthemassinaspringmasssystemismaximumattheequilibriumposition.Attheequilibriumposition,theelasticpotentialenergyofthespringisataminimumandthekineticenergyisatamaximumaccordingtotheconservationofmechanicalenergy.

19.AplanetorbitstheSuninanellipticalorbit.Atwhichpointinitsorbitisitskineticenergymaximum?

Answer:

Accordingtothelawofconservationofangularmomentum(L=mvr)(where(L)isangularmomentum,(m)ismass,(v)isvelocityand(r)isthedistancefromtheSun).Theangularmomentumisconservedthroughouttheorbit.

TheplanetisclosesttotheSunattheperihelion.Since(L)isconstantand(r)isminimumatperihelion,thevelocity(v)ismaximumatperihelion.Andsince(E_k=frac{1}{2}mv^{2}),thekineticenergyismaximumattheperihelion.

20.Arigidbodyisrotatingwithanangularacceleration(alpha).Ifthemomentofinertiaofthebodyis(I),whatisthetorqueactingonthebody?

Answer:

Therelationshipbetweentorque(tau),momentofinertia(I)andangularacceleration(alpha)isgivenby(tau=Ialpha).

21.Aballisthrownverticallyupwardswithaninitialvelocity(v_0).Whatisthemaximumheightitreaches?

Answer:

Atthemaximumheight,thefinalvelocity(v=0).Usingthekinematicequation(v^{2}v_0^{2}=2gh)(takingtheupwarddirectionaspositiveand(g)actsdownwards).

(0v_0^{2}=2gh),so(h=frac{v_0^{2}}{2g}).

22.Acarismovingonabankedcirculartrackofradius(r)andbankingangle(theta).Thecoefficientoffrictionbetweenthetiresandthetrackis(mu).Whatisthemaximumspeedatwhichthecarcanmovewithoutslipping?

Answer:

Theforcesactingonthecararethegravitationalforce(mg),thenormalforce(N)andthefrictionalforce(F_f).

Resolvingtheforcesalongthehorizontalandverticaldirectionsandusingthecentripetalforceequation(frac{mv^{2}}{r}=Nsintheta+F_fcostheta)and(Ncostheta=mg+F_fsintheta)with(F_f=muN).

Aftersomealgebraicmanipulations,themaximumspeed(v=sqrt{frac{rg(sintheta+mucostheta)}{costhetamusintheta}}).

23.Aparticleismovinginastraightlinewithanacceleration(a)whichisproportionaltoitsvelocity(v),i.e.,(a=kv)where(k)isapositiveconstant.Iftheinitialvelocityis(v_0),whatisthevelocityasafunctionoftime(t)?

Answer:

Weknowthat(a=frac{dv}{dt}=kv).

Separatingthevariables(frac{dv}{v}=kdt).

Integratingbothsides(int_{v_0}^{v}frac{dv}{v}=int_{0}^{t}kdt).

(ln(frac{v}{v_0})=kt),so(v=v_0e^{kt}).

24.Awavehasafrequency(f)andawavelength(lambda).Whatisitsspeed?

Answer:

Thespeedofawave(v)isgivenbytheformula(v=flambda).

25.Asoundwavetravelsthroughair.Thedisplacementofairmoleculesisgivenby(y=Asin(kxomegat)).Whatisthephasedifferencebetweentwopointsseparatedbyadistance(Deltax)?

Answer:

Thephaseofthewaveis(phi=kxomegat).

Fortwopointsseparatedbyadistance(Deltax),thephasedifference(Deltaphi=kDeltax),where(k=frac{2pi}{lambda}).

26.Agasiscompressedadiabatically.Whathappenstoitstemperature?

Answer:

Foranadiabaticprocess,(Q=0).Accordingtothefirstlawofthermodynamics(DeltaU=QW),so(DeltaU=W).

Whenthegasiscompressed,workisdoneonthegas((W>0)),so(DeltaU>0).Sincetheinternalenergyofanidealgasisdirectlyproportionaltoitstemperature((U=frac{f}{2}nRT)foranidealgas,where(f)isthedegreesoffreedom,(n)isthenumberofmolesand(R)isthegasconstant),thetemperatureofthegasincreases.

27.Aheatengineoperatesbetweenahotreservoirattemperature(T_H)andacoldreservoirattemperature(T_C).Whatisitsmaximumefficiency?

Answer:

Themaximumefficiencyofaheatengine(Carnotefficiency)isgivenby(eta=1frac{T_C}{T_H}),where(T_H)and(T_C)areinKelvin.

28.Acapacitorofcapacitance(C)ischargedtoapotentialdifference(V).Whatistheenergystoredinthecapacitor?

Answer:

Theenergystoredinacapacitoris(E=frac{1}{2}CV^{2}).

29.Aninductorofinductance(L)hasacurrent(I)flowingthroughit.Whatistheenergystoredintheinductor?

Answer:

Theenergystoredinaninductoris(E=frac{1}{2}LI^{2}).

30.Acurrent(I)flowsthroughawireofresistance(R).Whatisthepowerdissipatedinthewire?

Answer:

Thepowerdissipatedinawireisgivenby(P=I^{2}R)(using(P=VI)and(V=IR)).

31.Amagneticfield(B)actsperpendiculartoawireoflength(L)carryingacurrent(I).Whatistheforceonthewire?

Answer:

Theforceonacurrentcarryingwireinamagneticfieldis(F=BIL).

32.Achargedparticleofcharge(q)andmass(m)ismovingwithavelocity(v)inamagneticfield(B).Ifthevelocityisperpendiculartothemagneticfield,whatistheradiusofthecircularpathoftheparticle?

Answer:

Themagneticforce(F=qvB)providesthecentripetalforce(frac{mv^{2}}{r}).

Equatingthem(qvB=frac{mv^{2}}{r}),so(r=frac{mv}{qB}).

33.Anelectromagneticwavehasanelectricfieldamplitude(E_0)andamagneticfieldamplitude(B_0).Whatistherelationshipbetween(E_0)and(B_0)?

Answer:

Therelationshipis(E_0=cB_0),where(c)isthespeedoflightinvacuum.

34.Arayoflighttravelsfromamediumofrefractiveindex(n_1)toamediumofrefractiveindex(n_2).Iftheangleofincidenceis(theta_1)andtheangleofrefractionis(theta_2),whatisSnell'slaw?

Answer:

Snell'slawis(n_1sintheta_1=n_2sintheta_2).

35.Athinlenshasafocallength(f).Anobjectisplacedatadistance(u)fromthelens.Whatistheimagedistance(v)givenbythelensformula?

Answer:

Thelensformulais(frac{1}{f}=frac{1}{u}+frac{1}{v}),so(v=frac{uf}{uf})(foraconverginglens,withappropriatesignconventions).

36.Adoubleslitexperimentisperformedwithlightofwavelength(lambda).Theslitsareseparatedbyadistance(d)andthescreenisatadistance(D)fromtheslits.Whatisthepositionofthe(n)thbrightfringefromthecentralmaximum?

Answer:

Foradoubleslitexperiment,thepositionofthe(n)thbrightfringefromthecentralmaximumis(y=nfrac{lambdaD}6mgawiw),where(n=0,1,2,cdots).

37.Aphotonhasanenergy(E).Whatisitsmomentum?

Answer:

Theenergyofaphotonis(E=hf)and(f=frac{c}{lambda}),andthemomentumofaphoton(p=frac{h}{lambda}).So(p=frac{E}{c}).

38.AnelectronhasadeBrogliewavelength(lambda).Whatisitsmomentum?

Answer:

ThedeBrogliewavelengthisgivenby(lambda=frac{h}{p}),sothemomentum(p=frac{h}{lambda}),where(h)isPlanck'sconstant.

39.Inaphotoelectriceffectexperiment,theworkfunctionofametalis(phi).Iftheincidentphotonhasanenergy(E),whatisthemaximumkineticenergyoftheemittedphotoelectrons?

Answer:

Accordingtothephotoelectriceffectequation(E_k=Ephi),where(E_k)isthemaximumkineticenergyofthephotoelectrons.

40.Aradioactivesubstancehasahalflife(T_{1/2}).Iftheinitialnumberofradioactivenucleiis(N_0),whatisthenumberofnucleiremainingaftertime(t)?

Answer:

Thenumberofnucleiremaining(N=N_0(frac{1}{2})^{frac{t}{T_{1/2}}}).

41.Anucleusofmassnumber(A)andatomicnumber(Z)undergoes(beta^)decay.Whatarethemassnumberandatomicnumberofthedaughternucleus?

Answer:

In(beta^)decay,aneutroninthenucleusisconvertedintoaproton,anelectronandanantineutrino.

Themassnumber(A)remainsthesame,andtheatomicnumber(Z)increasesby1.Sothedaughternucleushasmassnumber(A)andatomicnumber(Z+1).

42.Anuclearreactionreleasesanenergy(E).Ifthemassdefectis(Deltam),whatistherelationshipbetween(E)and(Deltam)accordingtoEinstein'smassenergyequivalence?

Answer:

AccordingtoEinstein'smassenergyequivalence(E=Deltamc^{2}),where(c)isthespeedoflightinvacuum.

43.Acircuitconsistsofaresistor(R),aninductor(L)andacapacitor(C)inserieswithanalternatingvoltagesourceofvoltage(V=V_0sin(omegat)).Whatistheimpedance(Z)ofthecircuit?

Answer:

Theimpedance(Z=sqrt{R^{2}+(X_LX_C)^{2}}),where(X_L=omegaL)istheinductivereactanceand(X_C=frac{1}{omegaC})isthecapacitivereactance.

44.Atransformerhasaprimarycoilwith(N_1)turnsandasecondarycoilwith(N_2)turns.Iftheinputvoltageis(V_1),whatistheoutputvoltage(V_2)?

Answer:

Foranidealtransformer,(frac{V_2}{V_1}=frac{N_2}{N_1}),so(V_2=frac{N_2}{N_1}V_1).

45.Aprotonandanelectronareplacedinauniformelectricfield.Whichparticleexperiencesagreateracceleration?

Answer:

Theforceonachargedparticleinanelectricfieldis(F=qE).Theacceleration(a=frac{F}{m}=frac{qE}{m}).

Thechargeofaprotonandanelectronhasthe