《耶魯大學(xué)開放課程：基礎(chǔ)物理》第5-8課英中字幕

目錄

第5課...........................................................................2

第6課.........................................................................42

第7課.........................................................................81

第8課........................................................................125

第5課

1wasgoingtodoabrandnewtopic我們要討論一個(gè)全新的話題

but1didn'twanttogowithoutmakingsure但在此之前我要確認(rèn)一下

everyoneunderstandstheloop-the-loopproblem.每個(gè)同學(xué)都已經(jīng)理解過山車問題了

Becausethat'soneproblemwhere因?yàn)檫@是一個(gè)典型的例子

ifyoudidn'tlearnNewton'slawsandapplythem如果你不能正確地理解和運(yùn)用牛頓定律

properly,

youjustcouldn'tfigureitout,okay?你就無法解決這個(gè)問題

Alltheotherthingswedo,我們討論的一些其它問題

youknow,blocksslidingdownplanes,friction,比如石塊從斜面上滑落摩擦力滑輪

pulleys,

youhaveanintuitivefeelingforthat,你們對這些會(huì)有直覺的認(rèn)識

butthisonereallyisverynon-intuitive,right?但對這個(gè)問題真不知道如何下手

Hereisatrolleyatthetopofthetrack.假設(shè)小車在軌道的頂端

Theforcesonthetrolley,bothgravityandthe小車受的力有重力和軌道給的作用力

forceofthetrack,

arebothpushingdown.而且兩個(gè)力都是向下的

Andthequestionis,"Whyisn'tthatfalling問題是"為什么小車沒有掉下來"

down?"

Everythingispushingdown.所有的力都是向下的

Whyisn'titfallingdown?為什么它不會(huì)落下來

That*sthemystery,這是還沒有解答的問題

sowhat'stheansweryou'dgivetosomebody你要怎么回答如果別人問你

whosays,

"Whyisn'titfallingdown?"Yes?"為什么它沒掉下來"請講

Yes,sotheymightsay,是的他們可能會(huì)說

"Well,ifyousayit'sfallingdown,"如果你說它會(huì)掉下來

howcomeitdoesn'tendupheretothecenterof為什么沒有落在圓圈的中心"

thecircle?"

Yes,somebodyhaveananswertothat,yes?有人能回答這個(gè)問題嗎請說

Pardonme?請?jiān)僬f一遍

No,therearenoforces不對在水平方向上

inthehorizontaldirection;theforcesareall-yes?沒有受力力全都是你請說

It'snotnegligible.它不是可忽略的

Thereisanaccelerationthat'sproducingachange這里確實(shí)有加速度確實(shí)在讓速度改變

invelocity.

Butwhenwesayanapplehasadownward但我們說蘋果有個(gè)向下的加速度

acceleration,

youletitgoanditfallsdown;你一放手它就下落了

that'swhatwethinkisdownwardacceleration.我們就認(rèn)為有個(gè)向下的加速度

Hereissomethingwhichisdownward現(xiàn)在這里也有一個(gè)向下的加速度

acceleration,

butit'snotfallingdown.Yes?但是物體卻沒有下落請說

OK.Yes,I'lltakeonemoreanswer;yes?好的我再聽一個(gè)回答請講

Alright.So,letmesummarizeallthedifferent好的讓我把所有不同的答案總結(jié)起來

answers.

Firstofall,首先

onethingyoushouldremember;你要牢記一件事

therearenohorizontalforceshere.這里沒有任何水平方向的力

TheforcesarewhatI'vedrawn;所有的作用力我都畫出來了

that'swhyyoushouldbevery,verycareful.這就是你們要格外格外注意的

Justbecausesomethinghasatendencyto因?yàn)橐粋€(gè)物體只有向左的運(yùn)動(dòng)趨勢

movetotheleftdoesn'tmeanthere'saforceto不代表它受到了向左的力

theleft.

That'sthewholepointofNewton'slaw,這是牛頓定律的核心思想

forceisconnectedtoaccelerationnottovelocity.力只與加速度有關(guān)與速度無關(guān)

Soyes,thiswillhaveanacceleration,youcannot沒錯(cuò)它確實(shí)有加速度你不能回避這

avoidit.一點(diǎn)

F=m?aw川tellyouT+m?g=m?a,F=m?a說明T+m>g=m*a

andtheathatyouwantinthisproblem,這個(gè)a就是在這個(gè)問題里你所需要的

ifit'smovinginacircle,ism?vA2/R.若物體做圓周運(yùn)動(dòng)這個(gè)a等于m?

vA2/R

So,itisaccelerating,它確實(shí)有加速度

andthedifferencebetweenthisoneaccelerating,和這個(gè)加速度不同的是

letussay,1gototheloop-the-loopand1dropan比如說我坐在過山車上掉了一個(gè)蘋果

apple,

forgetthetrackit*sgotm?gacting,不考慮軌道它受到m?g的作用

it'sacceleratingtoo.蘋果也在加速

What'sthedifferencebetweenthesetwo?這兩種情況有什么區(qū)別呢

Here,westartwithzerovelocity;youaccelerate在這里的初速度為零向下加速運(yùn)動(dòng)

downwards.

Thatmeansyoupickupadownwardvelocity.也就是說你獲得了向下的速度

Asitacceleratesyoukeep隨著速度增加你會(huì)持續(xù)下落

fallingdownandpickingupmoreandmore速度也越來越大

speed.

Thesameacceleration'sexperiencedbythecart這里的車也有同樣的加速度

also,

andalittlemorebecauseTisalsopushingdown.而且加速度還大點(diǎn)因?yàn)門也是向下的

Buttherethechangeinvelocityoveratiny但這個(gè)在短時(shí)間內(nèi)變化的速度

amountoftime

isnotaddedtozerovelocity,并不是加在零速度上

buttothislargevelocityithasatthatinstant.而是加在此時(shí)的這個(gè)很大的速度上

Andthetinyincrementinvelocityifaddedthis如果速度像這樣有了一個(gè)小的增量

way

willproduceavelocityinthatdirection,則將產(chǎn)生一個(gè)在這個(gè)方向上的速度

andwhatthatreallymeansifyouwaitasmall這就表明如果你再等一小會(huì)兒

amountoftime,

thisthingisgoingalongthenewvelocity這個(gè)物體就會(huì)沿著新的速度方向運(yùn)動(dòng)

direction.

So,that'sthedifferencebetween-這就是兩種情況的不同

Sotheaccelerationisthesameforafallingapple在蘋果問題和過山車問題里

asfortheloop-the-loopcart.加速度的情況是相同的

Butintheapple'scase,但在蘋果模型里

itisaddedtozeroinitialvelocity加速度是加在為零的初速度上

sotheaccelerationandthevelocityarethesame,因此加速度和速度一樣即都是向下的

namelydown.

Ifyouhaveahugevelocity如果初始速度就很大

towhichyouaddatinychangeinvelocity,又加上了一個(gè)小的速度改變

thedownwarddirection是向下方向的改變

yourotatethevelocityvector就需要旋轉(zhuǎn)速度矢量的方向了

thatjustmeansthatasitgoesaroundthecircle,這就意味著在它做圓周運(yùn)動(dòng)時(shí)

thevelocityvectorisconstantlytangenttothe速度矢量一直與圓軌道相切

circle.

There'sanotherwayinwhichpeopleunderstand還有另一種理解方式

this.

IfyouareontheEarthandyouarestanding,say,假如你在地球上站在比如說一棟大

onabuilding樓上

andyoufireagun,然后開了一槍

itgoeslikethatandithitstheground子彈應(yīng)該是這樣打到地上的

becauseofaccelerationduetogravity.因?yàn)橛兄亓σ鸬募铀俣?/p>

Ifyoufireitalittlefasterit'llgothere.若子彈的速度再快一些它就落在這里

There'llbeacertainspeed總會(huì)有一個(gè)速度

atwhichitkeepsfalling使得子彈一直在下落

butdoesn'tgetanyclosertothecenter.但不會(huì)接近圓心

Thatisinfacthowyoulaunchabulletintoorbit.這就是把子彈打進(jìn)軌道的方法

So,what'sthefirstthingyoushoulddowhenyou那么當(dāng)你開槍時(shí)你首先要做什么

firethisgun?

Moveawaybecauseit'sgoingtocomeback快閃開因?yàn)樗鼤?huì)飛回來

andgetyoufrombehind.并從后面擊中你

Alrightthat'saveryusefullesson.所以這是非常重要的一課

So,youmightthink,"Hey,1shotitinthat你可能想"我是向那個(gè)方向發(fā)射的

direction,

it'snotmyproblem."不關(guān)我事"

Likeeverythingelseit'llcomeback像所有事情一樣它會(huì)返回來

andgetyouwhenyouleastexpectit.在你意想不到之時(shí)擊中你

Thisishowyouhave";這就是你得到的結(jié)果

So,isthisthingfallingornot?那么物體是不是在下落呢

Itisconstantlyfalling,它確實(shí)在不斷下落

buttheEarthisfallingunderit,但地球也在它下方下落

soit'sendlesssearchtocometotheEarth.因此這是落回地球的無窮之旅

Whereasifyoudropthebullet,當(dāng)你向下扔子彈

it'lljustfallandit'llhittheground;它只會(huì)下落打在地上

that'swhatwealwaysthinkasdrop.和我們通常所認(rèn)為的"下落"

Okay,sothisissomething1didinarush,剛才這些內(nèi)容我上節(jié)課講得很匆忙

and1didn'twantthat我本不想這樣的

becausethisisreallythepartofphysicswhichis因?yàn)檫@的確是個(gè)不太直觀的物理問題

notintuitive,

and1thinkweallknowwhatthemainpointis.我想我們都知道關(guān)鍵問題在哪

Themainpointisvelocityisavector關(guān)鍵在于速度是一個(gè)矢量

anditsvaluecanchangeduetochangein可以因方向的變化而變化

direction,

notchangeinmagnitude.不一定需要有模長的變化

Yessir?請講

Okay,ifyouwaitacertaintime△t,假設(shè)你等待了At的忖間

thentheaccelerationisvA2/r,加速度為vA2/r

putthev,putther,getsomenumber,代入v代入r得到某個(gè)值

multiplybythe△t,1/10ofasecond.乘以At比如0.1秒

That'llbethechangeinthattime.就得到了這段時(shí)間對應(yīng)的變化量

So,inthattime因此在這段時(shí)間里

alsotheguywouldhavemovedoverinthecircle車應(yīng)該在圓圈內(nèi)挪動(dòng)一些

alittlebit,

justpreciselysothatatthatpoint正好使得那一點(diǎn)的

yournewvelocityvectoristangenttoyournew新的速度矢量與你的新位置相切

location.

Sothislittle△v1drewisnotafixednumber;我畫的這個(gè)小Av并不是某個(gè)確定值

itdependsonhowlongyouwanttowait.這取決于你打算等多久時(shí)間

Calculussayswaitaninfinitesimalamountof微積分要求經(jīng)過一個(gè)無窮短的時(shí)間

time,

inpracticewait1/10ofasecondandit'llbefine.實(shí)際中等0.1秒就夠了

Todaythereisavery,veryimportantconcept1今天我要引入的概念非常非常重要

wanttointroduce.

It'saveryrobustandpowerfulconcept;這是一個(gè)非常有活力非常強(qiáng)大的概念

ithastodowithenergy.是關(guān)于能量問題的

Itispowerful說它強(qiáng)大

becauseafterthelawsofquantummechanics是因?yàn)樵趤喸邮澜缋?/p>

werediscoveredinthesubatomicworld,發(fā)現(xiàn)量子力學(xué)的定律之后

webasicallygaveuponmanycherishednotions.我們原則上放棄了很多原來重視的理論

Youmusthaveheard你肯定聽說過

thatparticlescannothaveadefiniteposition在給定的某個(gè)時(shí)刻粒子不可能

anddefinitevelocityatagiventime.同時(shí)具有確定的位置和速度

Theydon'thaveanytrajectories.它們沒有確定的運(yùn)動(dòng)軌跡

Youseeaparticleherenow,你現(xiàn)在在這里看見一個(gè)粒子

thenyouseeitthere.然后你又在那兒看到它

That'struebecausethey'reobservedevents.這是真的因?yàn)榇_實(shí)有人觀察到過

Whathappensinbetween?在這期間發(fā)生了什么

Youmightthinktheyhaveatrajectorybutthey你可能認(rèn)為有軌跡其實(shí)沒有

don't.

Theforcesarenotwelldefined我們無法恰當(dāng)?shù)囟x作用力

becauseifyoudon'tknowwheretheguyis,因?yàn)槿绻悴恢牢矬w的位置

youdon'tknowwhatthevelocityis,就沒法知道它的速度

youdon'tknowwhattheaccelerationis.你不知道加速度是什么

SomostoftheideasofNewtonianmechanicsare因此大多數(shù)牛頓力學(xué)的思想都基本上被

basicallysurrendered,摒棄了

butthenotionofenergyturnsouttobevery而能量概念卻依然很有生命力

robust

andsurvivesallthequantumrevolution.并在量子革命中存活了下來

Infact,thereisaperiod事實(shí)上曾經(jīng)有段時(shí)間

whenpeoplearestudyingnuclearreactions當(dāng)人們研究核反應(yīng)時(shí)

andtheenergyyoubeganwithdidn'tseemtobe發(fā)現(xiàn)初態(tài)的能量似乎并不等于末態(tài)的能

theenergyyouendedupwith.hl

Sosomethingwasmissing.說明我們漏掉了什么

SoNeilsBohr,FatheroftheAtomsaid,因此尼爾斯?玻爾原子結(jié)構(gòu)學(xué)說之父說

過

"MaybetheLawofConservationofEnergyisnot"也許能量守恒定律是錯(cuò)的"

valid."

ThenPaulisaid,但是泡利說

"1think1putmymoneyontheLawof"我賭能量守恒定律是對的

ConservationofEnergy.

1wouldpostulatethat我假設(shè)

someothertinyparticlesthatyouguyscannot有某種肉眼看不到的小質(zhì)點(diǎn)

see,

that'scarryingawayallthemissingenergy."是它們帶走了所有損失的能量"

Thatwasaveryradicalthingtodointhosedays這在當(dāng)時(shí)是個(gè)非常激進(jìn)的觀點(diǎn)

whenpeopledidnotlightlypostulatenew那時(shí)人們不輕易假設(shè)新粒子的存在

particles.

Nowadays,ifyoudon'tpostulateaparticle若放在當(dāng)今你不假設(shè)存在新的粒子

youdon'tgetyourPhDinparticlephysics.你就拿不到粒子物理的博士學(xué)位

Inthosedaysitwasveryradical,但在那時(shí)這是很激進(jìn)的

andtheparticlewasnotseenformany,many泡利說的這種粒子很多年都沒有被發(fā)現(xiàn)

years.

Itwascalledaneutrino.人們稱它為中微子

Nowadays,neutrinosareoneofthemostexciting當(dāng)今社會(huì)中微子是最熱門的研究之一

thingsonecouldstudy.

There'salotofmysteriesintheuniverse有很多宇宙的未解之謎與中微子有關(guān)

connectedwithneutrinos.

Okay,soenergyisaconceptyou'regoingtolearn.那么能量是你們今天要學(xué)習(xí)的概念

Here'sagainaplacewhere1askyou,這里又要我要求你們

don'tmemorizetheformulasandstartplugging不要只是記方程代數(shù)據(jù)

themin.

Trytofollowthelogicbywhichyouaredrivento試著跟學(xué)會(huì)這種概念推導(dǎo)的邏輯

thisnotion,

becausethenit'llbelessluggageforyoutocarry因?yàn)檫@樣你的記憶負(fù)擔(dān)要小一點(diǎn)

inyourhead.

AndI'lltrytomakethenotionasnaturalas我試著盡量自然地把這個(gè)概念傳授給你

possible們

ratherthansaying,"Thisismydefinitionof而不是只說"能量是這樣定義的

energy,

thisisthedefinitionofkineticenergy."動(dòng)能是這樣定義的"

Let'saskwheredoesallthatcomefrom.讓我們探求一下結(jié)果是從何而來的

Howmanypeoplehaveseenkineticandpotential有多少人之前就知道動(dòng)能和勢能

energiesinyourpast?

Okay,ifyouhaven'tseenitthat'sfine,若你之前不知道沒關(guān)系

thisisnotsomethingyou'resupposedtoknow.不要求你一定要知道這個(gè)

Ifyouknowitandyoulearneditproperlyit'llbe若你知道并正確地理解它會(huì)很有幫助

helpful,

butyoudon'thavetoknowthat.不過你不必非得知道

Youjusthavetoknowwhatwehavedonesofar.你只需要知道現(xiàn)在為止我們做了什么

Solet*sgobackandaskasimplequestion.那么我們來回顧一下問個(gè)簡單的問題

Whenaforceactsonabody,當(dāng)一個(gè)外力作用在物體上時(shí)

whatisitsmaineffect?對它主要有什么影響

Whatdoesitdo?可以讓物體做什么

Yes?請講

Bemoreprecisebyitwhenyousay"changesits更精確點(diǎn)闡述"改變它的運(yùn)動(dòng)狀態(tài)"

motion."

Whatdoesitmeantochangeitsmotion?改變運(yùn)動(dòng)狀態(tài)意味著什么

Yes?請講

Right,that'sfromNewton'sLaw;對了這是依據(jù)牛頓定律

thatmeansitchangesthevelocity.說明這個(gè)力改變了物體的速度

Solet'sforamoment,讓我們小結(jié)一下

infactfortheentirelecturetoday,事實(shí)上今天的全部課程內(nèi)容

gobacktoonedimension.都回到了i維空間

Then,wewilldothewholebusinessinhigher之后我們將在多維里進(jìn)行整體的探討

dimensions.

Inonedimensionchangingthevelocitysimply在一維空間里改變速度意味著提速

meansspeedingup.

Sobodiesaregoingtospeedupandforcesacton因此外力作用于物體物體會(huì)加速

them.

Thatweknow.這是我們清楚的

We'regoingtofindtherelation我們現(xiàn)在要弄清楚

betweenhowmuchspeedisaccumulated當(dāng)物體受到力的作用時(shí)

whenaforceactsonabody;速度是如何增加的

We'regoingtoderivethat.我們要把它推導(dǎo)出來

So,letustakeanexample舉一個(gè)例子

wheretheforceisinthexdirectionanditissome假設(shè)一個(gè)在X軸方向的恒定外力

constant.

"Constant"meaningit'snotvaryingwithtime;"恒定"的意思是不隨時(shí)間變化

somebodyjusthastojustapply10newtons有人用10牛的恒力推物體

constantlytoabody.

Whatdoesthatdo?這會(huì)怎么樣

ThatproducesanaccelerationwhichisF/m,這會(huì)產(chǎn)生一個(gè)加速度即F/m

right?That'sNewton'sLaw.對吧這是牛頓定律

That*saconstantacceleration.這是個(gè)恒定的加速度

So,let'sgobacktoLectureNo.l.我們回顧一下第一課的內(nèi)容

LectureNo.lwelearned在第一課里我們知道

thatifabodyhasconstantacceleration,如果一個(gè)物體有著恒定的加速度

thenthevelocityattheendisthevelocity那么末速度等于初速度的平方

squaredatthebeginning

plus2timesatimesthedistancetraveled.加上2乘以a乘以運(yùn)動(dòng)路程

That*swhatwelearnedlongback,那是我們很久以前就學(xué)過的

thisisjustkinematics.這是從運(yùn)動(dòng)學(xué)角度講的

Inthosedayswedidn'task,在那時(shí)我們從來不問

"Whyisithavingaconstantacceleration?""為什么一定是一個(gè)恒定的加速度"

Youwerejusttold,"It^aconstantacceleration,你只知道"加速度恒定繼續(xù)講"

justgoahead."

Nowthatwelearneddynamics,現(xiàn)在我們學(xué)了動(dòng)力學(xué)

weknowaccelerationhasacause,我們知道加速度的產(chǎn)生是有原因的

namelyacertainforce.就是一個(gè)恒定的外力

So,I'mgoingtowritehere我要寫在這里

asyoucanimagineF/mintheplaceofa,你可以想到F/m就是a

butI'mgoingtomakeonemorecosmeticchange但我現(xiàn)在要對這個(gè)符號進(jìn)行一下改動(dòng)

innotation.

Inalltheseenergyformulas,在所有能量方程中

ratherthancallingtheinitialvelocityorinitial初速度或任何初始量的下標(biāo)為0

anything

withsubscriptzeroandthefinalisnosubscript,而末態(tài)量沒有下標(biāo)

wearegoingtouseanotationwhere我們要用符號標(biāo)注一下

thefinalquantitiesaredenotedwithasubscript令末態(tài)量的下標(biāo)為2

2,

initialquantitiesaredenotedbyasubscript1.初始量的下標(biāo)為1

Inotherwords,initially,thebodyhadavelocityof換種說法開始物體的速度為

vl,v{\fsl0}l{\fsl8}

andthenithadanaccelerationaforadistanced,然后在d距離里有一個(gè)加速度a

thentherelationbetweenfinalinitialvelocitiesis那么初速度和末速度的關(guān)系是這樣的

this.

Solet'sputthatF/mhereandlet'sputad.代入F/m再乘以d

Sinceweareinterestedintheeffectoftheforce,因?yàn)槲覀儗ν饬λa(chǎn)生的影響有興趣

wewanttoseewhattheforcedidtothevelocity.于是我們想看看外力對速度有什么影響

Whatyoufindhere在這里你發(fā)現(xiàn)

isthatifyouwanttofindthechangeinthe如果你想知道速度的改變量

velocity,

ofcourseyoushouldtakethisguytotheleft-hand當(dāng)然你得把這?項(xiàng)弄到左邊來是吧

side,right?

Takeittotheothersideandyouwanttoisolate把它移到另一邊你需要求出力F

theforce,

sobringm/2totheleft-handside.將m/2乘到左邊

So,dothatinyourheadandwhatdoyouget?那么心算一下你會(huì)得到什么結(jié)果

(m/2)?v2A2-(m/2)?vlA2=F,d.(m/2)?v{\fslO}2{\fsl8)A2-(m/2)?

v{\fslO}l{\fsl8}A2=F?d

Thisisaveryimportantconcept.這是非常重要的概念

Itsays,whentheforceactsonabody這表明當(dāng)力作用于物體上

itchangesthevelocity,它會(huì)改變物體的運(yùn)動(dòng)速度

anditdependsonhowfartheforcehasbeen這還取決于在力的作用下運(yùn)動(dòng)了多少距

acting,離

howmanymetersyou'vebeenpushingthe你推著物體走了多少米

object.

It*sclearthatifyoudidn'tpushitatall,當(dāng)你一點(diǎn)都不推時(shí)情況很清楚

theneventheforcesacting,即使仍有外力作用

velocityhasn'thadtimetochange.速度也來不及發(fā)生變化

So,itseemstodependonhowfartheforceacted,看起來它取決于外力作用了多遠(yuǎn)

andthechangeisnotsimplyinvelocitybutin改變的不是速度本身而是速度平方

velocitysquared.

That*swhatcomesout;這就是結(jié)果

thenyourealizethatit'sthemostnaturalthingin你發(fā)現(xiàn)給這一項(xiàng)命名是世界上

theworld

togivethiscombinationanamebecausethat's最自然的事因?yàn)檫@就是我們得到的結(jié)

whatcomesoutofthis.果

Thatcombinationiscalledkineticenergy,這一部分就叫做動(dòng)能

andI'mgoingtocallitK2,我稱它為K{\fsl0}2{\fsl8}

meaningKattheendminusKatthebeginningis就是末態(tài)的K減去初始的K等于F?

Ftimesd.d

SoFtimesdisdenotedbythesymbolW,用W來表示F?d

andthat'scalledtheworkdonebyF.這個(gè)W表示F做的功

That'showweintroduceanotion這樣我們就介紹了一種表述

ofaforcetimesdistanceasthedefinitionofwork.功的定義就是力乘以距離

Thisishowpeople-youcansay,這就是人們怎樣也許你會(huì)問

"Whydidanyonethinkoftakingthecombination"為什么會(huì)有人想到讓F乘以d呢"

Ftimesd?H

Well,thisishowithappens.他就是這么想到的

So,youneedtohaveaunitforthis,你需要給這個(gè)一個(gè)單位

whichisactuallynewtonmeters,這實(shí)際上是牛頓米

butwearetiredofcallingeverythinganewton但每次都講牛頓米很煩

meter;

sowe'regoingtocallitajoule.所以我們叫它焦耳

SojouleorJ,isjustashorthandfornewton所以焦耳或者J是牛頓米的簡寫

meters.

Alright,soletusnowwritedownthisfollowing現(xiàn)在讓我們把接下來的表達(dá)式寫下來

expressionhere.

△K,△inanythingwedoisthechangeinanyAK任何符號中的A都是指數(shù)量的變

quantity,化

it'scalleddeltaofthatquantity,它叫這個(gè)量的變化量

AKisachangeinK△K是K的變化量

isequaltotheFtimesthedistancetraveled.它等于F乘以走過的距離

Letthedistancetraveledbesometinyamount△我們讓這個(gè)距離等于一個(gè)微小量Ax

X.

Thisformulaisvalidevenifdis1km,這條公式即使在d等于一千米時(shí)都適用

aslongasforceisconstant.只要力為恒定就行

ButI'msaying,let'sfocusonaproblem,但我是說讓我們專注于一個(gè)問題

whethertheforceisconstantornot,即物體在很短的時(shí)間里走過Ax的距離

overatinyintervalinwhichthebodymovesthe不管力是否恒等

distanceAx.

Then,wefindthatthechangeinkineticenergyis然后我們發(fā)現(xiàn)動(dòng)能變化量等于力乘以△

forcetimes△x.X

Now,whatifthereare36forcesactingonthe如果有三十六個(gè)力作用于物體上怎么辦

body?

Whichoneshould1use?我應(yīng)該用哪一個(gè)

I'mpullingandyou'repushingand—我在拉你在推還有

It'sgottobethenetforce我們要用合力

becauseNewton'sLawconnectsthenetforceto因?yàn)榕ｎD定律是把合力和加速度聯(lián)系起

theacceleration.來

Ifyouand1haveatugofwar如果我們拔河

andwecanceleachotherouttozero我們相互抵消使合力為零

thenthere'snoacceleration.那就沒有加速度

So,you'vegottogobacktothedefinition;所以你要回到定義上來

thisisvery,veryimportant.這非常非常重要

ThisFbetterbethetotalforce.這個(gè)F最好是合力

1shouldreallyuseasubscriptTfortotal,這里我應(yīng)該標(biāo)個(gè)下標(biāo)T表示全部

but1don'twanttodothatandwewriteF=但我不想這樣做我們寫F=m?a時(shí)

m?a;

itisunderstoodFiseverything.就應(yīng)該明白F是指合力

We'llcomebacklatertothenotionofwhat我們晚點(diǎn)會(huì)講

happens

ifFismadeupoftwoparts.F由兩部分組成時(shí)怎么辦

So,maybeI'lljustgiveyouasimpleexample.舉個(gè)簡單的例子

Here'sabody,right,這有個(gè)物體

andI'mpushingitthiswayandyou'repushingit我從這個(gè)方向推你從那個(gè)方向推

thatway,

let'ssaywithequalforce.兩個(gè)力相等

Andthebodycouldbeatrest這個(gè)物體可能靜止

orifthebodyhadinitialvelocity或者如果這個(gè)物體有初速度

it'llmaintaintheinitialvelocity它會(huì)保持以這個(gè)速度運(yùn)動(dòng)

becausevelocityisforfree.因?yàn)楸３炙俣仁遣恍枰饬Φ?/p>

Then,youcanask,"Whatdoesthistheoremsay?"你會(huì)問"這個(gè)定理說明了什么"

ThisiscalledaWorkEnergyTheorem.這叫功能原理

TheWorkEnergyTheoremsays,功能原理講的是

"Thechangeinenergyis"能量的變化量

isequaltotheworkdonebyalltheforces."等于所有力做功的總和"

Inthiscase,thereisnochangeinenergy,在這個(gè)問題里能量沒有改變

buttherearetwoforcesatworkthatcancel.而是做功的兩個(gè)力相互抵消

Then,itturnsout我們發(fā)現(xiàn)原來

thatitissensibletodefinetheworkdonebyme,分別定義你我所做的功會(huì)很有用

whichisequaltoFtimesthedistancetraveled,我做的功等于F乘以移動(dòng)的距離

andtheworkdonebyyouwillbeinfactgivena而你做的功則用負(fù)號表示

minussign.

So,whendoweattachaplussign那么我們什么情況下加正號

andwhendoweattachaminussign?什么情況下加負(fù)號呢

Ifyougobacktothewholederivation,如果你看回整個(gè)推導(dǎo)

itwasunderstoodthatawasapositivequantity.大家知道a是正數(shù)

Theneverythingworks.然后一切正常運(yùn)作

Butifthebodyismovingtotheright,但如果物體向右運(yùn)動(dòng)

andI'mpushingtotheright,我向右推

thentheworkdonebymeispositive.那么我做的功就是正功

Andifyouwerepushingtotheleft而如果你向左推

andthebodystillmovedtotheright,但物體依然向右運(yùn)動(dòng)

theworkdonebyyouisnegative.你做的就是負(fù)功

Inotherwords,換句話說

ifyougetyourway,namely,如果你能隨便推也就是

thingsmovethewayyou'repushing,你向哪邊推物體就向哪邊運(yùn)動(dòng)

theworkdonebyyouispositive.你做的功就是正功

Ifpeoplearepushingyoucountertoyourwill,如果有人跟你對著干

intheoppositedirectiontoyourforce,向你的反方向推

workdonebyyouisnegative.做的功就是負(fù)功

See,hereisasimpleexample.我們看一個(gè)簡單的例子

1haveapieceofchalk我有一支粉筆

andI'mliftingitatconstantvelocityfromthe我把它從地面勻速抬起

ground.

Itskineticenergyisnotchanging.它的動(dòng)能不變

Thatmeansthetotalworkdoneonitiszero,說明總功為零

butit'snotbecausetherearenoforcesonit.但那并不是因?yàn)闆]有作用力存在

Thereisgravityactingdown有方向向下的重力

andthereismecounteringgravitywithexactly還有我施加的與重力等大反向的力

therightamount.

I'mapplyingtheforcem,g.我施的力等于m-g

So,workdonebymewouldbepositive所以我做的功是正功

because1wantittogoup,因?yàn)槲蚁胨蛏弦苿?dòng)

anditgoesup.而它也確實(shí)在向上運(yùn)動(dòng)

So,ifitgoesupbyanamounth所以如果它向上移動(dòng)了距離h

workdonebymeism?g?h.我做的功就是m?g?h

Workdonebygravityinthemeantimeisminus同時(shí)重力所做的功等于負(fù)的m?g?h

m?g?h,

sothetotalworkdoneiszero.于是總功為零

So,thisformulasaysthisisthetotalforce,所以在這條公式里這是合力

butI'mtellingyouifyoutaketheworkdoneand但如果你把所做的功定義為AW

callitAW

itmaybearesultofmanyworks,它可能是許多力共同做功的結(jié)果

workdonebyme,workdoneyou,workdoneby我做的功你做的功和別人做的功

somebodyelse,

somearepositive,somearenegative,有些是正功有些是負(fù)功

youaddthemallup.你把它們都加起來

So,thatisthenotionofwork所以這表示

donebyoneofmanyforcesactingonabody.作用在物體上的其中一個(gè)力所做的功

Andthetotalworkdone而總功就是

isthealgebraicsumoftheworkdonebyallthe所有力所做的功的代數(shù)和

forces.

Alright,sowe'vegotthisresult,好了得到這個(gè)結(jié)果后

let'sdothefollowing.我們繼續(xù)下面的推導(dǎo)

Letusimagineallofthisoccursinatime△t.假設(shè)這些都發(fā)生在時(shí)間At里

Then△K/At,那么AK/At

whichwilleventuallybecomedK/dt會(huì)最終變成dK/dt

ifyoutakealltheproperlimits取適當(dāng)?shù)臉O限

becomesFtimesdx/dt,會(huì)得到F?(dx/dt)

whichisforcetimesvelocityandthat'scalled也就是力乘以速度我們稱之為功率

power.

So,poweristherateatwhichworkisdone.所以功率就是做功的快慢

Forexample,例如

if1climba12-storybuildingI'vedonesomework;如果我爬十二層高的樓我做的功

mymtimesgtimestheheightofthebuilding.等于我的m-g乘以樓高

1canclimbthebuildinginoneminute,我可能一分種內(nèi)爬上樓

1canclimbthebuildinginonehour;也可能要一小時(shí)

theworkdoneisthesame,做的功是一樣的

butthepowerisameasureofhowrapidlyworkis但功率是用來表達(dá)做功的快慢

done.

That'swhyit'stheproductofforceandvelocity.這就是為什么它是力和速度的乘積

So,theunitsforpowerarejoulespersecond;因此功率的單位是焦耳每秒

now,thatalsohasanewname,它也有一個(gè)新名字

whichiswattorjustW.就是瓦特或者W

Youmayuseakilowatt,你也會(huì)用到千瓦

whichiskW,也就是kW

littlekthenWisthousandwatts.k接上W就是一千瓦特

So,ifyou'vegota60-wattbulb,所以如果你有一個(gè)六十瓦的燈泡

it'sconsumingenergyattherateof60joulesper表示它每秒消耗六十焦耳的能量

second.

That'sthemeaningofa60-wattbulb.這就是六十瓦燈泡的意思

Okay,nowI'mgoingtodothenextgeneralization,我接下來要把這些內(nèi)容再推廣一下

whichiswhentheforceisnotaconstantbut就是當(dāng)力隨著X改變而不是恒定值的情

varieswithx.