MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physcs 8.2 Revew C: Work and Knetc Energy C. Energy... 2 C.. The Concept o Energy... 2 C..2 Knetc Energy... 3 C.2 Work and Power... 4 C.2. Work Done by Constant Forces... 4 C.2.2 Work and the Dot Product... 5 C.2.3 Work done by Non-Constant Forces... 6 C.2.4 Work Done Along an Arbtrary Path... 7 C.2.5 Power... C.3 Work and Energy... C.3. Work-Knetc Energy Theorem... C.3.2 Work-Knetc Energy Theorem or Non-Constant Forces... C.3.3 Work-Knetc Energy Theorem or a Non-Constant Force n Three Dmensons... 2 C.3.4 Tme Rate o Change o Knetc Energy... 3 C-
Work and Knetc Energy C. Energy C.. The Concept o Energy The concept o energy helps us descrbe many processes n the world around us. Fallng water releases stored gravtatonal potental energy turnng nto a knetc energy o moton. Ths mechancal energy can be used to spn turbnes and alternators dong work to generate electrcal energy. It's sent to you along power lnes. When you use any electrcal devce such as a rergerator, the electrcal energy turns nto mechancal energy to make the rergerant low to remove heat (the knetc moton o atoms), rom the nsde to the outsde. Human bengs transorm the stored chemcal energy o ood nto varous orms necessary or the mantenance o the unctons o the varous organ system, tssues and cells n the body. Ths catabolc energy s used by the human to do work on the surroundngs (or eample pedalng a bcycle) and release heat. Burnng gasolne n car engnes converts chemcal energy stored n the atomc bonds o the consttuent atoms o gasolne nto heat that then drves a pston. Wth gearng and road rcton, ths moton s converted nto the movement o the automoble. Stretchng or compressng a sprng stores elastc potental energy that can be released as knetc energy. The process o vson begns wth stored atomc energy released as electromagnetc radaton (lght) that s detected by ectng atoms n the eye, creatng chemcal energy. When a proton uses wth deuterum, (deuterum s a hydrogen atom that has an etra neutron along wth the proton n the nucleus), helum three s ormed (two protons and one neutron) along wth radant energy n the orm o photons. The mass o the proton and deuterum are greater than the mass o the helum. Ths mass energy s carred away by the photon. These energy transormatons are gong on all the tme n the manmade world and the natural world nvolvng derent orms o energy: knetc energy, gravtatonal energy, heat energy, elastc energy, electrcal energy, chemcal energy, electromagnetc energy, George B. Benedk and Fel M.H. Vllars, Physcs wth Illustratve Eamples rom Medcne and Bology Volume Mechancs, Addson-Wesley, Readng, 973, p. 5-6. C-2
nuclear energy, or mass energy. Energy s always conserved n these processes although t may be converted rom one orm nto another. Any physcal process can be characterzed by an ntal state that transorms nto a nal state. Each orm o energy undergoes a change durng ths transormaton, E = (C..) E Enal, Ental, Conservaton o energy means that the sum o these changes s zero, E + E + = E = 2 N = (C..2) Two crtcal ponts emerge. The rst s that only change n energy has meanng. The ntal or nal energy s actually a meanngless concept. What we need to count s the change o energy and so we search or physcal laws that determne how each orm o energy changes. The second pont s that we must account or all the ways energy can change. I we observe a process, and the changes n energy do not add up to zero, then the laws or energy transormatons are ether wrong or there s a new type o change o energy that we had not prevously dscovered. Some quantty s conserved n all processes and we call that energy. I we can quanty the changes o orms o energes then we have a very powerul tool to understand nature. We wll begn our analyss o conservaton o energy by consderng processes nvolvng only a ew orms o changng energy. We wll make assumptons such as gnore the eects o rcton. Ths means that rom the outset we assume that the change n heat energy s zero. Energy s always conserved but sometmes we preer to restrct our attenton to a set o objects that we dene to be our system. The rest o the unverse acts as the surroundngs. Our conservaton o energy then becomes E + = (C..3) system Esurroundngs C..2 Knetc Energy Our rst orm o energy that we wll study s the knetc energy K, an energy assocated wth the moton o an object wth mass m. Let s consder a car movng along a straght road (call ths road the -as) wth velocty v = v ˆ. The speed v o the car s the magntude o the velocty. The knetc energy o the car s dened to be the postve scalar quantty K 2 2 = mv (C..4) C-3
Note that the knetc energy s proportonal to the square o the speed o the car. The SI 2-2 unt or knetc energy s kg m s ; ths combnaton unts s dened to be a joule and 2 2 s denoted by [J]. Thus J kg m s. Let s consder a case n whch our car changes velocty. For our ntal state, the car moves wth an ntal velocty v = v,ˆ along the -as. For the nal state (some tme later), the car has changed ts velocty and now moves wth a nal velocty v = v ˆ. Thereore the change n the knetc energy s, 2 K = mv mv 2 2 2 (C..5) C.2 Work and Power C.2. Work Done by Constant Forces We begn our dscusson o the concept o work by analyzng the moton o a rgd body n one dmenson acted on by constant orces. Let s consder an eample o ths type o moton: pushng a cup orward wth a constant orce along a desktop. When the cup changes velocty and hence knetc energy, the sum o the orces actng on the cup must be non-zero accordng to Newton s Second Law. There are three orces nvolved n ths moton, the appled pushng orce F appled, the contact orce C= N+ k, and gravty, F = m g. The orce dagram s shown n Fgure C.2.. grav Fgure C.2.: Force dagram on a cup. Let s choose our coordnate system so that the + -drecton s the drecton o moton o the cup orward. Then the pushng orce can be descrbed by, F appled = F appled, ˆ (C.2.) C-4
Denton: Work done by a Constant Force Suppose a body moves n a straght lne rom an ntal pont to a nal pont so that the dsplacement o the cup s postve, >. The work W done by the constant orce F appled actng on the body s the product o the component o the orce Fappled, and the dsplacement, W = F (C.2.2) appled appled, Work s a scalar quantty; t s not a vector quantty. The SI unts or work are joules snce[ N m] = [ J]. Note that work has the same dmenson as knetc energy. I our appled orce s along the drecton o moton, both F > and >, so the appled, work done s just the product o the magntude o the appled orce wth the dstance moved and s postve. We can etend the concept o work to orces that oppose the moton, such as rcton. In our eample o the movng cup, the rcton orce s = ˆ = µ Nˆ = µ mg ˆ k k k (C.2.3) Here the component o orce s n the opposte drecton as the dsplacement. The work done by the rcton orce s negatve, W rcton = µ mg (C.2.4) k Snce the gravtatonal orce s perpendcular to the moton o the cup, t has no component along the lne o moton. Thereore, gravty does zero work on the cup when the cup s sld orward n the horzontal drecton. The normal orce s also perpendcular to the moton, hence does no work. In summary, the gravtatonal orce and the normal orce do zero work, the pushng orce does postve work, and the rcton orce does negatve work. C.2.2 Work and the Dot Product A very mportant physcal eample o the dot product o two vectors s work. Recall that when a constant orce acts on a mass that s movng along the -as, only the component o the orce along that drecton contrbutes to the work, W = F (C.2.5) C-5
For eample, suppose we are pullng a mass along a horzontal surace wth a orce F. Choose coordnates such that horzontal drecton s the -as and the orce F orms an angle β wth the postve -drecton. In Fgure C.2.2 we show the orce vector ˆ F= F + F ĵ and the dsplacement vector = ˆ. Note that s the component o y the dsplacement and hence can be greater, equal, or less than zero. Fgure C.2.2 Force and dsplacement vectors Then the dot product between the orce vector F and the dsplacement vector s F = ( F ˆ+ F ˆj) ( ˆ) = F y (C.2.7) Ths s the work done by the orce, W = F (C.2.9) The orce F orms an angle β wth the postve -drecton. The angle β takes values wthn the range π β π. Snce the -component o the orce s F = Fcosβ where F = F denotes the magntude o F, the work done by the orce s W = F = ( Fcos β ) (C.2.) C.2.3 Work done by Non-Constant Forces Consder a mass movng n the -drecton under the nluence o a non-unorm orce that s pontng n the -drecton, F= Fˆ. The mass moves rom an ntal poston to a nal poston.in order to calculate the work done by a non-unorm orce, we wll dvde up the dsplacement nto a large number N o small dsplacements where the nde denotes the th dsplacement and takes on nteger values rom to N, wth C-6
N =. Let ( F ) denote the average value o the -component o the orce n the nterval [ ]. For the th dsplacement, the contrbuton to the work s, W = ( F ) (C.2.) Ths contrbuton s a scalar so we add up these scalar quanttes to get the total work; = N = N N = = ( ) (C.2.3) = = W W F Ths depends on the number o dvsons N. In order to dene a quantty that s ndependent o the dvsons, take the lmt as N and. Then the work s = N W = lm ( F ) = = F d N = = (C.2.5) Ths last epresson s the denton o the ntegral o the -component o the orce wth respect to the parameter. In Fgure C.2.3, the graph o the -component o the orce wth respect to the parameter s shown. The work ntegral s the area under ths curve. Fgure C.2.3 Graph o -component o the orce as a uncton o C.2.4 Work Done Along an Arbtrary Path Now suppose that a non-constant orce F acts on an object o mass m whle the object s movng on a three-dmensonal curved path. The poston vector o the partcle at tme t wth respect to a choce o orgn s r () t. In Fgure C.2.4, the orbt o the object s shown o the object or a tme nterval, t t, movng rom an ntal poston r r( t = t) at tme t = t to a nal poston r r ( t = t ) at tme t = t. C-7
Fgure C.2.4 Orbt o the mass. We dvde the tme nterval t, t nto N small peces wth t N = t. Each ndvdual pece s labeled by the nde takng on nteger values rom to N. Consder two poston vectors r r ( t = t) and r ) r( t = t markng the and poston. The dsplacement r s then r = r r. Let F denote the average orce actng on the mass durng the nterval [ t, t]. We can locate the orce n the mddle o the path between r and r. The average work W done by the orce durng the tme nterval [ t, t] s the dot product between the average orce vector and the dsplacement vector correspondng to the product o the component o the average orce n the drecton o the dsplacement wth the dsplacement, W = F r (C.2.6) The orce and the dsplacement vectors or the tme nterval [ t t ] Fgure C.2.5., are shown n Fgure C.2.5 Inntesmal work dagram The work done s ound by addng these scalar contrbutons to the work or each nterval [ t, t], or = to N, W = N = N = W = F r N = = (C.2.7) C-8
We would lke to dene work n a manner that s ndependent o the way we dvde the nterval so we take the lmt as N and r. In ths lmt as the ntervals become smaller and smaller, the dstncton between the average orce and the actual orce becomes vanshngly small. Thus ths lmt ests and s well dened, then the work done by the orce s W = N r = lm F r = F dr N r = r (C.2.9) Notce that ths summaton nvolves addng scalar quanttes. Ths lmt s called the lne ntegral o the tangental component o the orce F. The symbol d r s called the nntesmal vector lne element. At tme t, dr s tangent to the orbt o the mass and s the lmt o the dsplacement vector r = r ( t+ t) r ( t) as t approaches zero. In general, ths lne ntegral depends on the partcular path the object takes between the ntal poston r and the nal poston r. The reason s that the orce F s non-constant n space and the contrbuton to the work can vary over derent paths n space. We can represent ths ntegral eplctly n a coordnate system by specyng the nntesmal vector lne element d r and then eplctly computng the dot product. For eample n Cartesan coordnates the lne element s dr = dˆ+ dyˆj + dzkˆ (C.2.2) where d, dy, and dz represent arbtrary dsplacements n the, y, and z -drectons respectvely as seen n Fgure C.2.6. Fgure C.2.6 Lne element n Cartesan coordnates The orce vector can be represented n vector notaton by F= F ˆ+ F ˆj+ F kˆ y z (C.2.2) Then the nntesmal work s the dot product C-9
dw = F dr = ( F ˆ+ F ˆj+ F kˆ) ( dˆ+ dyˆj+ dzkˆ) y z (C.2.23) dw = Fd + Fy dy + Fz dz (C.2.24) so the total work s r= r r= r r= r r= r r= r W = F dr = F d + F dy + F dz = F d + F dy + F dz (C.2.25) ( ) y z y z r= r r= r r= r r= r r= r The above equaton shows that W conssts o three separate ntegrals. In order to calculate these ntegrals n general we need to know the specc path the object takes. C.2.5 Power Denton: Power by a Constant Force: Suppose that an appled orce F appled acts on a body durng a tme nterval t, and dsplaces the body n the -drecton by an amount. The work done, W, durng ths nterval s W = Fappled, (C.2.26) where F appled, s the -component o the appled orce. The average power o the appled orce s dened to be the rate o dong work. W t F appled, ave = = = Fappled, v,ave P t (C.2.27) The average power delvered to the body s equal to the component o the orce n the drecton o moton tmes the average velocty o the body. Power s a scalar quantty and can be postve, zero, or negatve dependng on the sgn o 2 3 work. The SI unts o power are called watts [ W ] and [ W] J s kg m s. The nstantaneous power at tme t s dened to be the lmt o the average power as the t, t approaches zero, tme nterval [ ] W Fappled, P= lm = lm = Fappled, lm = F t t t t t t appled, v (C.2.28) C-
the nstantaneous power o a constant appled orce s the product o the orce and the nstantaneous velocty o the movng object. C.3 Work and Energy C.3. Work-Knetc Energy Theorem There s a connecton between the total work done on an object and the change o knetc energy. Non-zero total work mples that the total orce actng on the object s non-zero. Thereore the object wll accelerate. When the total work done on an object s postve the object wll ncrease ts speed. When the work done s negatve, the object wll decrease ts speed. When the total work done s zero, the object wll mantan a constant speed. In act we have a more precse result, the total work done by all the appled orces on an object s equal to the change n knetc energy o the object. W = K = mv mv 2 2 2 2 total (C.3.) C.3.2 Work-Knetc Energy Theorem or Non-Constant Forces The work-knetc energy theorem holds as well or a non-constant orce. Recall that the denton o work done by a non-constant orce n movng an object along the -as rom an ntal poston to the nal poston s gven by W = F d (C.3.2) F where s the component o the orce n the -drecton. Accordng to Newton s Second Law, F dv m dt = (C.3.3) Thereore the work ntegral can be wrtten as dv d W = Fd = m d = m dv dt dt (C.3.4) Snce the -component o the velocty s dened as v becomes = d dt, the work ntegral d W = m dv = mv dv dt v, v, v (C.3.5), v, C-
Note that the lmts o the ntegral have now be changed. Instead o ntegratng rom the ntal poston to the nal poston, the lmts o ntegraton are rom the ntal - component o the velocty v, to the nal -component o the velocty v,. Snce 2 d mv = mvdv 2 (C.3.6) the ntegral s It ollows that W = v, mv dv = v, d 2 v, v, 2 mv (C.3.7) W v, mv dv v, d 2 2 2 v,,, v, 2 mv = = = 2 mv 2 mv = K (C.3.8) C.3.3 Work-Knetc Energy Theorem or a Non-Constant Force n Three Dmensons The work energy theorem generalzed to three-dmensonal moton. Suppose under the acton o an appled orce, an object changes ts velocty rom an ntal velocty v = v ˆ + v ˆ j+ v k ˆ, y, z, (C.3.9) to a nal velocty v = v ˆ + v ˆ j+ v k ˆ, y, z, (C.3.) The knetc energy s 2 2 2 2 K = mv = m( v + vy + vz ) (C.3.) 2 2 Thereore the change n knetc energy s 2 2 2 2 2 2 2 2 K = mv mv = m( v, + vy, + vz, ) m( v, + vy, + v z, ) (C.3.2) 2 2 2 2 The work done by the orce n three dmensons s C-2
r= r W = F dr = F d+ F dy+ F dz z r= r r r r r y r r (C.3.3) As beore,we can apply Newton s Second Law to each ntegral separately usng The work s then dv dv dv F m, F m, F m dt dt dt y z = y = z = (C.3.4) dv dv dv dt dt dt r d r dy r dz = mdv + mdvy + mdv dt r r dt r z dt r r y r z W = m d + m dy + m dz r r r (C.3.5) Ths becomes r r r r y y r r z z W = m dv v + m dv v + m dv v (C.3.6) These ntegrals can be ntegrated eplctly yeldng the work-knetc energy theorem 2 2 2 2 2 2 W = mv, mv, + mvy, mvy, + mvz, mvz, = K (C.3.7) 2 2 2 2 2 2 C.3.4 Tme Rate o Change o Knetc Energy In one dmenson, the tme rate o change o the knetc energy, dk d 2 dv = mv = mv = mva = Fv dt dt 2 dt (C.3.8) snce by Newton s Second Law, total F = m a (C.3.9) the tme dervatve o the knetc energy s equal to the nstantaneous power delvered to the body, dk = Fv dt = P (C.3.2) The generalzaton to three dmensons becomes C-3
dk = Fv + Fv Fv P y y + z z = F v= dt (C.3.2) C-4