A.P. Physics Fis Semese Reiew Shee Fll, D. Wicks Chpe : Inducin Physics Reiew ypes zes nd he ules signiicn digis Reiew mss s. weigh, pecisin s. ccucy, nd dimensinl nlysis pblem sling. Chpe : One-Dimensinl Kinemics A. Velciy Dx x - xi Equins ege elciy: e = = D - i nd e = ( + i ) In psiin-esus-ime gph cnsn elciy, he slpe he line gies he ege elciy. See Tble. Insnneus elciy cn be deemined m he slpe line ngen he cue picul pin n psiin-esus-ime gph. Use e = Δx Tl Δ Tl ius legs he juney. B. Accelein Equin ege ccelein: clcule he ege elciy n enie juney i gien inmin bu he e D - i = = D - In elciy-esus-ime gph cnsn ccelein, he slpe he line gies ccelein nd he e unde he line gies displcemen. See Tble. Accelein due giy = g = 9.8 m/s. (Recll = - g = -9.8 m/s ) i Psiin Vesus Time: Tble : Gphing Chnges in Psiin, Velciy, nd Accelein Cnsn Psiin Cnsn Velciy Cnsn Accelein Bll Thwn Upwd x x Slpe = e x x Velciy Vesus Time: Slpe = e Slpe = -9.8 m/s Accele -in Vesus Time: = -9.8m/s
A.P. Physics Fis Semese Reiew Shee, Pge Tble : Cmping he Kinemic Equins Kinemic Equins Missing Vible x= x + e = + D x x= x + + inl = + D x D Chpe 3: Vecs in Physics A. Vecs Vecs he bh mgniude nd diecin whees scls he mgniude bu n diecin. Exmples ecs e psiin, displcemen, elciy, line ccelein, ngenil ccelein, cenipel ccelein, pplied ce, weigh, nml ce, icinl ce, ensin, sping ce, mmenum, giinl ce, nd elecsic ce. Vecs cn be med pllel hemseles in digm. Vecs cn be dded in ny de. See Tble 3 ec ddiin. F ec ngle q he x-xis, he x- nd y-cmpnens cn be clculed m Δx = csθ nd Δy = sinθ. The mgniude ec is = Δx + Δy nd he diecin ngle elie he Δy nees x-xis is θ = n Δx. T subc ec, dd is ppsie. Muliplying diiding ecs by scls esuls in ecs. In ddiin dding ecs mhemiclly s shwn in he ble, ecs cn be dded gphiclly. Vecs cn be dwn scle nd med pllel hei iginl psiins in digm s h hey e ll psiined hed--il. The lengh nd diecin ngle he esuln cn be mesued wih ule nd pc, especiely. B. Relie Min Relie min pblems e sled by specil ype ec ddiin. F exmple, he elciy bjec elie bjec 3 is gien by 3 = + 3 whee bjec cn be nyhing. Subscips n elciy cn be eesed by chnging he ec s diecin: =
A.P. Physics Fis Semese Reiew Shee, Pge 3 Vec Oienin Vecs e pllel: Vecs e pependicul: Tble 3: Vec Addiin Clculinl Segy Used Add subc he mgniudes (lues) ge he esuln. Deemine he diecin by inspecin. Use he Pyhgen Theem, D x +D y =, ge he esuln,, whee D x is pllel he x-xis nd D y is pllel he y-xis. - ædy ö Use q = n ç ge he ngle, q, mde wih he x-xis. èdx ø Vecs e neihe pllel n pependicul: Adding Vecs Limied useulness () Use he lw csines deemine he esuln: c = + b - bcsq () Use he lw sines help deemine diecin: = b = c sin A sin B sin C Adding Me Vecs (Vec Resluin Mehd) Used by ms physiciss () Mke digm. () Cnsuc ec ble. (Use ec, x-diecin, nd y-diecin he clumn hedings.) (3) Resle ecs using Δx = csθ nd Δy = sinθ when needed. (4) Deemine he signs. (5) Deemine he sum he ecs ech diecin, D x l nd D y. l (6) Use he Pyhgen Thm ge he esuln, : D x +D y = l l æd - y ö l (7) Use q = n ç èdxl ø ge he ngle, q.
A.P. Physics Fis Semese Reiew Shee, Pge 4 Chpe 4: Tw-Dimensinl Kinemics A. Pjecile Min See Tble 4 bee undesnd hw he pjecile min equins cn be deied m he kinemic equins. The kinemic equins inle ne-dimensinl min whees he pjecile min equins inle w-dimensinl min. Tw-dimensinl min mens hee is min in bh he hiznl nd eicl diecins. Recll h he equin hiznl min (ex. D x= xd ) nd he equins eicl min (ex. y, =-gd, ( ) D y=- g D, y, =-gd y) e independen m ech he. Recll h elciy is cnsn nd ccelein is ze in he hiznl diecin. Recll h ccelein is g = 9.8 m/s in he eicl diecin. When pjeciles e lunched n ngle, he nge he pjecile is en clculed m Δx = ( i csθ)δ nd is ime ligh is en clcule m D y= ( i sin q) D- g( D ). æ g ö Pjeciles llw pblic phwy gened by y= h-ç x è ø Tble 4: Relinship Beween he Kinemic Equins nd Pjecile Min Equins Kinemic Equins Missing Vible Pjecile Min, Ze Lunch Angle Pjecile Min, Genel Lunch Angle Assumpins mde: Assumpins mde: =- g nd, = 0 =- g,, = csq, y x nd, = sinq y x= x + e D x= x whee = cns. x D x= ( cs q) whee = cns. x = + D x y =- g = sinq - g y x= x + + inl D y =- g D y= ( sin q) - g = + D x D y =-gd y = sin q -gd y y F n bjec in ee ll, he bjec sps cceleing when he ce i esisnce, F Ai, equls he weigh, W. The bjec hs eched is mximum elciy, he eminl elciy. When quebck hws bll, he ngle high, lb pss is eled he ngle lw, bulle pss. When bh blls e cugh by eceie snding in he sme plce, he sum he lunch ngles is 90. In disnce cness pjeciles lunched by cnnns, cpuls, ebuches, nd simil deices, pjeciles chiee he hes disnce when lunched 45 ngle.
A.P. Physics Fis Semese Reiew Shee, Pge 5 The nge pjecile lunched iniil elciy nd ngle q is The mximum heigh pjecile be is lunch sie is y mx = æ ö R = ç sin q è g ø sin q g Chpe 5: Newn s Lws Min Newn s Fis Lw: (Lw Inei) Recll h mss is mesue inei. Newn s Secnd Lw: Newn s Thid Lw: Recll cin-ecin pis Tble 5: Newn s Lws Min Mden Semen Lw I he ne ce n n bjec is ze, is elciy is cnsn. An bjec mss m hs n ccelein gien by he ne ce F diided by m. Th is F = m F eey ce h cs n n bjec, hee is ecin ce cing n dieen bjec h is equl in mgniude nd ppsie in diecin. Tnslin An bjec es will emin es. An bjec in min will emin in min cnsn elciy unless ced upn by n exenl ce. F = m ne F eey cin, hee is n equl bu ppsie ecin. A. Suey Fces A ce is push pull. The uni ce is he Newn (N); N = kg-m/s See Newn s lws min in Tble 5. Cmmn ces n ming bjec include n pplied ce, icinl ce, weigh, nd nml ce. Cnc ces e cin-ecin pis ces pduced by physicl cnc w bjecs. Reiew clculins egding cnc ces beween w me bxes. Field ces like giinl ces, elecsic ces, nd mgneic ces d n equie diec cnc. They e sudied in le chpes. Fces n bjecs e epesened in ee-bdy digms. They e dwn wih he ils he ecs igining n bjec s cene mss. Weigh, W, is he giinl ce exeed by Eh n n bjec whees mss, m, is mesue he quniy me in n bjec (W = mg). Mss des n depend n giy. Appen weigh, W, is he ce el m cnc wih he l scle in n cceleing sysem. F exmple, he sensin eeling heie lighe in n cceleing ele. The nml ce, N, is pependicul he cnc suce lng which n bjec mes is cpble ming. Thus, n bjec n leel suce, N nd W e equl in size bu ppsie in diecin. Hwee, n bjec n mp, his semen is n ue becuse N is pependicul he suce he mp. Tensin, T, is he ce nsmied hugh sing. The ensin is he sme hughu he lengh n idel sing.
A.P. Physics Fis Semese Reiew Shee, Pge 6 The ce n idel sping seched cmpessed by n mun x is gien by Hke s Lw, F x = kx. Ne h i we e nly ineesed in mgniude, we use F = kx whee k is he sping ce cnsn. Hke s Lw is ls used ubbe bnds, bungee cds, ec. Chpe 6: Applicins Newn s Lws A. Ficin FS,mx Ceicien sic icin = µ S = whee N F S,mx is he mx. ce due sic icin. F K Ceicien kineic icin = µ K = whee F K is he ce due kineic icin. N A cmmn lb expeimen inles inding he ngle which n bjec jus begins slide dwn mp. In his cse, simple expessin cn be deied deemine he ceicien sic icin: µ = nq. Ne h his expessin is independen he mss he bjec. S B. Newn s Secnd Lw Pblems (Includes Rmp Pblems). Dw ee-bdy digm epesen he pblem.. I he pblem inles mp, e he x- nd y-xes s h he x-xis cespnds he suce he mp. 3. Cnsuc ec ble including ll he ces in he ee-bdy digm. F he ec ble s clumn hedings, use ec, x-diecin, nd y-diecin. 4. Deemine he clumn l in ech diecin:. I he bjec mes in h diecin, he l is m. b. I he bjec des n me in h diecin, he l is ze. c. Since his is Newn s Secnd Lw pblem, n he chices besides ze nd m e pssible. 5. Wie he mh equins he sum he ces in he x- nd y-diecins, nd sle he pblem. I is en helpul begin wih he y-diecin since useul expessins e deied h e smeimes helpul le in he pblem. Recll h he mh equins egding icin nd weigh e en subsiued in he mh equins help sle he pblem. C. Equilibium An bjec is in nslinl equilibium i he ne ce cing n i is ze, F = 0. Equilenly, n bjec is in equilibium i i hs ze ccelein. I ec ble is needed n bjec in equilibium, hen F x = 0 nd F y = 0. Typicl pblems inle ce clculins bjecs pessed gins wlls nd ensin clculins picues n wlls, lundy n clhesline, hnging bskes, pulley sysems, cin sysems, cnneced bjecs, ec. D. Cnneced Objecs Cnneced bjecs e linked physiclly, nd hus, hey e ls linked mhemiclly. F exmple, bjecs cnneced by sings he he sme mgniude ccelein. When pulley is inled, he x-y cdine xes e en ed und he pulley s h he bjecs e cnneced lng he x-xis. A clssic exmple cnneced bjec is n Awd s Mchine, which cnsiss w msses cnneced by sing h psses e single pulley. The ccelein his sysem is gien by æm - m ö = ç g. èm+ m ø
A.P. Physics Fis Semese Reiew Shee, Pge 7 Chpe 7: Wk nd Kineic Enegy A. Wk A ce exeed hugh disnce pems mechnicl wk. When ce nd disnce e pllel, W = Fd wih Jules (J) Nm s he uni wk. When ce nd disnce e n ngle, nly he cmpnen ce in he diecin min is used cmpue he wk: W = ( Fcs q) d = Fdcsq Wk is negie i he ce ppses he min (q >90 ). Als, J = Nm = kg-m /s. I me hn ne ce des he wk, hen W Tl n = å Wi The wk-kineic enegy heem ses h WTl =D K = K - Ki = m - mi See Tble 6 me inmin bu kineic enegy. æ Aö æf ö In hemdynmics, W = Fd = Fdç = ç ( Ad) = PDV wk dne n by gs. è Aø è Aø i= Tble 6: Kineic Enegy Kineic Enegy Type Equin Cmmens Kineic Enegy s Funcin Min: K = m Used epesen kineic enegy in ms cnsein mechnicl enegy pblems. Kineic Enegy s Funcin Tempeue: æ ö 3 ç è m = Ke = kt øe Kineic hey eles he ege kineic enegy he mlecules in gs he Kelin empeue he gs. B. Deemining Wk m Pl Fce Vesus Psiin In pl ce esus psiin, wk is equl he e beween he ce cue nd he displcemen n he x-xis. F exmple, wk cn be esily cmpued using W = Fd when ecngles e pesen in he digm. F he cse sping ce, he wk sech cmpess disnce x m equilibium is W = kx. On pl ce esus psiin, wk is he e ingle wih bse x (displcemen) nd heigh kx (mgniude ce using Hke s Lw, F = kx). C. Deemining Wk in Blck nd Tckle Lb The expeimenl wk dne gins giy, W, is he sme s he heeicl wk dne by he Ld sping scle, W. Scle W Oupu = W Ld = Fd Ld = W d Ld = mgd Ld whee d = disnce he ld is ised. Ld WInpu = WScle = FdScle whee F = ce ed m he sping scle nd d = disnce he scled Scle med m is iginl psiin. Ne h he ce ed m he scle is ½ he weigh when w sings e used he pulley sysem, nd he ce ed is ¼ he weigh when u sings e used.
A.P. Physics Fis Semese Reiew Shee, Pge 8 D. Pwe W P = P = F wih Ws (W) s he uni Pwe. W = J/s nd 746 W = hp whee hp is he bbeiin hsepwe. Chpe 8: Penil Enegy nd Cnsein Enegy A. Cnseie Fces Vesus Nncnseie Fces. Cnseie Fces A cnseie ce des ze l wk n ny clsed ph. In ddiin, he wk dne by cnseie ce in ging m pin A pin B is independen he ph m A B. In he wds, we cn use he cnsein mechnicl enegy pinciple sle cmplex pblems becuse he pblems nly depend n he iniil nd inl ses he sysem. In cnseie sysem, he l mechnicl enegy emins cnsn: Ei = E. Since E = U + K, i llws h Ui + Ki = U + K. See Tble 6 kineic enegy, K, nd Tble 7 penil enegy, U, ddiinl inmin. F bll hwn upwds, descibe he shpe he kineic enegy, penil enegy, nd l enegy cues n pl enegy esus ime. Exmples cnseie ces e giy nd spings.. Nncnseie Fces The wk dne by nncnseie ce n clsed ph is n ze. In ddiin, he wk depends n he ph ging m pin A pin B. In nncnseie sysem, he l mechnicl enegy is n cnsn. The wk dne by nncnseie ce is equl he chnge in he mechnicl enegy sysem: W = W =D E = E - E. Nncnseie nc i Exmples nncnseie ces include icin, i esisnce, ensin in pes nd cbles, nd ces exeed by muscles nd ms. Tble 7: Penil Enegy Penil Enegy Type Equin Cmmens Giinl Penil Enegy: U = mgh Gd ppximin n bjec ne se leel n he Eh s suce. Giinl Penil Enegy Beween Tw Pin Msses: mm U =- G whee G = 6.67 x 0 - Nm /kg = Uniesl Giin Cnsn Wks well ny liude disnce beween bjecs in he uniese; ecll h is he disnce beween he cenes he bjecs. Elsic Penil Enegy: U = kx whee k is he ce (sping) cnsn nd x is he disnce he sping is seched cmpessed m equilibium. Useul spings, ubbe bnds, bungee cds, nd he sechble meils.
A.P. Physics Fis Semese Reiew Shee, Pge 9 Chpe 9: Line Mmenum nd Cllisins A. Mmenum Line mmenum is gien by p = m wih kg-m/s s he uni mmenum. In sysem hing seel bjecs, ptl = pi. n i= Newn s secnd lw cn be expessed in ems mmenum. The ne ce cing n n bjec is equl he e chnge in is mmenum: F ne = Δ p Δ When mss is cnsn, F ne = Δ p Δ = p pi = m mi = m( Δ Δ Δ i) = mδ Δ = m. The impulse-mmenum heem ses h I = F eδ = Δ p whee he quniy F eδ is clled he impulse, I. F pblem-sling pupses, me useul m he impulse-mmenum heem is FΔ = m( i ). A pcicl pplicin he impulse-mmenum heem, F!" eδ = Δ p!", inles u i bgs. In n umbile cllisin, he chnge in mmenum, Δp, emins cnsn. Thus, n incese in cllisin ime, D, will esul in decesed ce impc, F, educing pesnl injuy. B. Cllisins F sysem bjecs, he cnsein line mmenum pinciple ses h he ne mmenum is cnseed i he ne exenl ce cing n he sysem is ze. In he wds, pi = p. F n elsic cllisin, m i + m i = m + m nd peecly inelsic cllisin, m + m = ( m + m ). See he cllisin ypes in Tble 8. i i In n elsic cllisin in ne dimensin whee mss m is ming wih n iniil elciy, nd mss m is iniilly es, he elciies he msses e he cllisin e: æm- m ö, = ç èm+ m ø nd æ m ö, = ç. èm+ m ø Cllisin Type nd Exmple Elsic: N pemnen demin ccus; billid blls cllide. Inelsic: Pemnen demin ccus; Ms umbile cllisins. Peecly Inelsic: Pemnen demin ccus nd bjecs lck gehe ming s single uni; in cs cllide nd lck gehe. Tble 8: Cllisin Types D he Objecs Is Mmenum Sick Tgehe? Cnseed? N Yes Yes N Yes N Yes Yes N Is Kineic Enegy Cnseed?
A.P. Physics Fis Semese Reiew Shee, Pge 0 Chpe 0: Rinl Kinemics nd Enegy A. Rinl Min Ds Angul psiin, Dq, in dins is gien by D q = whee Ds is c lengh nd is dius. Recll h θ(d) = π dins 360 θ(deg). Cuneclckwise (CCW) ins e psiie, nd clckwise (CW) ins e negie. In inl min, hee e w ypes speeds (ngul speed nd ngenil speed) nd hee ypes cceleins (ngul ccelein, ngenil ccelein, nd cenipel ccelein). See Tble 9 cmpisn. Since elciy is ec, hee e w wys h n ccelein cn be pduced: () chnging he elciy s mgniude nd () chnging he elciy s diecin. In cenipel ccelein, he elciy s diecin chnges. When pesn dies c in cicle cnsn speed, he c hs cenipel ccelein due is chnging diecin, bu i hs n ngenil ccelein due is cnsn speed. The l ccelein ing bjec is he ec sum is ngenil nd cenipel cceleins. Tble 9: Cmping Angul nd Tngenil Speed nd Angul, Tngenil, nd Cenipel Accelein Clculin Equins Unis, Cmmens Angul Speed: Dq dins/s we = D Sme lue hses A nd B, side-by-side n meyg-und. Tngenil Speed: m/s = w Dieen lues hses A nd B, side-by-side n mey-g-und. Angul Accelein: e Dw = D dins/s Sme lue hses A nd B, side-by-side n meyg-und. Tngenil Accelein: = m/s Dieen lues hses A nd B, side-by-side n mey-g-und. Cenipel Accelein: c = = w m/s is pependicul c wih c dieced wd he cene he cicle nd ngen i.
A.P. Physics Fis Semese Reiew Shee, Pge m Cenipel ce, F C, is ce h minins cicul min: FC = mc = = mw The peid, T, is he ime equied cmplee ne ull in. I he ngul speed is cnsn, hen p T =. w The equins inl kinemics e he sme s he equins line kinemics. See Tble 0 cmpisn. e Tble 0: Kinemic equins Rinl Min Line Equins Angul Equins x= x + θ = θ + ω e = + ω = ω + α x x = + + θ = θ + ω + α = + D x ω = ω + α(δθ) A cmpisn line nd ngul inei, elciy, ccelein, Newn s secnd lw, wk, kineic enegy, nd mmenum e pesened in Tble. The mmen inei, I, is he inl nlg mss in line sysems. I depends n he shpe mss disibuin he bjec. In picul, n bjec wih lge mmen inei is diicul s ing nd diicul sp ing. See Tble 0- n p.98 mmens inei unim, igid bjecs ius shpes nd l mss. The gee he mmen inei, he gee n bjec s inl kineic enegy. An bjec dius, lling wihu slipping, nsles wih line speed nd es wih ngul speed w =.
A.P. Physics Fis Semese Reiew Shee, Pge Tble : Cmping Equins Line Min nd Rinl Min Mesuemen Clculin Line Equins Angul Equins Inei: Mss, m I = km whee k = cnsn Aege Velciy: e Dx = we D Dq = D Aege Accelein: e D = e D Dw = D Newn s Secnd Lw: Fne = m = I = F^ Wk: W = Fd csq W = q Kineic Enegy: K = m K = Iw Mmenum: p = m L= Iw Chpe : Rinl Dynmics nd Sic Equilibium A. Tque A ngenil ce, F, pplied disnce,, m he xis in pduces que = F in Nm. (Since F is pependicul, his is smeimes wien s = F ^.) A ce pplied n ngle he dil diecin pduces he que = F sinq Cuneclckwise ques e psiie, nd clckwise ques e negie. The inl nlg ce, F = m, is que, = I, whee I = mmen inei nd = ngul ccelein. The cndiins n bjec be in sic equilibium e h he l ce nd he l que cing n he bjec mus be ze: å F,,. Reled pblems en inle x = 0 å F = y 0 å = 0 bidges, sclds, signs, nd ds held by wies. B. Sling Sic Equilibium Pblems. Cnsuc digm shwing ll he ces.. Cee n equin dding he ces gehe. Remembe ene cec signs in yu ce equin. F exmple, upwd ces e psiie nd dwnwd ces e negie. Since he bjec is n ming, se he ce equin equl ze. ( F = 0 )
A.P. Physics Fis Semese Reiew Shee, Pge 3 3. Cee n equin dding he ques gehe. I yu e n sue wh d: S by wiing he ces gin leing sme spce beween hem. Muliply ech ce by n pppie disnce. Ech ce nd disnce pi will he he sme subscip. Chse n xis in in yu digm. I is helpul i yu chice elimines ne he w unknwn ces. Then yu cn sle he he ce. Remembe ene cec signs in yu que equin. Cnside whehe ech ce will cee cuneclckwise clckwise in esuling in psiie negie que, especiely. Since he bjec is n ming, se he que equin equl ze. ( å = 0 ) 4. Subsiue numbes in yu que equin nd sle he unknwn ce. 5. Subsiue he ce yu deemined in he ls sep in he ce equin nd sle he he unknwn ce. C. Angul Mmenum The inl nlg mmenum, p= m, is ngul mmenum, L= Iw in kg-m /s, whee I = mmen inei nd w = ngul elciy. I he ne exenl que cing n sysem is ze, is ngul mmenum is cnseed nd L = L. i D. Simple Mchines All mchines e cmbinins mdiicins six undmenl ypes mchines clled simple mchines. Simple mchines include he lee, inclined plne, wheel nd xle, wedge, pulley(s), nd scew. Fu din Mechnicl dnge, MA, is deined s MA = = Fin du. I is numbe descibing hw much ce disnce is muliplied by mchine. Eiciency is mesue hw well mchine wks, nd % Eiciency is clculed using % Eiciency = W u 00 W in ( ) Wu W whee is he wk upu nd is he wk inpu. in Chpe : Giy A. Genel Cnceps Abu Giy nd Keple s Lws Newn s Lw Uniesl Giin shws h he ce giy beween w pin msses, m mm nd m, seped by disnce is F = G whee G is he uniesl giin cnsn, G = 6.67x0 - Nm /kg. Remembe h is he disnce beween he cenes he pin msses. In Newn s Lw Uniesl Giin, nice h he ce giy deceses wih disnce,, s. This is eeed s n inese sque dependence. The supepsiin pinciple cn be pplied giinl ce. I me hn ne mss exes giinl ce n gien bjec, he ne ce is simply he ec sum ech indiidul ce. (The supepsiin pinciple is ls used elecsic ces, elecic ields, elecic penils, elecic penil enegies, nd we ineeence.)
A.P. Physics Fis Semese Reiew Shee, Pge 4 Replcing he Eh wih pin mss is cene, he ccelein due giy he suce he Eh is gien by GM Eh g =. REh A simil equin is used clcule he ccelein due giy he suce he plnes nd mns in he Sl Sysem. M Eh A sme liude, h, be he Eh, gh = G R + h cn be used clcule he ( ) ccelein due giy. In 798, Heny Cendish is deemined he lue G, which llwed him clcule he mss greh he Eh using M Eh =. G Using Tych Bhe s bseins cncening he plnes, Jhnnes Keple muled hee lws bil min s shwn in Tble. Newn le shwed h ech Keple s lws llws s diec cnsequence he uniesl lw giin. As peiusly menined in he penil enegy ble, he giinl penil enegy, U, beween mm w pin msses m nd m seped by disnce is U =- G. This equin is used in mechnicl enegy cnsein pblems snmicl siuins. Enegy cnsein cnsideins llw he escpe speed be clculed n bjec lunched m GM Eh he suce he Eh: e = =,00 m/s» 5,000 mi/h. R Eh Eh Tble : Keple s Lws Obil Min Lw Mden Semen Lw Alene Descipin s Lw: Plnes llw ellipicl bis, wih he Sun ne cus he ellipse. The phs he plnes e ellipses, wih he cene he Sun ne cus. nd Lw: As plne mes in is bi, i sweeps u n equl mun e in n equl mun ime. 3 d Lw: The peid, T, plne inceses s is men disnce m he Sun,, ised he 3/ pwe. Th is, æ p ö 3/ 3/ T = = (cnsn) ç GM è S ø An imginy line m he Sun plne sweeps u equl es in equl ime inels. The sque plne s peid, T, is 3 ppinl he cube is dius,. 4p 3 Th is, T = =(cnsn) 3 GM S whee = mss he Sun. M S