Central Forces: Circular Motion and Gravitation

Size: px

Start display at page:

Download "Central Forces: Circular Motion and Gravitation"

George Sims
5 years ago
Views:

1 CF-1 Centl Foces: Cicul Motion nd Gittion Cicul motion: object moing in cicle of dius, with constnt speed. T = peiod = time fo 1 complete eolution, 1 cycle ( Don't confuse tension T with peiod T.) speed distnce = = = time π T An object moing in cicle is cceleting, becuse its elocity is chnging, -- chnging diection. ecll the definition of cceletion: 1 = t t, elocity cn chnge is two wys: Mgnitude cn chnge o diection cn chnge: 1 1 Fo cicul motion with constnt speed, one cn show tht 1) the mgnitude of the cceletion is = ) the diection of the cceletion is lwys towds the cente of motion. This is centipetl cceletion. "centipetl" = "towd cente" Notice tht the diection of cceletion ecto is lwys chnging, theefoe this is not cse of constnt cceletion (cn't use the "constnt cceletion fomuls") Is clim (1) sensible? m ( m ) s m Check units: s = = = = m m s [ ] Yep. Think: to get big, we must he pidly chnging elocity. Hee, we need to pidly chnge the diection of ecto need to get ound cicle quickly need eithe lge speed o smll dius. = / mkes sense. (Poof is gien in the Appendix.) Is clim () sensible? Obsee tht ecto is towd cente of cicle. 9/9/006 Uniesity of Colodo t Boulde

2 CF- 1 Diection of = diection towd which elocity is chnging 1 xmple: cceletion on mey-go-ound. dius = 5 m, peiod T = 3 s π π() 5 = = = m/s T 3 ( 10. 5) = = = m/s =. 3 g's! 5 A humn cn withstnd ~ 5 g's fo few minutes o ~10 g's fo few seconds without losing consciousness. Foces nd cicul motion NII: F net = m To mke something ccelete, we need foce in the sme diection s the cceletion. Centipetl cceletion is lwys cused by centipetl foce (foce towd cente). xmple: ock twiled on sting. (Assume no gity) F T m = 0.1 kg, T (peiod) = 1 s, dius = 1 m Wht is tension F T in the sting? F T F T is the only foce cting. (No such thing s "centifugl foce"!) π FT = m = m, =, T ( π / T) 1 FT = m = 4π m = 4π (.) 0 1 = 3. 9 N 1 pound T 1 9/9/006 Uniesity of Colodo t Boulde

3 CF-3 Wht bout the outwd "centifugl foce"? A peson on mey-go-ound (o twiled on ope by gint) "feels" n outwd foce. This is n illusion! Thee is no outwd foce on the peson. Ou intuition is filing us. Ou intuition bout foces ws deeloped oe lifetime of expeiences in inetil (non-cceleting) efeence fmes. If we e suddenly plced in n cceleting efeence fme, ou bins (wongly) intepet ou sense impessions s if we wee still in non-cceleting fme. The esult is tht the diection of the peceied foce is exctly opposite the diection of the tue foce. xmple: A peson in c cceleting fowd. The chi pushes the die fowd. The foce on the die is in the fowd diection. But the die "feels" heself pessed bck into the set. It seems thee is some foce pushing the die bckwd. WONG! "Centifugl foce" (not to be confused with "centipetl foce") is lso clled "pseudo-foce" o "fictitious foce". Newton's Lws e only lid in non-cceleting efeence fme (n inetil fme). If we ty to nlyze motion in non-inetil fme (fo instnce, in otting fme) then Newton's Lws don't hold. Howee, we cn petend tht Newton's Lws hold in n cceleting fme if we petend tht "pseudo-foces" exist. Tht is, we cn get the ight nswe if we mkes two mistkes. In my opinion, this is Deil's bgin. Computtionl conenience hs come t the pice of endless confusion of millions of physics students (nd mny pofessionl enginees!). My dice: If you he choice, NV do clcultions in non-inetil fmes. Aoid using fictitious foces. Conside the ock on the sting gin (still no gity). If the sting beks, then thee is no longe ny foce on the ock nd it will moe in stight line with constnt elocity. [ccoding to NI: if F net = 0, then = constnt]. The eson the ock does not moe in stight line is becuse the sting keeps pulling it inwd, tuning it wy fom its stight-line pth. Thee is NO outwd foce on the ock. 9/9/006 Uniesity of Colodo t Boulde

4 CF-4 xmple: ottion with fiction. A c ounds cue on flt od (not bnked). The dius of the cicul cue is = 100 m, nd the speed of the c is = 30 m/s ( 68 mph). How lge sttic fiction coefficient (µ S, not µ K!) is needed fo the c to not skid off the od? F fic N mg F fic Top iew View fom e of c F net = F fic = m µ S N = m / (c bout to skid F fic = µ S N ) N = mg µ S m g = m / (m's cncel) ( 30m/s) µ S = = = 09. (no units) g (. 9 8m/s )( 100m) So, need µ S 0.9 o else c will skid. (Fo ubbe on dy sphlt µ S 1.0, fo ubbe on wet sphlt µ S 0.7. So c will skid if od wet.) Gity! Newton's Uniesl Lw of Gittion (fist stted by Newton): ny two msses m 1 nd m exet n ttctie gittionl foce on ech othe ccoding to m m F = G 1 m 1 F F m G = uniesl constnt of gittion = N m / kg difficult to mesue!) (G is ey smll, so it is ey Don't confuse G with g. 9/9/006 Uniesity of Colodo t Boulde

5 CF-5 Newton showed tht the foce of gity must ct ccoding to this ule in ode to poduce the obseed motions of the plnets ound the sun, of the moon ound the eth, nd of pojectiles ne the eth. He then hd the get insight to elize tht this sme foce cts between ll msses. [Tht gity cts between ll msses, een smll ones, ws expeimentlly eified in 1798 by Cendish.] Newton couldn't sy why gity cted this wy, only how. instein (1915) Genel Theoy of eltiity, explined why gity cted like this. xmple: Foce of ttction between two humns. people with msses m 1 m 70 kg distnce = 1 m pt. m 11 1 m ( )( 70) 7 F = G = = N 1 This is ey tiny foce! It is the weight of gm mss. A hi weighs 10 3 gms the foce of gity between two people tlking is bout 1/60 the weight of single hi. Computtion of g Impotnt fct bout the gittionl foce fom spheicl msses: spheicl body exets gittionl foce on suounding bodies tht is the sme s if ll the sphee's mss wee concentted t its cente. This is difficult to poe (Newton woied bout this fo 0 yes.) test mss m F g sphee, mss M test mss m point mss M F g (sme s with sphee) We cn now compute the cceletion of gity g! (Befoe, g ws expeimentlly detemined, nd it ws mystey why g ws the sme fo ll msses.) F g = m = m g th mss m, dopped ne sufce Mm G = m g (since = is distnce fom m to cente of th) mss M m's cncel! g = GM 9/9/006 Uniesity of Colodo t Boulde

6 CF-6 Newton's Theoy explins why ll objects ne the th's sufce fll with the sme cceletion. F net = m sys tht bigge foce is equied to ccelete bigge mss m. F g = GMm/ sys tht bigge mss m feels bigge foce. So ne the eth, bigge msses expeience bigge foce in wy tht poduces the sme cceletion fo ll msses. Newton's theoy lso mkes quntittie pediction fo the lue of g, which is coect. xmple: g on Plnet X. Plnet X hs the sme mss s eth (M X = M ) but hs ½ the dius ( X = 0.5 ). Wht is g x, the cceletion of gity on plnet X? Plnet X is dense thn eth, so expect g x lge thn g. GMX GM 1 GM gx = = = = 4g. Don't need lues of G, M, nd! X O set up tio: GM / ( ) ( 1/ ) 4 g of eth X x X X g M = = 1 4, gx 4g g GM = = = / M X * At height h boe the sufce of the eth, g is less, since we e futhe fom the sufce, futhe fom the eth's cente. = + h h GM GM g = = eth ( + h) The spce shuttle obits eth t n ltitude of bout 00 mi 1.6 km/mi 30 km. th's dius = 6380 km. So the spce shuttle is only bout 5% futhe fom the eth's cente thn we e. If is 5% lge, then is bout 10% lge, nd M m F ( on mss m in shuttle) = G bout 10% less thn on eth's sufce g ( + h) Astonuts on the shuttle expeience lmost the sme F g s when on eth. So why do we sy the stonuts e weightless?? 9/9/006 Uniesity of Colodo t Boulde

7 CF-7 "Weightless" does not men "no weight". "Weightless" mens "feefll" mens the only foce cting is gity. If you fll down n iless eleto shft, you will feel exctly like the stonuts. You will be weightless, you will be in fee-fll. th stonut F g N F g An stonut flls towd the eth, s she moes fowd, just s bullet fied hoizontlly fom gun flls towd eth. Obits Conside plnet like th, but with no i. Fie pojectiles hoizontlly fom mountin top, with fste nd fste initil speeds. Plnet would go stight, if no gity The obit of stellite ound the eth, o of plnet ound the sun obeys Keple's 3 Lws. Keple, Gemn ( ). Befoe Newton. Using obsetionl dt fom Dnish stonome Tycho Bhe ("B-hy"), Keple discoeed tht the obits of the plnets obey 3 ules. obits! KI : A plnet's obit is n ellipse with the Sun t one focus. KII : A line dwn fom plnet P to sun S sweeps out equl es in equl times. fste S sme time intels sme es slowe Sun Plnet 9/9/006 Uniesity of Colodo t Boulde

8 CF-8 KIII: Fo plnets ound the sun, the peiod T nd the men distnce fom the sun e elted T TA T by = constnt. Tht is fo ny two plnets A nd B, =. This mens tht 3 3 B3 A B plnets futhe fom the sun (lge ) he longe obitl peiods (longe T). * Keple's Lws wee empiicl ules, bsed on obsetions of the motions of the plnets in the sky. Keple hd no theoy to explin these ules. Newton ( ) stted with Keple's Lws nd NII (F net = m) nd deduced tht MS mp Fg = G. Newton pplied simil esoning to the motion of the th-moon ( Sun plnet ) SP M m system (nd to n th-pple system) nd deduced tht Fg = G. ( th-mss m ) Newton then mde mentl lep, nd elized tht this lw pplied to ny msses, not just to the Sun-plnet, the th-moon, nd th-pojectile systems. Stting with F net = m nd F g = G Mm /, Newton ws ble to deie Keple's Lws (nd much moe!). Newton could explin the motion of eeything! Deition of KIII (fo specil cse of cicul obits). Conside smll mss m in cicul obit bout lge mss M, with obitl dius nd peiod T. We im to show tht 3 T / = const. Stt with NII: F net = m M peiod T The only foce cting is gity, nd fo cicul motion = / m m / M π = = = / / Mm G m G / T [ecll the = dist / time = π / T ] M 4π T 4π G = = = constnt, independent of m 3 T GM (Deiing this esult fo ellipticl obits is much hde, but Newton did it.) An ext esult of this clcultion is fomul fo the speed of stellite in cicul obit: GM =. Fo low-eth obit (few hunded miles up), this obitl speed is bout 7.8 km/s 5 miles/second. 9/9/006 Uniesity of Colodo t Boulde

9 CF-9 Mesuement of Big G The lue of G ("big G") ws not known until In tht ye, Heny Cendish (nglish) mesued the ey tiny F g between led sphees, using deice clled tosion blnce. m1 m Fg Fg = G G = m m 1 ( If F g,, nd m's known, cn compute G.) Befoe Cendish's expeiment, g nd wee known, so using g = GM compute the poduct G M, but G nd M could not be detemined septely., one could With Cendish's mesuement of G, one could then compute M. Hence, Cendish "weighed the eth". Appendix: Poof of = / fo cicul motion The poof inoles geomety (simil tingles). It is mthemticlly simple, but subtle. Conside the motion of pticle on cicle of dius with constnt speed. And conside the position of the pticle t two times septely by shot time intel t. (In the end we will tke the limit s t 0.) We cn dw ecto digms epesenting nd 1 + = 1 + = : = t 1 Notice tht these e simil tingles (sme ngles, sme length tios) Also, note tht 1 = = nd 1 = =. Becuse the tingles e simil, we cn wite =, which is the sme s =. t t = =. A little lgeb gies 9/9/006 Uniesity of Colodo t Boulde

Physics 111. Uniform circular motion. Ch 6. v = constant. v constant. Wednesday, 8-9 pm in NSC 128/119 Sunday, 6:30-8 pm in CCLIR 468

Physics 111. Uniform circular motion. Ch 6. v = constant. v constant. Wednesday, 8-9 pm in NSC 128/119 Sunday, 6:30-8 pm in CCLIR 468 ics Announcements dy, embe 28, 2004 Ch 6: Cicul Motion - centipetl cceletion Fiction Tension - the mssless sting Help this week: Wednesdy, 8-9 pm in NSC 128/119 Sundy, 6:30-8 pm in CCLIR 468 Announcements