DO PHYSICS ONLINE SPACE LENGTH CONTRACTION
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1 DO PHYSICS ONLINE SPACE LENGTH CONTRACTION RELATIVE LENGTH: LENGTH CONTRACTION When measuring he lengh of an obje i is neessary o be able o deermine he exa posiion of he ends of he obje simulaneously. If, howeer, obserers in differen referene frames may disagree on he simulaneiy of wo eens, hey may also disagree abou he lengh of objes he lengh of a moing rod appears o onra in he direion of moion relaie o a saionary obserer. Equaion (1) is known as he Lorenz-Fizgerald Conraion Equaion of a moing lengh (1) L L res lengh or proper lengh, L eloiy of rod relaie o he saionary obserer, Conraion akes plae in he direion of moion only You are a saionary obserer in an inerial frame of referene. A rain was iniially a res in your frame of referene and you measure is lengh. Howeer, when he rain is in moion your measuremen of is lengh is shorer. There is a onraion in is lengh. The rain is shorer when i is in moion, bu jus as high and wide as i was a res (figure 1). rain a res w.r.. obserer rain in moion w.r.. obserer rain is shorer in direion in moion bu jus as high and wide as i was a res Fig. 1. How long is a rain? I depends on he relaie moion of he obserer and he rain. This is a real differene in lengh of he obje when i is moion relaie o an obserer. For a person in he rain, here is no onraion in lengh (figure ).
2 Earh obserer: ime for spaeraf raelling a speed o rael a disane L o red plane red plane a res w.r.. Earh L = L Spaeship obserer: ime for spaeraf raelling a speed o rael a disane L o red plane red plane moing a speed w.r.. spaeraf L L = L L L L Fig.. A lengh is measured o be shorer when i is moing relaie o he obserer han when i is a res. lengh onraion A New Sandard of Lengh Lengh is one of he fundamenal quaniies in Physis beause is definiion does no depend on oher physial quaniies. The SI uni of lengh, he mere was originally defined as one en-millionh of he disane from he equaor o he geographi Norh Pole. The firs ruly inernaional sandard of lengh was a bar of plainum-iridium alloy alled he sandard mere and kep in Paris. The bar was suppored mehanially in a presribed way and kep in an airigh abine a o C. The disane beween wo fine lines engraed on gold plugs near he ends of he bar was defined o be one mere. In 1961 an aomi sandard of lengh was adoped by inernaional agreemen. The mere was defined o be imes he waelengh of he orange-red ligh from he isoope krypon-86. This sandard had many adanages oer he original inreased preision in lengh measuremens, greaer aessibiliy and greaer inariabiliy o lis a few. In 1983 he mere was re-defined in erms of he speed of ligh in a auum. The mere is now defined as he disane ligh raels in a auum in 1/ of a seond as measured by a aesium lok. Sine he speed of ligh is onsan and we an measure ime more auraely han lengh, his sandard proides inreased preision oer preious sandards. The reason for ha pariular fraion (1/ ) is ha he sandard hen orresponds o he hisorial definiion of he mere he lengh on he bar in Paris. So, our urren sandard of lengh is aually defined in erms of ime in onras o he original sandard mere, whih was defined direly in erms of lengh (disane).
3 Muon deay Experimenal eidene for ime dilaion and lengh onraion Muons are unsable pariles wih a res mass of 7 imes ha of an eleron and a harge of ±1.6x1-19 C. Muons deay ino elerons or posirons wih an aerage lifeime of. s as measured in heir frame of referene. When high energy pariles suh as proons alled osmi rays ener he amosphere from ouer spae, hey inera wih air moleules in he upper amosphere reaing a osmi ray shower of pariles inluding muons ha reah he Earh s surfae. The muons reaed in hese osmi ray showers rael a.98 w.r. o he Earh. Newonian (lassial) poin of iew: Speed of muons =.98 = (.98)(3.x1 8 ) m.s -1 =.94x1 8 m.s -1 Aerage lifeime of muons (proper ime) =. s =.x1-6 s Aerage disane raelled by muon before deaying L = = (.94x1 8 )(.x1-6 ) m = 65 m Hene, from a Newonian poin of iew, muons would no be able o reah he Earh s surfae from he upper amosphere where hey are produed. Howeer, experimens show ha a large number of muons do reah he Earh s surfae in osmi ray showers. Speial relaiiy poin of iew: Heigh aboe Earh s surfae: muon produed L = 1 km = 1.x1 5 (proper lengh) m From poin of iew of muons he disane o he Earh s surfae is onraed o a shorer lengh L L L =.x1 3 m This onraed disane is muh less and so many muons will be able o reah he Earh s surfae. From an obserer iewing he muon approahing, ime inerals for moing lok will be dilaed. Aerage lifeime of muons (proper ime) =. s =.x1-6 s Dilaed aerage lifeime w.r.. Earh obserer =? s s Aerage disane raelled by muons L = = (.98)(1.11x1-5 ) m = 3.x1 3 m Aerage disane is now shor enough so ha many muons will reah he Earh s surfae.
4 Example The diameer of our galaxy is 6.x1 m. (a) If he speed of he spaeship was.99999, how long would i ake o ross he galaxy as measured from he frame of referene of he spaeship? (b) How muh ime would elapse on Earh for his journey aross he galaxy? Soluion (a) A raeller in he spaeraf would be a res in he spaeship and would see he galaxy approahing a speed = The raeller would see he galaxy onraed in he direion of moion. The onraed lengh L is L = 6.x1 m = L L x m If he ime is measured in he spaeraf, he ime for he journey aross he galaxy is =? s =? years L =.7x1 18 m L = L 9 8.9x1 s 3 years Een a spaeship raelling a speeds i sill akes 3 years o ross he galaxy. (b) An Earh bound obserer will iew he moing lok on he spaeship and measure a dilaed ime ineral Spaeship ime ineral = 3 years (proper ime) Dilaed ime ineral for Earh obserer =? years = x1 years WOW!!! For he Earh obserer, 64 years pass for he spaeship o ross he galaxy bu on he spaeship only 3 years pass. Only, spaeships raelling a speeds exremely lose o he speed of ligh an raerse he huge asronomial disanes need for spae rael in reasonable imes.
5 QUESTIONS AND PROBLEMS VIEW ANSWERS How o answer a quesion: problem soling View periodi able (ied Aug 1) Numerial alues for onsans and useful physial quaniies P5791 P5935 P5176 P5615 P5791 A missile raelling a 9% of he speed of ligh has a res lengh of 1 m. Calulae he lengh of he moing missile as measured by a saionary obserer direly under he fligh pah of he missile. P5935 A spaeraf in he shape of a square box wih sides 3 mm long moes away from he Earh a a eloiy of.5. Find he olume of he box as measured from Earh. P5176 Two loks are synhronized and hen one is sen ino spae a.45 for one hour of is ime. (a) Calulae he ime passed for he lok lef behind in he saionary frame of referene. (b) Desribe and draw he spae of he moing lok as obsered from he saionary lok. Assume hey are irular o sar. () Wha shape will he moing lok o an obserer moing wih he lok? (d) If he moing lok had a mass of 94 g when saionary, alulae he mass when i is moing a.45. P5615 The radius of our galaxy is 3 1 m, or abou ligh years. Assume he speed of a spaeraf is.99. (a) (b) () (d) (d) From he poin of iew of an obserer on Earh, alulae he ime o rael o he edge of our galaxy? Use he lengh onraion formula o alulae he ime for a person o rael from he enre o he edge of our galaxy. Use he ime dilaion effe o alulae he ime for a person o rael from he enre o he edge of our galaxy. How fas would a spaeship hae o rael o go from he enre o he edge of our galaxy in 3 years as measured from wihin he spaeship? How muh ime would elapse on Earh during his journey? Wha does his ell us abou he fuure of raeling o edge of our galaxy?
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