Test 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?
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1 Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What s a pojecton of a ecto? d) What does t mean that multplcaton of a ecto by a numbe s dstbute oe addton of numbes? (Use unambguous symbols to dstngush opeatons on ectos fom opeatons on numbes.) I am on the ege of a majo beakthough, but I m also at that pont whee chemsty leaes off and physcs begns, so I ll hae to dop the whole thng. In a Catesan coodnate system, ectos and B ae gen by the followng lnea combnatons of the base ectos: + j and B - + j. a) ssgn (somophcally) a pa of scala components to each ecto. b) Fnd the scala poduct of the two ectos. c) Detemne the angle between the ectos.. What mass of a mateal wth densty ρ s equed to make a hollow sphecal shell hang nne adus and oute adus o?. Two ectos and B hae pecsely equal magntudes. Fo the magntude of +B to be 100 tmes geate than the magntude of -B what must the angle 1 between them be? (Hnt: sn 1 ( 1 ; cos ( 1+ )
2 - 1 - a) In a measuement, a mathematcal quantty (numbe, ecto, ) s assgned to a physcal quantty. b) The followng condtons ae satsfed by scala poduct: 1. o B B o (commutate). ( α ) o B α( o B) (mxed assocate) B o C o C + Bo C (dstbute oe addton of ectos). ( ) ( ) ( ). 0 0 f and only f 0 c) The pojecton, of ecto, n the decton of a unt ectoe s defned as o ˆ ( e ) eˆ d) If α andβ ae two abtay numbes, and s an abtay ecto then ( α + β) α β
3 - - B j a) The coeffcents n the lnea combnaton of the base ectos consttute the scala components of a ecto. s both ectos ae gen n the fom of the lnea combnaton, the scala components ae pactcally gen [,] and B [ 1,] b) The scala poduct of ectos and B s equal to the scala poduct of the R n ectos assgned n a Catesan coodnate system o B o B o [,] [ 1,] + 6 c) Followng the defnton of the angle between the ectos, the angle between ectos and B s o B ϕ accos accos accos 60 B 1 5 [,] o [ 1,] [,] [ 1,]
4 - - (Fom the defnton of densty, we can elate the mass dm of a small (nfntesnal) segment to the appopate (dffeental) olume of the segment: o C (1) dm ρdv To fnd the mass of the ente object, we hae to ntegate ( add ) the mass of all the segments. ddng the mass s based on the scala popety of ths quantty.) Snce the densty s unfom oe the olume of the body the ntegaton s sgnfcantly smplfed. () m ρdv ρ dv ρv shell shell Knowng the ad, we can fnd the olume of the shell dectly fom the defntton of olume, caclulatng the appopate ntegal. It s ease though to use the scala popety of olume and the geneally known fomula fo the olume of a sphee (theoem). The olume of the shell s equal to the dffeence n olume of the oute and the nne sphee: () V π o π Equatons () and () fom a set wth two unknowns {m,v}. Solng (by substtuton) we fnd the mass of the shell (the answe): m ρv π( o )ρ
5 - - +B B -B -B Soluton 1 Let epesent the angle between the dectons of the two ectos. Fom the defnton of magntude ( ) + B + B + ( 1+ + B o B ( ) B + B B ( 1 cos B o sn The ato gen n the poblem yelds the followng tgonometc equaton cos 1 B tan B sn fom whch 1 actan
6 Soluton (fo oented segments) Snce and B hae the same magntudes (lengths),, B, and +B fom an sosceles tangle n whch the angles ae (π-), / and /. Smlaly, B, and -B fom an sosceles tangle n whch the angles ae, (π-)/ and (π-)/. Fom the law of cosnes, the magntude of both +B and -B can be expessed n tems of the magntude of ecto (and B ) + B B + B B cos ( π ) ( 1+ + B ( 1 cos B cos sn
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