Homework Key #7 - Phy 375R

Transcription

1 HMWK7-75R.nb Homewor Ke #7 - Ph 75R Problem #: See Ke for Homewor # Problem #: Transform the lne element... The nverse transformaton s t' = ÅÅÅÅ t tanh- ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ ' = $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ÅÅÅÅ - ÅÅÅÅ - t - ÅÅÅÅÅÅ ' = ' = The ranes of ' and t' are ' > - ÅÅÅÅÅ and - < t' < and ±I ÅÅÅÅ HDsL = HD'L + HD'L + HD'L - II ÅÅÅÅ + ÅÅÅÅÅ ' M Dt'M. The metr s - ÅÅÅÅ M > t. The lne element s = The nverse metr s - I ÅÅÅ + ÅÅÅÅÅ ' M - I ÅÅÅ ÅÅÅÅÅ ' M- = The onl non-vanshn dervatve of s wth respet to '. Thus the onl non-vanshn Chrstofel smbols are G = ÅÅÅÅÅÅ + ' G = ÅÅÅÅÅÅ + ' G = + ' ÅÅÅÅÅÅÅÅÅÅÅÅÅ The urvature s ero. In the lmt ÅÅÅÅÅ t' `, t s eas to see that - -

2 HMWK7-75R.nb t = ÅÅÅÅ + ÅÅÅÅÅÅÅ ' t' ÅÅÅÅÅÅÅÅÅ + O ÅÅÅÅÅÅÅÅÅ t' +... = ÅÅÅÅ + ÅÅÅÅÅÅÅ ' + ÅÅÅÅ ÅÅÅÅÅÅÅÅÅ t' Keepn tra to seond order we have t = t ' + ÅÅÅÅÅÅÅÅÅÅ ' = ' + ÅÅÅÅÅÅÅÅÅÅÅ t' + ÅÅÅÅÅÅÅÅÅÅ ' + O ÅÅÅÅÅÅÅÅÅ t' Obvuosl, we need to restrt the rane of ' to ÅÅÅÅÅÅÅÅ ' `. Problem #: (a.) Draw the traetores n a spae tme... What s happenn s shown below: ÅÅÅÅÅÅ t n ears 5 5 n lhtears Sne ÅÅÅÅÅ m s about ÅÅÅÅÅÅÅ ltr, the aeleraton n ths ase s ÅÅÅÅ ÅÅÅÅÅÅÅÅ ltr. The last tme that the omover an ever hane hs moton s r r and ath up to the aelerated observer s as he/she passes the lht ome. Thus n hs/her tme ths s ears. To the aelerated observer t taes forever for the omover to reah ths event. An event horon s the boundar of the reon n spae tme that one an obet oes behnd t, the obet annot be retreved. Ths lht one s an event horon for the aelerated observer. He/she an send messaes to the omover but the omover annot move to on the aelerated observer. If the omover dedes after one lht ear to ath the aelerated observer, he/she an turn on a roet so that the sta wthn the event horon, whh s the lht one. In other words, he/she wants to aelerate so that he/she has the same asmtope. If he/she starts at (,), the ma pont s at (,) so that the proper dstane to the ma pont s ltr. Thus he/she needs an aeleraton at least equal to ÅÅÅÅÅÅÅÅ ltr or ÅÅÅÅÅ m. r s Below, I show the two events n queston. The frst event, (.5,.5) s n the seton of spae tme that our frend an oordnate. The other event, (,-), s not and thus annot be oordnated b hm/her.

3 HMWK7-75R.nb t n ears n lht-ears t n ears n lht-ears To fnd the oordnates of the frst event, (.5,.5), note that the lne of smultanet for our frend oes throuh the ma pont, (,). Thus the slope of the lne of smultanet that passes throuh that event has slope ÅÅÅÅÅ = ÅÅÅÅ and ths s our frends ÅÅÅÅ velot when the event ours aordn to hm/her and thus the proper tme to our frend s µ tanh - H ÅÅÅÅ L =.7 rs And the dstane s the proper dstane alon the lne of smultanet whh s - $%%%%%%%%%%%%%%%% H ÅÅÅÅ %%%%%%%%% L - H ÅÅÅÅ L = - è!!! =.ltrs. To fnd the past and future of eah event onstrut lht ones. The future lht lne from the frst event has the equaton, t = -, and t ntersets the hperbola, = è!!!!!!!!!!! + t, at t = r. From the event I è!!!, M on the frst event s n the past of our frend. B hs/her lo ths s from µ tanh - I ÅÅÅÅÅÅÅÅ M =. rs on. The past lht lne from our event hts the hperbola at è!!!! (,) and thus ths event s n our frends future for all tmes before ero. Construtn lht lnes on the seond event, we see that ths event s never n the future of our frend and s n the past after the event that s the smultaneous soluton to the equatons, t=-5 and = è!!!!!!!!!!! + t. That s the event, $%%%%%%%%%%%%%%%%% + H ÅÅÅÅÅÅ L, - ÅÅÅÅÅÅ = H.9, -.L. Ths s a neat stuaton. Ths event s never n hs/her future but does appear n hs/her past. The proper tme of ths event to our frend s µ tanh - H- ÅÅÅÅÅÅÅ..9 L = -.8rs Problem #: Consder the two surfae enerated revolvn...

4 HMWK7-75R.nb The urvature s neatve on the surfaes onteuous to the hole. The top and bottom are flat and the outsde s postve. A oordnate sstem that wors s n the notes. It s based on usn the anle, f, of revoluton about the as and the anle, q, around the ornal rle as measured from the top: = H - Sn@qDL Cos@fD = H - Sn@qDL Sn@fD = Cos@qD The dervatves of the poston wth respet to the oordnates are and q q q f f f -Cos@qD Cos@fD = -Cos@qD Sn@fD -Sn@qD = Thus the metr s -H - Sn@qDL Sn@fD H - Sn@qDL Cos@fD = H - Sn@qDL The nverse metr s = H - Sn@qDL - To et the Chrstofel smbols, we need the seond dervatves. These are qq qq qq qf qf qf ff ff ff The onnetons are Sn@qD Cos@fD = Sn@qD Sn@fD -Cos@qD -Cos@qD Cos@fD = -Cos@qD Sn@fD = -Sn@qD fq fq fq -H - Sn@qDL Cos@fD = -H - Sn@qDL Sn@fD Sn@qD Cos@fD SmplfATransposeA Sn@qD Sn@fD E. -Cos@qD 88<< -H - Sn@qDL Sn@fD H - Sn@qDL Cos@fD E

5 HMWK7-75R.nb 5 G qq,q = G qf,q = G ff,q = H - Sn@qDL Cos@qD G qq,f = G qf,f = G ff,f = and usn G = l G,l, G q qq = G q qf = G q ff = H - Sn@qDL Cos@qD G f qq = G f qf = G f ff = Problem #5: How man Calores do ou need... You put out about 7 Watts. There are p µ 7 ses per ear. µ das per ear and thus ou put out ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 7µpµ7 ÅÅÅÅÅÅ º 7 µ J.µ or.75 µ C. Sne dal onsumpton s about C, most of our alores o nto heat atvt ounts for ver lttle. Ths s shown learl wth a short stnt on the eerse mahnes. Ten mnutes of hard effort elds onl C. The real effort of waln s rasn our bod or a fraton of our bod a few entmeters wth eah step. M mass s about 5 and thus the wor per step s ÅÅÅÅ 5 µ µ µ - º 5 J. Thus waln a mle whh s about µ steps onsumes about C. Even ven the estmate s low t s no more than C. Thus dal atvt s onl of the order of hundreds of C whh s onsstent. Sne the heat loss s the prmar ener onsumpton and that s proportonal to the mass of metabol materal n the bod a C reduton wll ome to a bod weht equlbrum the s ÅÅÅÅÅÅ of the ornal weht. Thus our new weht would be redued b ÅÅÅÅÅÅ µ º #.