Assignment 1

School of Science and Technology

COURSEWORK ASSESSMENT SPECIFICATION

MODULE CODE : PHYS22123 MODULE TITLE : Fundamental Forces MODULE LEADER : Dr Christopher Castleton TITLE : Problem based assignment. LEARNING OUTCOMES ASSESSED : K2, K3, K6, S1, S2, S3, S4 CONTRIBUTION TO ELEMENT : 64%

DATE SET : Monday 20th October 2014 (NTU Week 13) DATE OF SUBMISSION : Monday 10th November 2014 (NTU Week 16) SUBMISSION METHOD : Student Info Desk FEEDBACK DATE : Monday 1st December 2014 (NTU Week 19) FEEDBACK METHOD : Individual paper-based + group feedback during lecture. NOTE : The usual University penalties apply for late submission

and plagiarism. Please consult your student handbook for further details.

NTU Copyright 2014

PHYS22123: Fundamental Forces

Lost In (Vector) Space

2

I. Assessment Requirements

Each student should, individually, work through the problems set out below, providing their answers and working in written form. This may be done by hand or typed, and should be handed in, in paper form, at the coursework receipting desk.

II. Assessment Scenario/Problem

Note on the aims of this assignment This assignment is intended to allow you to develop your mathematical and physical problem solving skills, as applied to the area of classical gravity and planetary motion, and to gain practice in scientific reasoning and deduction in a context in which background knowledge and experience will not enable you to guess the answers from the start. It should also encourage you to look again at the processes by which our understanding of classical gravity and planetary motion were obtained, and to ponder more deeply about the relationship between the fundamental forces operating within our universe and the behaviour and structure of objects within it.

Lost%In%(Vector)%Space%%You$ have$ been$ travelling$ around$ the$ galaxy$ with$ a$ human$ shaped$ alien$ with$ flame8red$feathers$instead$of$hair,$known$simply$as$The$Professor.$The$Professor$travels$in$what$looks$from$ the$ outside$ like$ a$ wardrobe,$ with$ the$ symbol$ $ engraved$ on$ the$ door.$ However,$anyone$stepping$in$through$that$door$finds$the$interior$of$a$very$large$space$ship,$called$the$GRAD,$short$for$General$Relativity$Avoidance$Device.$$

The$last$few$days$have$been$spent$trying$to$find$the$source$of$a$storm$of$new$small$asteroids$mostly$around$108100$m$ in$diameter,$which$keep$materialising$ just$outside$ the$Kuiper$belt.$They$ appear$ from$no8where,$ and$promptly$ implode,$ spreading$debris$ far$ and$wide.$Having$failed$to$find$the$source,$The$Professor$has$set$course$for$Nottingham$to$refuel.$There$ is$a$planetary$ co8rotational$ space8time$ rift$ running$ through$Nottingham,$ he$ explains.$ We$ just$need$ to$ land$ somewhere$ for$ a$ few$ minutes.$ I$ know$ an$ empty$ warehouse$ we$ can$ use$ in$Lenton.$So$he$adjusts$the$switches$and$dials$a$little,$and$with$the$now$familiar$wurring$and$clicking$you$land$with$a$slight$bump.$$

A$glance$at$the$screen$showing$the$outside$reveals$a$surprise,$however.$Instead$of$the$inside$of$ a$warehouse,$ you$ see$ the$ inside$of$ another$ spaceship.$ Krondian$Planetary$ construction$vessel,$Andromeda$Class,$Mark$ IV$ says$The$Professor.$ But$what$ is$ it$doing$here?$At$ that$moment,$a$deep$roar$comes$from$outside$the$GRAD.$Then$the$floor$shakes,$and$everything$around$you$shimmers$for$a$few$seconds,$colours$shifting$and$perspective$twisting.$For$a$short$while,$every$molecule$in$your$body$feels$like$it$is$trying$to$explode.$When$things$settle$again,$The$ Professor$ studies$ the$ display$ screens$ for$ a$ few$ seconds$ before$ saying$ Not$ good.$NOT$GOOD$ and$ takes$ a$ step$ towards$ the$ door,$ only$ to$ hit$ his$ head$ on$ the$ ceiling.$ Feeling$strangely$light,$like$astronauts$on$an$asteroid,$you$follow$as$best$you$can,$mostly$floating,$out$into$ the$ control$ room$ of$ the$ large$ spaceship,$ where$ a$ 4$ meter$ long$ bright$ blue$ slug$ has$tethered$ itself$by$ the$ tail$ to$a$control$console$middle$of$ the$ floor,$ six$of$ its$eight$eye8stalks$pointed$intently$at$a$computer$screen,$the$other$two$pointed$at$you$and$The$Professor.$$

Welcome$aboard,$says$the$slug.$I$ think..$Something$ is$badly$wrong.$My$ships$computer$just$set$up$an$emergency$force$field$to$prevent$the$ship$$and$us,$from$exploding.$The$artificial$



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gravity$ plates$ have$ gone$haywire,$ and$ I$ can$barely$ keep$ things$ under$ control.$ Allow$me$says$The$Professor,$his$hands$moving$rapidly$over$the$controls.$The$out$comes$his$Ultrasonic$spanner$for$some$finishing$touches.$There,$that$should$hold$things$for$a$while,$until$we$have$figured$out$what$it$going$on.$Im$The$Professor,$by$the$way.$Professor$who?$(The$slug$asks$the$normal$question.)$Just$The$Professor.$Well,$thank$you$for$your$help.$Im$Flartyglublut,$senior$construction$engineer$for$Trans8Krondian$Construction$Ltd.$Noticing$your$puzzled$look$it$adds$I$custom8build$planets$for$rich$clients.$I$just$built$one.$I$think.$The$clients$are$living$on$Earth$at$the$moment,$and$I$found$a$rift$to$provide$the$material$for$the$planet.$Of$course,$I$placed$the$planet$well$outside$you$solar$system.$Dont$want$the$humans$to$spot$it.$$

So$who$are$the$clients?$you$ask.$The$pigeons,$of$course.$Primary$species$on$this$planet,$but$they$got$fed$up$with$the$humans$thinking$they$were$ in$charge$and$then$messing$everything$up.$So$they$wanted$an$exact$duplicate$of$Earth$complete$with$a$duplicate$of$the$moon,$but$minus$the$primates.$So$thats$why$ Im$here,$except$something$seems$to$have$gone$wrong.$The$Professor$cuts$in:$I$was$refuelling$from$the$rift$at$the$same$moment.$It$seems$to$have$got$overloaded$and$become$unstable.$Flartyglublut$turns$a$paler$shade$of$blue,$and$looks$at$her$display$screens.$Oops.$That$may$explain$why$explain$things$a$bit.$$

Flartyglublut$turns$back$to$the$controls$for$a$moment,$before$saying$I$think$we$actually$got$sucked$through$the$rift,$and$I$cant$identify$where$we$are$now.$None$of$the$nearby$stars$are$in$the$charts.$Actually,$not$even$the$galaxies.$And$they$all$look$.$wrong.$Far$too$big,$and$much$too$close$together.$Theres$a$whole$supercluster$ in$the$space$normally$occupied$by$ just$our$own$ galaxy.$ The$ Professor$ has$ also$ been$ examining$ the$ screens.$ We$ seem$ to$ be$ in$ orbit$around$ that$ bright$ blue$ star$ over$ there,$ close$ to$ the$ orbit$ of$ this$ blue/green$ and$ white$planet.$You$gaze$at$the$screen$yourself:$Thats$the$Earth!$Not$quite,$thats$my$new$Earth8II,$complete$with$Moon8II8$corrects$Flartyglublut.$and$according$to$these$reading,$Moon8II$is$in$a$beautiful$low$eccentricity$orbit.$But$neither$has$come$out$right.$Correct$mass,$yes,$and$the$continents$and$craters$are$in$the$right$places,$but$the$radii$are$too$small.$Theres$something$odd$ about$ the$ star$ too$ interjects$ The$ Professor$ Its$ producing$ too$ much$ energy$ for$ its$apparent$ size.$ And$ look$ at$ the$ orbits$ of$ these$ other$ planets$ $ they$ are$ all$ crazy,$ even$ the$circular$ones.$Look$how$fast$everything$is$moving!$$

Finally,$ you$ start$ to$ piece$ things$ together.$ So$ gravity$ itself$ has$ changed!$ you$ exclaim.$Exactly!$agrees$the$Professor$Which$means$we$arent$even$in$our$own$universe!$We$need$to$find$our$way$back$as$soon$as$possible.$Our$bodies$wont$cope$with$the$additional$pressures$and$strains$for$long.$We$need$to$find$a$rift$with$the$opposite$chirality$in$order$to$get$home.$

Suddenly$ a$ siren$ goes$ off.$ Asteroid$ cloud$ approaching$ comes$ a$ metallic$ sounding$ voice$Impact$ in$3,$2,$1$seconds.$At$once$there$is$a$deafening$drumming$noise$from$the$hull,$the$ship$ shakes,$ and$ bits$ of$ equipment,$ girders,$ and$ other$ things$ you$ cant$ identify$ start$ flying$about.$The$bombardment$doesnt$last$long,$but$when$it$ends$you$have$a$shock.$The$Professor$is$lying$lifeless$on$the$ground.$You$check$for$a$pulse,$but$there$is$none.$He,$hes$dead!$you$say.$Bad$timing$says$Flartyglublut$but$his$species$are$part$phoenix.$Look,$the$feathers$are$starting$to$burn.$They$will$gradually$burn$down,$like$small$fuses.$Once$they$have$burnt$out,$his$whole$body$will$burst$into$flames,$and$he$will$come$back$to$life$with$a$whole$new$body.$Takes$quite$ a$while$ though,$ they$ say,$ and$ Ive$ just$ been$ checking:$ that$ asteroid$ cloud$wasnt$ the$only$one.$This$star$system$is$filled$with$them,$so$we$really$cant$wait$around.$We$need$to$fly$off$and$find$a$new$rift,$but$first$we$need$to$figure$out$how$gravity$works$in$this$new$universe,$so$I$can$re8calibrate$the$drive$system$of$my$ship.$Its$probably$safe$to$assume$that$the$gravity$force$ law$ in$ this$ universe$ will$ still$ be$ proportional$ to$ the$ product$ of$ the$ masses,$ but$ the$variation$ with$ distance$ is$ clearly$ different.$ Probably$ a$ different$ value$ for$ the$ universal$gravitational$constant$too.$$



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She$ half$ slithers,$ half$ floats$ over$ to$ another$ console.$ Tentacles$ emerge$ from$her$ front$ and$hover$over$the$controls$for$a$couple$of$minutes,$before$she$returns$and$hands$you$a$print$out.$Here$ is$ some$ data$ on$ local$ planets$ $ just$ some$ of$ the$ ones$with$ near8circular$ orbits.$ The$others$I$cant$figure$out$yet.$There$may$be$a$bit$of$scatter$in$the$data,$but$its$the$best$I$can$do$for$ now.$ Start$ by$ re8deriving$ Keplers$ third$ law,$ then$ derive$ the$ distance$ dependence$ of$gravity$in$this$universe.$Ive$added$in$the$data$for$Moon8II,$but$of$course,$that$wont$help$you$with$Kepler.$Could$be$useful$later$on$though.$While$you$do$that,$I$will$do$some$deeper$scans,$and$try$to$find$ourselves$a$suitable$rift.$And$off$she$slithers.$$

Question%1%(9%marks)%

Note:$ Here,$ and$ in$ all$ questions,$ assume$ all$ physics$ is$ the$ same$ as$ in$ our$ universe$except$for$the$specific$features$noted.$

The$new$form$for$Keplers$3rd$law$can$still$be$written$in$the$form$T = krx $where$T$is$the$orbital$period,$r$ is$the$orbital$radius,$k$ is$a$constant$of$proportionality,$and$x$ is$a$rational$number.$Using$a$suitable$graph,$find$the$value$of$x,$which$should$be$a$rational$number.$Please$show$your$working.$(Note,$in$our$universe,$x$=$6/4.)$

$

%

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*$Values$for$Moon8II$are$given$for$orbit$around$Earth8II,$not$around$the$star.$%Question%2%(8%marks)%

According$ to$ Flartyglublut$ ,$ Newtons$ law$ of$ Universal$ Gravitation$ should$ still$ look$

likeF = Gm1m2rn$ in$ this$universe,$except$ that$ n 2 .$Use$your$answer$to$question$1$to$

derive$the$new$value$for$n.$$


What$forms$do$you$expect$for$Keplers$second$law$in$this$universe?$Would$the$first$law$be$ the$ same$ as$ in$ our$ universe,$ or$ different?$ Would$ Lagrange$ points$ and$ orbital$precession$ work$ the$ same$ here,$ differently$ or$ not$ exist$ at$ all?$ Briefly$ explain$ your$answers.$

Once$you$have$finished,$Flartyglublut$returns$Ive$found$another$rift$in$orbit$around$this$star.$Its$ a$ long$ way$ out$ though:$ 4.311011$ m$ from$ the$ star.$ There$ is$ a$ large$ stream$ of$ small$asteroids$ flying$ into$ it$ and$ disappearing,$ so$ we$ will$ have$ to$ steer$ carefully.$We$ only$ have$about$ 1.491019$ J$ left$ in$ the$ energy$ tanks$ $ I$ was$ due$ to$ refuel$ once$ I$ had$ finished$ the$construction$ job.$And$with$The$Professor$dead$for$now$we$cant$use$his$GRAD$$even$ if$ its$vortex$core$functions$ in$this$universe$at$all.$So$we$need$to$figure$out$ if$my$ship$has$enough$energy$to$get$us$that$far$out.$She$looks$at$your$data,$then$points$out$that$We$need$to$know$

Planet% Orbital%Radius,"r%(m)% Period,%T%(s)%1$ 7.63108$$ 4.79102$2$ 7.13109$$ 8.22103$Earth8II$ 1.071010$ 1.31104$4$ 5.491010$ 1.14105$5$ 2.131011$$ 5.49105$Moon8II*$ 3.01108$$ 1.31105$



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the$ value$ of$ the$ Universal$ Gravitational$ constant,$ G,$ in$ this$ universe.$ I$ make$ it.$ she$pauses,$tentacles$moving$over$the$console$again$$5.97910813$$in$SI$units.$$


Use$the$additional$data$given$above$to$check$(derive$for$yourself!)$Flartyglubluts$value$for$ the$ Universal$ Gravitational$ constant,$ G.$ (Clue:$ Assume$ that$ the$ constant$ k$ in$$T = krx $8$and$hence$in$ T 2 = k2r2 x $takes$exactly$the$same$form$as$in$our$universe,$apart$from$the$numerical$values$of$G$itself.$Use$the$data$given$above.$You$may$need$to$look$up$some$other$data$elsewhere.)$

Flartyglublut$heads$off$to$re8calibrate$the$ships$drive$system,$leaving$you$to$calculate$whether$or$not$the$energy$reserves$will$get$you$out$as$far$as$the$rift.$$


Using$the$answers$to$the$previous$questions$(or$otherwise)$find$an$accurate$value$for$the$ mass$ of$ the$ star,$ with$ an$ error$ bar$ estimate.$ Show$ your$ working.$ Is$ there$ an$alternative$way$to$calculate$the$mass$of$the$star$from$the$graph$in$question$1?$Which$method$is$more$accurate,$and$why?$


Derive$the$formula$for$gravitational$potential$energy$E$in$terms$of$the$constant$G,$for$two$masses$m1$and$m2$distance$r$apart$in$this$universe.$(Take$the$potential$energy$to$be$zero$at$infinity.)$Use$this$formula,$and$other$data$from$above$to$calculate$the$total$energy$needed,$presuming$that$any$energy$losses$are$minimal.$The$mass$of$the$ship$is$4.25105$ kg$ and$ your$ current$ orbit$ is$ at$ 1.031010$m$ from$ the$ star.$ If$ Flartyglublut$empties$the$energy$reserves$completely,$will$there$be$energy$available$to$make$it?$If$there$will$be,$with$what$velocity$will$you$reach$the$rift?$If$not,$can$you$suggest$ways$of$making$up$the$difference?$

Finally,$everything$is$ready.$Suddenly,$there$is$a$roaring,$whooshing$sound$behind$you,$and$a$familiar$voice$says$Ah,$thats$better.$How$did$you$get$on$while$I$was$gone?$It$only$takes$The$Professor$a$few$moments$to$catch$up.$Youre$both$brilliant!$he$says$Couldnt$have$done$it$better$myself.$So$what$are$we$waiting$for?$Alons8y!$And$while$were$going,$lets$see$what$we$can$figure$out$about$this$new$universe..$


Showing$your$working,$calculate$the$ratio$of$the$strength$of$the$force$of$gravity$in$the$two$universes$ at$ the$distances$1$m$and$1,000,000$ km.$At$what$distance$ is$ the$ ratio$unity?$Hence$explain$as$many$of$the$early$observations$described$above$as$possible.$Are$ there$ any$ observations$ that$ you$ cannot$ explain$ from$ this,$which$might$ indicate$other$differences$in$physical$laws$between$the$two$universes?$


In$ our$ (real)$ universe$ one$ can$ argue$ for$ the$ inverse$ square$ law$ geometrically,$ by$considering$field$lines$passing$through$spherical$shells$around,$say,$a$planet.$Can$you$construct$this$argument?$What$would$it$imply$about$this$new$universe?$



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III. Assessment Criteria !See overleaf for GBA grid.

IV. Feedback Opportunities Summative (After youve submitted the coursework) You will receive specific (individual) feedback regarding your coursework submission together with your awarded grade when it is returned to you. There will also be a feedback session during one of the lectures, in which the answers will be presented and discussed. Clearly, feedback provided with your coursework is only for developmental purposes so that you can improve for the next assessment or subject-related module.

V. Moderation The Moderation Process All assessments are subject to a two-stage moderation process. Firstly, any details related to the assessment (e.g., clarity of information and the assessment criteria) are considered by an independent person (in this case the module shadow). Secondly, the grades awarded are considered by the module shadow to check for consistency and fairness across the cohort for the piece of work submitted.

VI. Aspects for Professional Development Portfolio

Which parts of this assessment are relevant to a professional portfolio will depend greatly on the chosen profession, and what you want to show to a potential employer. However, the derivations and numerical questions could be used to demonstrate analytical and mathematical skills, while the physical reasoning questions demonstrate ability to think laterally about physical problems, and interpret results in the light of prior knowledge of physical laws.

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ing!of!phy

sical!la

ws!

demon

strated.!

Key!ph

ysical!im

plications!of!the

!results

!!of!c

alcu

latio

ns!&!

deriv

ations!are!disc

ussed!an

d!ap

plied!co

rrectly

,!with

!basic!

unde

rstand

ing!of!key

!phy

sical!la

ws!

demon

strated.!M

isund

erstan

ding

s!of!so

me!seco

ndary!features.!

Most!o

f!the

!phy

sical!im

plications!of!

the!results

!misu

nderstoo

d!or!

inco

rrectly

!app

lied.!At!m

ost!lim

ited!

demon

stratio

n!of!und

erstan

ding

!of!

physical!la

ws.!

Work!of!no!

merit,!or!

absent!

Assignment 1

Documents

Transcript of Assignment 1