12bae031 assignment4

7/24/2019 12BAE031 assignment4

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FORCES ON

MISSILESPRASANNABALAJI.S 12BAE031

Forces are vector quantities having oth a !agnitu"e an"

a "irection. #hen "escriing the action o$ $orces% one

!ust account $or oth the !agnitu"e an" the "irection. In

&ight% a roc'et is su(ecte" to $our $orces) *eight% thrust%

an" the aero"+na!ic $orces% ,i$t an" "rag. -he !agnitu"e

o$ the *eight "een"s on the !ass o$ a,, o$ the arts o$

the roc'et. -he *eight $orce is a,*a+s "irecte" to*ar"s

the center o$ the earth an" acts through the center o$


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gravit+% the +e,,o* "ot on the /gure. -he !agnitu"e o$

the thrust "een"s on the !ass &o* rate through the

engine an" the ve,ocit+ an" ressure at the eit o$ the

no,e. -he thrust $orce nor!a,,+ acts a,ong the

,ongitu"ina, ais o$ the roc'et an" there$ore acts throughthe center o$ gravit+. So!e $u,, sca,e roc'ets can !ove% or

gi!a,% their no,es to ro"uce a $orce *hich is not

a,igne" *ith the center o$ gravit+. -he resu,ting torque

aout the center o$ gravit+ can e use" to !aneuver the

roc'et. -he !agnitu"e o$ the aero"+na!ic $orces "een"s

on the shae% sie% an" ve,ocit+ o$ the roc'et an" on

roerties o$ the at!oshere. -he aero"+na!ic $orces act

through the center o$ ressure% the ,ac' an" +e,,o* "oton the /gure. Aero"+na!ic $orces are ver+ i!ortant $or

!o"e, roc'ets% ut !a+ not e as i!ortant $or $u,, sca,e

roc'ets% "een"ing on the !ission o$ the roc'et. Fu,, sca,e

oosters usua,,+ sen" on,+ a short a!ount o$ ti!e in the

at!oshere.

In &ight the !agnitu"e% an" so!eti!es the "irection% o$

the $our $orces is constant,+ changing. -he resonse o$

the roc'et "een"s on the re,ative !agnitu"e an"

"irection o$ the $orces% !uch ,i'e the !otion o$ the roe in

a tugo$*ar contest. I$ *e a"" u the $orces% eing

care$u, to account $or the "irection% *e otain a net

eterna, $orce on the roc'et. -he resu,ting !otion o$ theroc'et is "escrie" + Ne*ton4s ,a*s o$ !otion.

A,though the sa!e $our $orces act on a roc'et as on an

air,ane% there are so!e i!ortant "i5erences in the

a,ication o$ the $orces6

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7n an air,ane% the ,i$t $orce 8the aero"+na!ic $orce

eren"icu,ar to the &ight "irection9 is use" to overco!e

the *eight. 7n a roc'et% thrust is use" in oosition to

*eight. 7n !an+ roc'ets% ,i$t is use" to stai,ie an"

contro, the "irection o$ &ight.

7n an air,ane% !ost o$ the aero"+na!ic $orces are

generate" + the *ings an" the tai, sur$aces. For a roc'et%

the aero"+na!ic $orces are generate" + the /ns% nose

cone% an" o"+ tue. For oth air,ane an" roc'et% the

aero"+na!ic $orces act through the center o$ ressure

8the +e,,o* "ot *ith the ,ac' center on the /gure9 *hi,e

the *eight acts through the center o$ gravit+ 8the +e,,o*"ot on the /gure9.

#hi,e !ost air,anes have a high ,i$t to "rag ratio% the

"rag o$ a roc'et is usua,,+ !uch greater than the ,i$t.

#hi,e the !agnitu"e an" "irection o$ the $orces re!ain

$air,+ constant $or an air,ane% the !agnitu"e an"

"irection o$ the $orces acting on a roc'et change

"ra!atica,,+ "uring a t+ica, &ight.

#eight is the $orce generate" + the gravitationa,

attraction on the roc'et. #e are !ore $a!i,iar *ith *eight

than *ith the other $orces acting on a roc'et% ecause

each o$ us have our o*n *eight *hich *e can !easure

ever+ !orning on the athroo! sca,e. #e 'no* *hen one

thing is heav+ an" *hen another thing is ,ight. But *eight%

the gravitationa, $orce% is $un"a!enta,,+ "i5erent $ro! the

other $orces acting on a roc'et in &ight. -he aero"+na!ic

$orces% ,i$t an" "rag% an" the thrust $orce are !echanica,

$orces. -he roc'et !ust e in h+sica, contact *ith the the

gases *hich generates these $orces. -he gravitationa,

$orce is a /e," $orce an" the roc'et "oes not have to e in

contact *ith the source o$ this $orce.

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-he nature o$ the gravitationa, $orce has een stu"ie" +

scientists $or !an+ +ears an" is sti,, eing investigate" +

theoretica, h+sicists. For an o(ect the sie o$ a roc'et%

the e,anation given three hun"re" +ears ago + Sir

Isaac Ne*ton is su:cient to "escrie the !otion o$ the

o(ect. Ne*ton "eve,oe" his theor+ o$ gravitation *hen

he *as on,+ 23 +ears o," an" u,ishe" the theories *ith

his ,a*s o$ !otion so!e +ears ,ater. As Ne*ton oserve"%

the gravitationa, $orce et*een t*o o(ects "een"s on

the !ass o$ the o(ects an" the inverse o$ the the square

o$ the "istance et*een the o(ects. ;ore !assiveo(ects create greater $orces an" the $arther aart the

o(ects are the *ea'er the attraction. Ne*ton *as a,e to

eress the re,ationshi in a sing,e *eight equation. -he

gravitationa, $orce% F% et*een t*o artic,es equa,s a

universa, constant%


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square o$ the ra"ius o$ the Earth to $or! the Earth4s

gravitationa, acce,eration% ge .

ge = < > ! Earth ? 8" Earth9@2

ge = .C !?sec@2 = 32.2 $t?sec@2

-he *eight #% or gravitationa, $orce% is then (ust the !ass

o$ an o(ect ti!es the gravitationa, acce,eration.

# = ! > ge

An o(ect4s !ass "oes not change $ro! ,ace to ,ace% ut

an o(ect4s *eight "oes change ecause the gravitationa,

acce,eration ge "een"s on the square o$ the "istance

$ro! the center o$ the Earth. Let4s "o a ca,cu,ation an"

"eter!ine the *eight o$ the Sace Shutt,e in ,o* Earth

orit. 7n the groun"% the oriter *eighs aout 2D0%000

oun"s. In orit% the Shutt,e is aout 200 !i,es aove thesur$ace o$ the Earth) the "istance $ro! the center o$ the

Earth is 200 !i,es. -hen6

! = #s ? ge = #o ? go

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#o = #s > go ? ge

*here #s = sur$ace *eight 82D0%000 oun"s9% #o is the

orita, *eight% an" go is the orita, va,ue o$ the

gravitationa, acce,eration. #e can ca,cu,ate the ratio o$

the orita, gravitationa, acce,eration to the va,ue at the

sur$ace o$ the Earth as the square o$ Earth ra"ius "ivi"e"

+ the square o$ the orita, ra"ius.

go ? ge = 8" Earth9@2 ? 8" orit9@2

go ? ge = 8000?2009@2 = .0

7n orit% the shutt,e *eighs 2D0%000 > .0 = 22%D

oun"s. Notice6 the *eight is not ero. -here is a ,arge

gravitationa, $orce acting on the Shutt,e at a "istance o$

200 !i,es. -he *eight,essness eerience" +

astronauts on oar" the Shutt,e is cause" + the $ree$a,,

o$ a,, o(ects in orit. -he Shutt,e is u,,e" to*ar"s theEarth ecause o$ gravit+. But the high orita, see"%

tangent to the sur$ace o$ the Earth% causes the $a,,

to*ar"s the sur$ace to e eact,+ !atche" + the

curvature o$ the Earth a*a+ $ro! the shutt,e. In essence%

the shutt,e is constant,+ $a,,ing a,, aroun" the Earth.


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Because the *eight o$ an o(ect "een"s on the !ass o$

the o(ect% the !ass o$ the attracting o(ect% an" the

square o$ the "istance et*een the!% the sur$ace *eight

o$ an o(ect varies $ro! ,anet to ,anet. #e have "erive"

a gravitationa, acce,eration $or the sur$ace o$ the Earth%ge = .C !?sec@2% ase" on the !ass o$ the Earth an"

the ra"ius o$ the Earth. -here are si!i,ar gravitationa,

acce,erations $or ever+ o(ect in the so,ar s+ste! *hich

"een" on the !ass o$ the o(ect an" the ra"ius o$ the

o(ect. 7$ articu,ar interest $or the Gision $or Sace

E,oration% the gravitationa, acce,eration o$ the ;oon g!

is given +6

g! = < > ! ;oon ? 8" ;oon9@2

g! = 1.1 !?sec@2 = D.3 $t?sec@2

an" the gravitationa, acce,eration o$ ;ars g!ar is given

+6

g!ar = < > ! ;ars ? 8" ;ars9@2

g!ar = 3.C !?sec@2 = 12.1 $t?sec@2


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-he !ass o$ a roc'et is the sa!e on the sur$ace o$ the

Earth% the ;oon an" ;ars. But on the sur$ace o$ the ;oon%

the *eight $orce is aroi!ate,+ 1? the *eight on Earth%

an" on ;ars% the *eight is aroi!ate,+ 1?3 the *eight

on Earth. Hou "on4t nee" as !uch thrust to ,aunch thesa!e roc'et $ro! the ;oon or ;ars% ecause the *eight is

,ess on these ,anets.

A,, $orces are vector quantities having oth a !agnitu"e

an" a "irection. For a roc'et% *eight is a $orce *hich is

a,*a+s "irecte" to*ar"s the center o$ the Earth. -he!agnitu"e o$ this $orce "een"s on the !ass o$ a,, o$ the

arts o$ the roc'et itse,$% ,us the a!ount o$ $ue,% ,us an+

a+,oa" on oar". -he *eight is "istriute" throughout

the roc'et% ut *e can o$ten thin' o$ it as co,,ecte" an"

acting through a sing,e oint ca,,e" the center o$ gravit+.

In &ight% the roc'et rotates aout the center o$ gravit+%

ut the "irection o$ the *eight $orce a,*a+s re!ains

to*ar" the center o$ the Earth.

uring ,aunch the roc'et urns u an" ehausts its $ue,%

so the *eight o$ the roc'et constant,+ changes. For a

!o"e, roc'et% the change is a s!a,, ercentage o$ the

tota, *eight an" *e can "eter!ine the roc'et *eight as

the su! o$ the co!onent *eights. For a $u,, sca,e roc'et

the change is ,arge an" !ust e inc,u"e" in the equations

o$ !otion. Engineers have esta,ishe" severa, !ass ratios

*hich he, to characterie the er$or!ance o$ a roc'et

*ith changing !ass. Fu,, sca,e roc'ets are o$ten stage" or

ro'en into s!a,,er roc'ets *hich are "iscar"e" "uring

&ight to increase the roc'et4s er$or!ance.

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-hrust is the $orce *hich !oves the roc'et through the

air% an" through sace. -hrust is generate" + the

rou,sion s+ste! o$ the roc'et through the a,ication o$

Ne*ton4s thir" ,a* o$ !otion) For ever+ action there is an

equa, an" oosite reaction. In the rou,sion s+ste!% anengine "oes *or' on a gas or ,iqui"% ca,,e" a *or'ing

&ui"% an" acce,erates the *or'ing &ui" through the

rou,sion s+ste!. -he reaction to the acce,eration o$ the

*or'ing &ui" ro"uces the thrust $orce on the engine. -he

*or'ing &ui" is ee,,e" $ro! the engine in one "irection

an" the thrust $orce is a,ie" to the engine in the

oosite "irection.

A,, roc'et engines ro"uce thrust + acce,erating a

*or'ing &ui". But there are !an+ "i5erent *a+s to

ro"uce the acce,eration% an" !an+ "i5erent avai,a,e

*or'ing &ui"s. Let4s ,oo' at so!e o$ the various t+es o$

roc'et engines an" ho* the+ ro"uce thrust.

-he si!,est roc'et engine uses air as the *or'ing &ui"%

an" ressure ro"uce" + a u! to acce,erate the air.

-his is the t+e o$ engine use" in a to+ a,,oon or a

sto! roc'et. Because the *eight &o* o$ air is so s!a,,%

this t+e o$ roc'et engine "oes not ro"uce !uch thrust.

A ott,e roc'et uses *ater as the *or'ing &ui" an"

ressurie" air to acce,erate the *or'ing &ui". Because

*ater is !uch heavier than air% ott,e roc'ets generate

!ore thrust than sto! roc'ets.

;o"e, roc'ets% an" !ost $u,, sca,e roc'ets use che!ica,

roc'et engines. he!ica, roc'et engines use the

co!ustion o$ roe,,ants to ro"uce ehaust gases as


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the *or'ing &ui". -he high ressures an" te!eratures o$

co!ustion are use" to acce,erate the ehaust gases

through a roc'et no,e to ro"uce thrust. -here are t*o

i!ortant arts o$ a che!ica, roc'et engine) the no,e%

an" the roe,,ants. -he no,e "esign "eter!ines the!ass &o* rate% ehaust ve,ocit+% an" eit ressure $or a

given initia, ressure an" te!erature. -he initia,

ressure an" te!erature are "eter!ine" + the

che!ica, roerties o$ the roe,,ants. Proe,,ants are

co!ose" o$ a $ue, to e urne" an" an oi"ier% or

source o$ o+gen% $or co!ustion. Kn"er nor!a,

te!erature con"itions% roe,,ants "o not urn% ut

require so!e source o$ heat% or igniter% to initiateco!ustion. he!ica, roc'et engines "o not t+ica,,+ re,+

on the surroun"ing at!oshere as a source o$ o+gen.

-here$ore% che!ica, roc'et engines can e use" in sace%

*here there is no at!oshere resent.

Aero"+na!ic $orces are generate" an" act on a roc'et as

it &ies through the air. Forces are vector quantities having

oth a !agnitu"e an" a "irection. -he !agnitu"e o$ the

aero"+na!ic $orces "een"s on the shae% sie an"

ve,ocit+ o$ the roc'et an" so!e roerties o$ the air

through *hich it &ies. B+ convention% the sing,e

aero"+na!ic $orce is ro'en into t*o co!onents6 the

"rag $orce *hich is oose" to the "irection o$ !otion%

an" the ,i$t $orce *hich acts eren"icu,ar to the "irection

o$ !otion. -he ,i$t an" "rag act through the center o$ressure *hich is the average ,ocation o$ the aero"+na!ic

$orces on an o(ect.

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Aero"+na!ic $orces are !echanica, $orces. -he+ are

generate" + the interaction an" contact o$ a so,i" o"+

*ith a &ui"% a ,iqui" or a gas. Aero"+na!ic $orces are not

generate" + a $orce /e,"% in the sense o$ the

gravitationa, /e,"%or an e,ectro!agnetic /e,". For ,i$t an"

"rag to e generate"% the roc'et !ust e in contact *ith

the air. So outsi"e the at!oshere there is no ,i$t an" no

"rag. Aero"+na!ic $orces are generate" + the "i5erence

in ve,ocit+ et*een the roc'et an" the air. -here !ust e

!otion et*een the roc'et an" the air. I$ there is no

re,ative !otion% there is no ,i$t an" no "rag. Aero"+na!ic

$orces are !ore i!ortant $or a !o"e, roc'et than $or a$u,, sca,e roc'et ecause the entire &ight ath o$ the

!o"e, roc'et ta'es ,ace in the at!oshere. A $u,, sca,e

roc'et c,i!s aove the at!oshere ver+ quic',+.

Aero"+na!ic $orces are use" "i5erent,+ on a roc'et than

on an air,ane. 7n an air,ane% ,i$t is use" to overco!e

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the *eight o$ the aircra$t% ut on a roc'et% thrust is use" in

oosition to *eight. Because the center o$ ressure is

not nor!a,,+ ,ocate" at the center o$ gravit+ o$ the roc'et%

aero"+na!ic $orces can cause the roc'et to rotate in

&ight. -he ,i$t o$ a roc'et is a si"e $orce use" to stai,iean" contro, the "irection o$ &ight. #hi,e !ost aircra$t have

a high ,i$t to "rag ratio% the "rag o$ a roc'et is usua,,+

!uch greater than the ,i$t.

#e can thin' o$ "rag as aero"+na!ic $riction% an" one o$

the sources o$ "rag is the s'in $riction et*een the!o,ecu,es o$ the air an" the so,i" sur$ace o$ the !oving

roc'et. Because the s'in $riction is an interaction et*een

a so,i" an" a gas% the !agnitu"e o$ the s'in $riction

"een"s on roerties o$ oth so,i" an" gas. For the so,i"%

a s!ooth% *ae" sur$ace ro"uces ,ess s'in $riction than a

roughene" sur$ace. For the gas% the !agnitu"e "een"s

on the viscosit+ o$ the air an" the re,ative !agnitu"e o$

the viscous $orces to the !otion o$ the &o*% eresse" as

the Re+no,"s nu!er. A,ong the sur$ace% a oun"ar+ ,a+er

o$ ,o* energ+ &o* is generate" an" the !agnitu"e o$ the

s'in $riction "een"s on the state o$ this &o*. #e can a,so

thin' o$ "rag as aero"+na!ic resistance to the !otion o$

the o(ect through the &ui". -his source o$ "rag "een"s

on the shae o$ the roc'et an" is ca,,e" $or! "rag. As air

&o*s aroun" a o"+% the ,oca, ve,ocit+ an" ressure arechange". Since ressure is a !easure o$ the !o!entu!

o$ the gas !o,ecu,es an" a change in !o!entu!

ro"uces a $orce% a var+ing ressure "istriution *i,,

ro"uce a $orce on the o"+. #e can "eter!ine the

!agnitu"e o$ the $orce + integrating% or a""ing u the

,oca, ressure ti!es the sur$ace area aroun" the entire

o"+. -he ase area o$ a !o"e, roc'et ro"uces $or!

"rag.

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occurs *hen a &o* o$ gas is turne" + a so,i" o(ect. -he

&o* is turne" in one "irection% an" the ,i$t is generate" in

the oosite "irection% accor"ing to Ne*ton4s thir" ,a* o$

action an" reaction. For a !o"e, roc'et% the nose cone%

o"+ tue% an" /ns can turn the &o* an" eco!e asource o$ ,i$t i$ the roc'et is inc,ine" to the &ight "irection.

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12bae031 assignment4

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