L4:4. motion from the accelerometer. to recover the simple flutter. Later, we will work out how. readings L4:3

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elave moon L4:1 To appl Newon's laws we need measuemens made fom a 'fed,' neal efeence fame (unacceleaed, non-oang) n man applcaons, measuemens ae made moe smpl fom movng efeence fames We hen need a

1 elave moon L4:1 To appl Newon's laws we need measuemens made fom a 'fed,' neal efeence fame (unacceleaed, non-oang) n man applcaons, measuemens ae made moe smpl fom movng efeence fames We hen need a wa o ecove he neal measuemens ang accoun of he moon of he measuemen fame n ohe suaons, we ma wsh o ecove moons elave o a movng fame fom measuemens of neal moon Thee nds of oseves neal oseve unacceleaed non-oang L4: Tanslang oseve acceleaed u non-oang oang oseve acceleaed and oang O See Dnamcs (vol 1) L Halfman ddson-wesle (196) Fo eample The flue moon elave o aes fed n he od of he acaf s elavel smple: flappng 'up' and 'down' We see o measue placng an acceleomee a he wng p wh s sensve as algned wh he -as of he acaf Howeve, he acceleomee measues he -componen of he neal acceleaon plus he acceleaon due o gav Lae, we wll wo ou how o ecove he smple flue moon fom he acceleomee eadngs L4: Noaon Tanslang oseve O O L4:4 O O O = O = neal oseve O We wll geneall use: Uppe case lees (,, ) fo neal measuemens Lowe-case lees (, v, a) fo nonneal measuemens oson of elave o oang oseve

2 Moon elave o anslang oseve neal poson of pacle O O X O YO Z O = O +.e., poson of elave o = poson of elave o O plus poson of O elave o d d O d neal veloc of pacle d d d = = + whee d/d denoes me ae of change seen neal oseve Oseve ma choose o desce usng Caesan coodnae ssem wh un vecos, whch he/she sees as consans: = + + eloc seen s v = = + + whee / denoes me ae of change seen movng oseve L4:5 Tanslang efeence fames Consde pacles,, C n moon Suppose and C ae oseved anslang oseve movng wh. Z Then asolue poson, veloc and acceleaon of, e.g., ae gven : = +, = + v, = + a C C C/ X Y /C / We could us as well ge o move wh and oseve : = +, = + v, = + a Cleal, hen: Thus, elav of osevaons of anslang oseves s saghfowad (ecause aes of change he see ae no affeced oaons) =, v = v, a = a f moves wh C: = C + v C (1) whee C = + vc () and = + v () (1), () = + v C + vc (4) (), (4) v = v C + v C L4:7 d O d = +, = + +, v = d d = + + Now, ecause ae non-oang, also sees hese un vecos as consan;.e., d = + + = = v d Hence, neal veloc of = O + v O O X O YO Z O.e., asolue veloc of = asolue veloc of O plus veloc of elave o anslang oseve neal acceleaon of d dv = = O d + d d v v = = + + = a d = O + a L4:6 Eample: Helcope landng on shp n coss wnd Shp has fowad speed of 0 nos, due Noh Ocean s saona Wnd s fom he Wes a 0 nos Conolle wans helcope o descend vecall a 10 nos Wha helcope veloc elave o he a s equed? L4:8 (Noh) veloc of shp: = S 0 veloc of a: = 0 eloc of helcope elave o shp: v = H S 10 veloc of helcope: H = S + vh S = + vh Hence: vh = S + vh S = (nos) Helcope speed elave o a: 0 v = H 0 = 0 nos n oseve v H

3 Moon elave o oang oseve L4:9 neal poson of pacle = +.e., poson of elave o = poson of elave o plus poson of elave o d d O d neal veloc of pacle d d d = = + whee d/d denoes me ae of change seen neal oseve Oseve ma choose o desce usng Caesan coodnae ssem wh un vecos, whch he/she sees as consans: = + + eloc seen s v = = + + whee / denoes me ae of change seen movng oseve d d d d = d d d d Thn of un veco as poson veco of he pon = 1 on he -as; hen d/d s he veloc of hs pon = α v = sn n = 1 sn Thus d d = d d Smlal =, = d d L4:11 d Hence = d = + ( + + ) d.e., = + Cools Theoem d (ue fo an veco) ae of change seen ae of change seen angula veloc of d e d d d = +, d d = + + Howeve, ecause ae now oang, sees hese un vecos changng wh me;.e., v = d d d d = d d d d, v = = + + L4:10 ecall he aes of change of un vecos n clndcal coodnaes: d e = e = e e = e = d e d e = d e e = e = e e = e e d d 1 e Noe ha he angula veloc of he gd ad e e e s a veco quan: = = e = e magnude s nsananeous ae of change of oenaon decon s nsananeous as of oaon gh-hand scew convenon e e e Moon elave o oang oseve L4:1 neal poson of pacle = + d d O d neal veloc of pacle = = + d d d.e., = + v+ vel. of el. of vel. of seen Coecon fo oseve s oaon

4 Moon elave o oang oseve = + v+ neal acceleaon of pacle d d dv d d = = d d d d d v = + + v acc. of cc. of seen Cools componen = + a+ + v + O acc. of Tangenal componen Cenpeal componen pplcaon o spal sun manoeuve acc. of = + a+ + v + acc. of Le's assemle he componens we need s. acceleaon of : = T ϕ n = = 4.55 m/s L4:1 L4:15 pplcaon o spal sun manoeuve ssume ha he acaf -as s algned wh he angen un veco lane mus oll aou hough an angle ϕ o manan a coodnaed un Z L4:14 Lsn ϕ = m Lcos ϕ = mgcos β an ϕ = cos β g ϕ = 17 ϕ n mg cosβ Lf L ew along -as Coodnae ansfomaon cosϕ snϕ = n 0 snϕ cosϕ.e., { } Tϕ { n } = s n e Y X e β e cc. of seen. componen s vaon we wan o measue! cceleomee { } { } Componens n acaf aes of asolue acceleaon of ngula veloc of nnsc ad n /σ / n = + σ ngula veloc of acaf: = n +ϕ 1 1 ϕ = an cos β, ϕ g = 1+ an ϕ g { } = + a+ + v + ecall ha and σ ae consan fo ccula hel = + + ϕ σ L4:16 T n = ad/s ngula acceleaon of acaf:.e., = + ϕ ϕ + + σ + n + n σ σ ϕ = + ϕ σ n σ { } T ad/s n = ϕ = ( ) ( 1+ an ) cos β ϕ anϕ g ϕ

= + a+ + v + L4:17 Cools acceleaon: a = co v We don now v, u we can we v = v 5 Hence v s pependcula o, so ha he -componen of a co s eo Tangenal acceleaon: a ang = We don now, u vaon amplude s small,

5 = + a+ + v + L4:17 Cools acceleaon: a = co v We don now v, u we can we v = v 5 Hence v s pependcula o, so ha he -componen of a co s eo Tangenal acceleaon: a ang = We don now, u vaon amplude s small, so we can we 5 Then 5 5 Le = α + α+ α = α+ α Hence, -componen of a ang s 5α T = T ϕ n = s = m/s Hence, -componen of a ang coodnae ansfom { } { } Cenpeal acceleaon: a = cenp ( ) = = 5+ 5 Hence, -componen cenp of a s = m/s { } T ϕ { n } = = T The acceleomee eadng L4:18 peoelecc acceleomee The acceleomee esponds o he -componen of he asolue acceleaon, and he -componen of he gavaonal acceleaon g (whch egses as a negave acceleaon) + ndcaed m Wng p suface g cos β cos cos g β ϕ osve acceleaon Componen of gav along sensve as of acceleomee 0 g= ge,.e., { g } = 0 g 0 cosβ snβ 0 g snβ β = = 0 = 0 0 snβ cosβ g gcosβ { g } T { g } n g snβ 0 cosϕ snϕ = = 0 0 snϕ cosϕ g cosβ g sn β g = gcos βsn ϕ g cos β cos ϕ as epeced! { g } Tϕ { g }.e., n { } s Z n e Y X g ϕ β e L4:19 e.e., he 'ndcaed' acceleaon s cos cos = g = g β ϕ ndcaed We can see hs nspecon, u le's wo ou fomall usng coodnae ansfomaons acc. of ung all ogehe! = + a+ + v + L4:0 acc. of = a = ndcaed + gcos β cos ϕ + + v + = ndcaed [ ] = ndcaed m/s equed coecon. hew!! cc. of seen. componen s vaon we wan o measue! cceleomee cos cos = g = g β ϕ ndcaed Hence, vaon acceleaon s

cles fom Meam & age (5h edn) elevan o hs lecue /8 elave moon (anslang aes) 5/1-5/ noducon, oaon conceps of anslaon and oaon of gd od; angula veloc 5/4 elave veloc smplfcaon fo plane moon 5/6 elave

6 cles fom Meam & age (5h edn) elevan o hs lecue /8 elave moon (anslang aes) 5/1-5/ noducon, oaon conceps of anslaon and oaon of gd od; angula veloc 5/4 elave veloc smplfcaon fo plane moon 5/6 elave acceleaon smplfcaon fo plane moon 5/7 Moon elave o oang aes smplfcaon fo plane moon 7/-7/6 eneal moon genealsaon o hee dmensons L4:1 s a oang oseve, wh angula veloc, who sees poson, veloc and acceleaon of as, v and a L4: v = + + = + a+ + v = = = 1. (m/s) = = (m/s).96 (m/s ) =.96 (m/s ) 4.14 = + (m) cos 45 = 60 = 0. (ad/s) Eample polem: Meam & age 4 h edn 5/164 L4: 5/170 n 5 h edn = + v+ v = v = ( + ) = ( ).e., v= (m/s) = + a+ + v + a= v v = + a 0 a L4:4.e., a= (m/s )

Chapter 3: Vectors and Two-Dimensional Motion

Chapter 3: Vectors and Two-Dimensional Motion Chape 3: Vecos and Two-Dmensonal Moon Vecos: magnude and decon Negae o a eco: eese s decon Mulplng o ddng a eco b a scala Vecos n he same decon (eaed lke numbes) Geneal Veco Addon: Tangle mehod o addon