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1 NSW BOS Mhics Esio Soluios 8 F dowlod d pi fo wwwiuco Do o phoocopy opyigh 8 iuco Q L u 5 d ( ) c u u 5 Q Qc ( ) ( ) d 5 u d c d d l c d [ ] [ ] ( ) d l ( ) l l Qd L u fo > ( ) u d Wh u ; wh u d d ( u ) d [ ] u Q ( ) ( )( ) 8 ( ) ( ) ( ) d d u d d ( ) [ l( ) ( ) l( )] l l d Qi i i i i i i i i i Qii i cis i cis i cis cis cis i cis Qiii i cis i i cos i si cos Hc cos Qiv i i cos cis cis Qc z z 8 iy y iy y 8 cis y Rcgul hypol cd h oigi wih sypos y ± Qdi M is ( ω z ωz) d icps ( ) d ( ) z z cos isi cos isi z cos Qdii Digols of plllog isc ch oh M is h idpoi of digol PS L h copl u psd y S ( z ) z Hc z Qi y g( ) Q ( i )( i) i i i i i 7 i i 7 opyigh 8 iuco NSW BOS Mhics Esio Soluios 8

2 Qii y g( ) Qcii I d θ I I I ( I) Qiii y f( ) g g( ) ( ) < Qd Rdius lsi Spd of picl P v ω ωlsi Viclly: T cos g () v ω l si Hoizolly: T si () lsi g Solv () d () siulously ω lcos Qi A of LOM k Qii A of KLM A k l ( k l ) P Qi Sic z d z fo l z p ( z) z z > fo l z Hc ( z) p hs o l zos Qiii Qii is zo of ( z) ( )( ) p Hc 8 Qiii p( ) ( ) ( ) Qci Fo I Fo is zo of ( z) θdθ p I I ( θ θ) ( θ ) θ dθ dθ θ sc θdθ L u θ sc θ u wh θ d u wh dθ θ I I I I u u opyigh 8 iuco NSW BOS Mhics Esio Soluios 8 d A P ( d ) 8 ( 8 ( d) ( d) ) d d uis fo h foo of h fc Qiv Us Pyhgos ho o fid h sid lghs 8 7 R 9 A P 9 8 ( 9 7 )R R d uis

3 y Qi y By iplici diffiio d Gdi of g ( ) y d P is y Equio of g is y y ( ) Epd d g o Sic ( ) y y y y y y y P is poi o h llips d h P y c pssd s quio of g ( ) y y Qii Siilly quio of g ( ) Q y c y pssd s y T u v h iscio of h wo gs L ( ) y y u v d u v y u v y u v Rg o ( ) ( ) y y u v y y ( u v) T lis o h li ( ) ( ) y y y Qiii M is h idpoi of PQ M ( ) ( ) y y y is li hough h oigi O M sisfis h li cus ( ) ( ) ( y y) ( y y) y y y y ( ) M lis o h li OT i O M d T colli Q5i dp d P P 7 P 7 7 Psisfis dp d P P Q5ii To: P 7 7 Q5iii Evully: P 7 dp P Q5iv To: P d P d Aul % of gowh o % % Q5i p( ) ( ) ( ) ( ) ( ) ( ) p doul zo p ( ) p( ) hs zo p h zo is Q5ii p ( ) ( ) ( ) ( ) ( ) p( ) is cocv upwd fo Also ( ) (doul zo) p ( ) fo h -is Q5iii ( ) ( ) p wh p hs doul zo p fo p ouchs ( ) ( ) q( ) ( )( ) ( ) ( ) ( ) p p Q5ci ( ) h h d ± h h Q5cii A of coss-scio h h h opyigh 8 iuco NSW BOS Mhics Esio Soluios 8

4 Q5ciii Volu of ous: V h dh 8 8 h dh No: d givs of h of cicl of dius Q Giv p ( z) z z z c ω d I ( ) > wh ω hs zos ω d ω Th oos of z li o h ui cicl d spd y ω cis d ω cis Hc ω ω cos d ω ω cis z z z c ( z)( zω )( zω) ( z ) z ( ω ω) ωω z z ( ) ( )( ) z z z z z SR S R ( scθ) θ sc θ ( scθ ) θ sc θ ( sc θ) ( sc θ) θ sc θ θ ( ) sc θ ( sc θ) ( sc θ θ sc θ) ( sc θ) ( sc θ) Qci ( )(!! ) (! )(! ) ( )!! (! ) ( )! (! )! ( )! ( )! ( )!!!! ( )! ( )( )! ( )!! ( )! ( )!! ( )!!!!! y Qi y By iplici diffiio d Gdi of g ( scθ θ) scθ d y θ P : d Equio of g: y θ ( scθ) y scθ θ Epd d siplify: scθ y θ sc θ θ ( θ ) sc θ sc θ y θ Qii Shos disc w li y c d poi ( u v) is u v c SR scθ scθ sc θ θ sc θ θ P scθ θ is o h igh ch of h hypol ( > ) ( ) sc θ scθ d Qiii S R scθ sc θ θ ( scθ ) θ sc θ ( scθ) SR θ sc θ scθ sc θ θ Qcii Fo Qci opyigh 8 iuco NSW BOS Mhics Esio Soluios 8 Qciii As Q7i s p ( ) Q7ii Q7iii p d p p! s pd : p : : : Q7iv ( ) : ( )( ) ( )! : : :! : : fo lg

5 Q7i Q φ S R Q7civ Giv v wh v 5 5 log 5 ( log 8 ) 9 θ θ φ P c T L PQR φ TSP θφ (Eio gl quls su of opposi iio gls) RPT φ (Equl gls i l sgs) TPS θφ (S dig) TSP TPS Q7ii TS TP c ( TSP is isoscls) RT c d QT c TRP θφ (Eio gl quls su of opposi iio gls) TPQ θφ (S dig) TRP TPQ TRP d TPQ siil c c Hc c ( c)( c ) c c Epd d siplify c c Boh sids dividd y c c v v v d R d v < Q7ci ( ) dv d ( v ) ( v) Q7cii pss h vlociy of h cu d v v d Q7ciii ( ) [ ( ) ] v d ( v ) A h s of h dif ( ) v Giv ( ) c c v d v v v v log v v d v v v v v v log v si θ Q8 Th s: cosθ cosθ cos( ) θ siθ si θ siθ cosθ Wh RHS cosθ LHS h siθ siθ s is u wh Assu i is u wh k si kθ i cosθ cosθ cos( k ) θ siθ Wh k cosθ cosθ cos kθ cos k LHS ( ) ( ( ) ) cos θ cosθ cos( k ) θ cos( k )θ si ( k ) θ si( kθ θ) RHS siθ siθ si k θ cosθ coskθ si θ siθ ( si θ) si k θ coskθ siθ cosθ siθ si kθ cos kθ siθ cosθ si kθ si θ siθ siθ si kθ cos kθ cosθ si kθ siθ siθ si kθ cos( kθ θ) siθ cos θ cosθ cos( k ) θ cos( k ) θ LHS h s is u fo k Hc i is u fo Q8i δ δ A Ak R siδcos cos cos k ( ) si δ δ δ si δ R siδ R si cos δ δ si si δ R cos si δ δ Sic δ d hc A R cos Q8ii As cos A of sph) δ R ( of h sufc opyigh 8 iuco NSW BOS Mhics Esio Soluios 8 5

6 Q8ci f( ) ( ) si si si( ) si wh > > < d f si si si si f f ( ) ( ) ( ) ( ) cos ( ) si si cos( ) ( ) si( ) si si si( ) ( si( ) si si ( ) ) f ( ) f( ) si Q8cii f( ) ( si cos cos si ) si si ( si cos cos si ) si cos si cos si si si si cos si cossi cos si si si cossi si cos cossi si ( ) si ( ) si si( ) si Q8ciii si( ) si( ) si si ( ) si si si( ) i ( ) si Sic si < < si( ) si d si( ) Hc k k wh k J d si Pls ifo hli@iuco cocpul hicl d/o ypig os opyigh 8 iuco NSW BOS Mhics Esio Soluios 8