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O r r r, F F r,r r = r r F = F (. ) r = r r 76

77

d r = F d r = F (. ) F + F = 0 d ( ) r + r = 0 (. 3) M = + MR = r + r (. 4) P G P MX = + MY = + MZ = z + z PG / PG = / M d R = 0 (. 5) 78

79

d r = F d r = F r = r r d ( r r) = F F d r = F + µ = (. 6) + µ d r = F (. 7) F r ( = r ) r 80

r r r = r r f () r f ()= r G r r r >> 8 P Q, kk ( > 0)

j F j F,, 3,... r, r, r 3,... d r = F + F + F3 +... 8 d r = F + F + F +... (. 8) 3... Fj + Fj = 0 d d r r + +... = F F + +... (. 9) M = + +... R MR = r + r +... (. 0) M d R = F + F +... = F (. ) M R F + F+...

83

84 F F O A B r l F O O l F l O F O O O F Fl N = r F ( r = OA) (. r (,, z) F ( F, F, Fz) N N = Fz zf N = zf Fz (. 3 N = F F z N = Fl = Frsnθ (. 4 r F r F θ = 0 π N = 0

85 C=A B AB C = ABsnθ θ AB θ < π ABsn A AB A B O B C AB A A C = A B (. 5 Bsn A B = ( B A) (. 6 B, j,k j = k, j = k, j k =, k j = (. 7 k = j, k = j, = j j = k k = 0 k= j A = A + A j + A k, B = B + B j + B k C = A B = A + A j + Azk B + B = ABz Az B + Az B A Bz j + A B = C + C j + C k ( ) ( j + Bzk ) ( ) ( ) ( A B ) z z k (. 8 ω [rad/s] k r v ω r z k O r P r rθ ω [rad/s]ωr r k k r v = k ( ω r) = ω( k r) k = ( 00,, ) r = (,, 0) v = ω, v = ω, v = 0 z A = 3j k B = + 4j k A B ( A+ B) ( A B) ( A B) + ( A B) = A B ( A+ B) ( A B) = ( B A) z S= j O v

v v p = v (. 9 l l = r p (. 0a r l = ( l, l, lz) r = (,, z) p = ( p, p, pz) (.8) P l = p zp, l = zp p, l = p p (.0b) z z z r pz N l z Nz = F F (. lz = p p d d p =, p = dl z d d = (. F = &&, F = && dl z = F F (. 3 dl z = N z (. 4 86 d l = N (. 5 N

dlz = d d d d d d d d d d ( ) p p = = + = d d ( fg)' = f ' g + fg' L L L = L = 0, Lz 0 pz zp = 0, zp pz = 0 pz zp + zp pz = 0 zp + zp = 0 zp ( p) = 0 L ( p p ) 0 z = z = 0 z = 0 O r p v r p O P v 87

v, v,... dv dv d + +... = ( v + v +...) = F + F +... p = v d ( p + p +...) = F + F +... (. 6) d p = F (. 7) P p (. 8) = t dr dr p = v = = M d P = M R (. 9) d P d R = M = F (. 30) P M dr / d P = M R dp / F dp d R = M = F P 88

89

F t ( dv / ) = F F F t t ( ) t dv t = v( t ) v( t) = t F t t p = v t t t p ( t ) p( t = F (. 3) ) t t F F, F, F t z F t t F e =/ > 0 e e e = I = 0< e < e = 0 I = v 90

9 g g d = g t o t d= g t d= g t0 t0 t t [ ] = g[ t] t 0 ( t) ( t ) = g( t t o t t 0 0 ) ( t) ( to ) = g( t t0 ) F = g t t F = g g( t t0) t = 0 t0 θ, ( > ), / 0 >

R = OG r r ' r r = R + r ' (. 3) = X + ' = Y + ' z = Z + z ' t dr dr dr ' = + v = V + v ' (. 33) v ' = + r R r (. 34) = M R = MR M R = r = + ' r r r r ' = 0 (. 35) t dr ' = 0 v ' = 0 (. 36) 9

O R G r r 93

94 ' ',, ' z z z V V V + = + = + = / v ( ) ( ) ( ) ' ' ' ' ' ' ' ' ' z z z z z z z V V V V V V V V V + + = + = + + = + = + + = + = + = + = + = ' ' ' z z z MV MV MV ( ) ( ) ( ) + + + + + = + + ' ' ' z z z V V M V + = ' M v V v (. 37) (.36)

3 (0,3) (-,0) (,0) 95, -, - + G = + K' K' = µ ( ) µ K'

dl/ = N l N = r F dl = r F + r F + r F3 +... dl = r F + r F + r F3 +... (. 38) dl3 = r 3 F3 + r 3 F3 + r 3 F3 +...... F = F r j j L = l + l + l +... = (. 39) 3 l dl = r F + r F + r 3 F3 +... = r F (. 40 dl = 0 t L = C ( C : ) L 96 F

j rj O Fj r Fj 97

98 [. 6 R O r r ' = r R L R L = r v r = L G + L' G L = R MV L' = r ' v ' (. 4 G, LG L' LG = R MV t M dv / = F dl d G V = R M = R F = R F (. 4 dl / = r F L = L G + L' r = R + r ' dl' = r ' F (. 43 N ' = r ' F N '= 0 L' = C ( C : ) L'

L = r v = ( r ' + R) ( v ' + V ) = ( R) V + ( r ') V + R ( v ') + ( r ' v ') r ' = 0 v ' = 0 L = L G + L' dl / = r F L = L G + L' r = R + r ' dl = r F dlg dl' + = ( R + r ') F dl' R F + = R F + r ' F dl' = r ' F Coffee Break r v L = vr M GM / r = v / r v = GM / r = L /( r) r = L /( GM ) L r 99

MgM F, F,..., F n r, r,..., r n F r F, F,..., F n n 00

0 G O F (=G ) r 3 r r3 F F3

0 d R M = F d R / = 0 F = 0 (. 44) L' L '= 0 L LG ' L L = L G + L' LG = 0 L = LG + L'= 0 dl = 0 d L / = r F r F = 0 (. 45 r F = 0, F = 0, Fz = 0 ( Fz zf ) = 0, ( zf Fz ) = 0, ( F F ) = 0

a M D l b S l A S a C b B Mg AB R F S sn θ = R S cosθ + F = Mg amg = bs cosθ F,R,S amg almg S = = bcosθ b l b amg sn θ amg R = = b cosθ l b amg a F = Mg S cosθ = Mg cos θ = Mg b cosθ b b>a F>0 b<a F<0 A N l G N Mg F B 03

04 z j =r j ϕ& z P j O z ϕ 0 r j j ϕ ϕ C j ϕ& z ϕ(t) t d ϕ / L z ϕ j j P j j z j P z PC P C d ϕ / PC = j + j = rj P j CP r j d ϕ / j l = (.3 (.0b)) z ( ) Lz = l jz = jr j ( dϕ / ) j j (. 46 = j j (. 47 j I dϕ L z = I (. 48 L ( = r P ) d Lz = ( F F ) d ϕ = ( F F ) I (. 49 z

j = rj ( dϕ / )snθ j = j ( dϕ / ) ( rj snθ j = j ) j = rj ( dϕ / )cosθ j = j ( dϕ / ) ( rj cosθ j = j ) j z L z C l = r p = r ( v ) j =r j ϕ& r j j j j j j P j 05 l l = z l z = = ( z ) z ( ) ( ) z l = jz = ( + ) j = j j j jrj j ( dϕ / ) j j j j ( dϕ / ) z Lz = l jz = jr j ( dϕ / ) j j

ϕ d ϕ / r j = ( dϕ / ) j r j ( / ) K = j j = jrj ( dϕ / ) j j I K = I dϕ (. 50 ϕ d ϕ / d / I I( dϕ / ) ( d / ) I ( d ϕ / ) = N z ( d / ) = F I( dϕ / ) ( d / ) 06

07 z z h M I h ϕ N z = ( Mg snϕ) h O ϕ h z d G ϕ I = ( F F ) g d ϕ I = Mghsnϕ ϕ snϕ ϕ d ϕ Mgh = ϕ I d k = ω, ω = ϕ(t) ϕ0 a ϕ ( t) = ϕ0 sn( ωt + α) Mgh ω = I T π I T = = π ω Mgh π l / g I l = Mh l

08 a ω l z l = r p = r ( v ) l z = a( ) = aω l z = a( ) = a( aω) = a ω I I = r j j j I = a I l z lz = Iω I = r (. 5 j j j I dv j ρdv ρ I I = + ρdddz (. 5 ( ) z d ϕ I = ( F F ) = N N d ϕ / I

09 M l a O l-a a d d ( M / l) d l = a l a I M M 3 l a M ρ d = d = [ ] a = ( l 3la + 3a ) a a l 3l 3 l M l l Ml a = I = l 3l 3 = 3 + 4 M Ml a = l( I = l 3l + 3l ) = 3 3 M a M / πa r r + dr I = j r j r j I = r j M r = πrdr r πa r a M a 3 M 4 a I = r dr = [ r ] 0 = Ma a 0 a 4

M a a r z z dz O z dz z r r = a z = ρ πr dz = ρπ( a z ) dz I = Ma / dz = Ma = ρπ( a z ) dz ( a z ) = ρπ( a z ) dz M 3M z ρ = = 3 V 4πa a a I = = ρπ ( a z ) dz = Ma a a 5 0

I I G M λ I = IG + Mλ (. 53 M I I = Mκ (. 54 κ = I / M a Ma / 5 / 5 a

I z z' I = r ρdv IG = r' ρdv r r ( ) ( ) z r = ' + X + ' + Y G z O = ' + X' + X + ' + Y' + Y ( ) ( ) λ = X + Y r = r' +λ + X' + Y' I = r' ρdv + λ ρdv + X ' ρdv + Y ' ρdv I G ρdv M v ' = 0, z σ ds σds z,, O ds I = r σds, I = σds, I σds z = r = + Iz = I + I (. 55 l z O a b

R F M d X F M d = Y = F, (. 56 Mg I G L z ' dϕ Lz'= IGω = IG dl' = r ' F d ϕ IG = ( F F ' ' ) (. 57 ( /3) g snθ g snθ ( /3) gt snθ gt snθ 4 g t sn θ g t sn θ 9 3

θ M a F R M d X Mg F = snθ M d Y R Mg = cosθ d ϕ I G = Fa F R X(), t Y(), t ϕ() t F R ϕ Y = 0 d X d ϕ = a F R Y = 0 R = Mgcosθ IG = Ma / d X = g snθ 3 d X t = 0 X = 0, = 0 dx = gt sn θ 3 X = gt sn θ 3 X = l l snθ Mgh Mgl snθ 4

IG = Ma / Ma d ϕ M d ϕ F = = a a d X d ϕ a = d d ϕ d X a = t M d X F = d X M d X M = Mg sn θ d X = g snθ 3 r r G V V ω 5

z M l N N NMg l sn (.64) N L L dl dln (.65) L z 6

7

8 L L P O P L P z L sn P P P OP P P dl PP z N P L sn z dl dl PP dll sn (.66) L sn Mg l sn Mg l L (.67) M l L L I LI (.68) Mg l I (.69)

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J M 0 M 0 J (.54) F l l af F dω I = ( l a) F af ' Iω = ( l a)j 0 0 I = Ma 5 a a( l a) 5( l a) a ω0 = J = J I am J 5( l a) 7a 5l 0 ' 0 aω0 = J = J M am am

l 7a/5 = aω 0 0 ' 0 0 <

J 5( l a) 7a 5l 0 ' 0 aω0 = J = J M am am l l 7a/5 ( ) l 7a/5 0 0 0 0

3 l 7a/5 = aω 0 0 ' 0 0 <

4 l 7a/5 0 0 Mg d dω M = µ Mg, I = µ Mga (.54) J = µ gt = µ gt M Mg (.55)(.57) dω I = aµ Mg 5 5 aω' = a aµ Mg t = a aµ Mg t = µ gt I Ma (.57) 5 5( l a) J 5 a ω = aω0 + aω' = aω0 + µ gt = + µ gt a M a (.60)(.6) ( 7a 5l ) J t = 7Mgaµ a (.6) 5l J = 7a M (.63) l a ( 0 0) l 7a/5

5 l 7a/5 = aω 0 0 ' 0 0 >

O G OG P OG Y 6 OG P(,0) O I Y I ( ) I Y X,Y M 0 0 u, M ux 0, M Y 0 Y OG h u0, h I X 0 0, Y 0 = ω Mh Y 0 0 = 0 I Mh 0 O P O O 0 P

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