2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )

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http://astr-www.kj.yamagata-u.ac.jp/~shibata f4a f4b 2 f4cone f4eki f4end 4 f5meanfp f6coin () f6a f7a f7b f7d f8a f8b f9a f9b f9c f9kep f0a f0bt version feqmo fvec4 fvec fvec6 fvec2 fvec3 f3a (-D) f3b (3-D) f3c f3d (-D) f3e f3f f3h f4ae f4b f4d f4c f4e f4f f8a f8b f8c f8d f8e f9a f9b f9c f9e f9f

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )

3 Chapter 6 (f6coin) (f4end)? Chapter 5 (f5meanfp) ( ( )? N [] σ e = 8π ( ) e 2 2 3 mc 2 = 6.65 0 25 cm 2 e m c (, Thomson cross secion). 6 8 8 /2 6 0 /2 /2 6 000 000 6 0 000 /2 x 0 + ( ) ( ) x 0 x D = x D 2 ( 4 0 6 ) (f6a) l ( l ) σ (= 6.65 0 25 cm 2 ) n(8.4 0 23 cm 3 ) l = σn

4 R = 6.96 0 0 cm c = 3.00 0 0 cm/sec Chapter 7 (f7d) name= F7D (f7a) : I II ( ) III : T a 3/2 III name= F7A (f7b). :T = 27.3 :R = 3.84 0 8 m Chapter 8 (f8a) x = A sin ωt + B cos ωt A, B, ω t (f8b) a = 5 exp[ t 20 ] 0 Chapter 9 2. g = 9.8m/s 2 R = 6.38 0 6 m (f9a) x = v 0 t, y = y 0 2 gt2 name= F7B m v 0, y 0, g

5 (f9b) (R, θ(t)) t R θ(t) ω(t) = dθ(t) (x, y) (R, θ) x = R cos θ y = R sin θ x y F x, F y m (r, θ) t : r = r(t), θ = θ(t). ( ) l 2. = C( + ϵ cos θ) r C, ϵ (< ) l l (l ) (f9c) (f9b) d 2 θ 2 = g R sin θ θ R g t (f9kep) I II ( ) III T a T a3/2 Chapter 0 (f0a) x (Center of Mass) X CM = m x + m 2 x 2 m + m 2 m,m 2 x,x 2. V,V 2 CM V CM = m V + m 2 V 2 m + m 2

6 2. V CM 3. T CM T CM T = T CM + 2 (m + m 2 )V 2 CM 4. F,F 2 () M dv CM = F M = m + m 2, F = F + F 2 5. Chapter (feqmo) (x, y, z) θ (x, y, z ) x = x cos θ + y sin θ () y = y cos θ x sin θ (2) z = z (3) F x = F x cos θ + F y sin θ (4) F y = F y cos θ F x sin θ (5) F z = F z (6) (f0bt) (=eject) ( ;satellite (). M V m ( eject) V + U V 0 ( assumed) U 2. U/V 0 m (fvec4) f A B (fvec) d df (fa) = A + f da d da (A B) = B + A db m = m v = 3i + 2j k m 2 = 2m v 2 = 2i + 2j + 4k

7 (fvec6) r(t) v = dr/ t v = v ˆv = v/v (fvec2) kg ( m sec r = ti + (t + t2 2 )j ( 4 πt sin π2 2 )k.. t = 0 t = 2. 3. t = (fvec3) a = dv v2 ˆv + R n (7) R n. v = vˆv 2. 3. dˆv ˆv dˆv n dˆv = v R n 4. (7), (λ, ϕ ) (λ 2, ϕ 2 ) Chapter 3 (f3a). 2 mv2 + mgh = ( ) m, v, h, g T () U( ) T + U = ( ) 2.

8 (f3b). m T = 2 mv2 r U(r) F v dt = F v. 2. (f3d) O kx m( m) k, x O m 2 mv2 + U(x) = ( ) U(x) T = T 2 T = 2 F ds mg ( ) 2 mv2 + mgh = ( ) h ( ) (f3e) 2 m ivi 2 + Gm im j = ( ) r ij i (f3f) i,j a ( ) G σ (f3c) M( M) m( m) M m 2 mv2 + U(r) = ( ) (f3h) M R m M ( r r < R r > R )

9 Chapter 4 (f4ae) m F r(t) v(t) ( ( ) ). F 2. F U : 2 mv2 + U = ( ) (f4b) (f4d) m Ψ = Gm r gradient( ) C r G (f4c) N m i (i =, 2,..., N) Ψ = N i= Gm i r i r i (f4e) m (a, 0, 0) ( a, 0, 0) (x, y, 0). 2. 2 mv2 GmM r = E (f4f). M R 2.

0 Chapter 8-9 (f8a) l m (f8e) R I F ω (f8b) τ = xf y yf x F θ r F r 0 F θ r F r 0 (f8c) R V (f9a) CM R CM ( O CM ). O CM (f8d) L = i m i r i V i ( ) ( M) ω a b l = i m i r i V i (r i, V i CM ) L = MR CM V CM + l ( M = m i ) 2. CM M dv CM = i F i

dl/ = ()= r i F i CM 2. dl = i r i F i 3. (f9b) M (f9c) M ( ) (f9e) F8D (f9f) I 0 (turn table) (mono rail) m ω r. dr dω ω

2 (a-f4a.tex). 2. x 8x 00 2 x = 200 8 = 25[kg ] (8) 3. l l/4 x l = mg l 4 x = 25g[N] (9) g (a-f4b.tex) L R F p. MgL sin θ T a cos θ = LF = RF p T = L Mg tan θ (0) a 2. l H T l Mg H T = Mg H l ()

3 l H x h x = a sin θ h = L cos θ θ dx dθ = a cos θ, dh dθ = L sin θ, (2) θ x = (a cos θ) θ h = (L sin θ) θ H l = h x = L sin θ a cos θ (3) (??) ( ) T = L Mg tan θ (4) a (a-f4cone.tex) l h T l Mg h T = Mg h (5) l h l l = 2πr θ () h h = r/ tan θ l = 2π r h = r/ tan θ T = Mg 2π tan θ ϕ M 2π m = M( ϕ/2π) F (6) F = m g tan θ (7) T F F T sin( ϕ) F T = Mg/2π tan θ

4 (a-f4eki.tex) t H E = ( ) ( ) = gh m (8) m A ρ m = ρa H E = E 2 v = 2gH E 2 = 2 ( m)v2 (9) m = ρa H t v t a ρav t m = ρa H = ρav t (20) t 0 dh = a A v = a 2gH (2) A t = 0 t H 0 (H ) /2 dh = a 2g H 0 A H t 0 (22) ( ) H /2 H /2 0 = a g A 2 t (23) ( H(t) = H /2 0 a ) 2 g A 2 t (24) (a-f5meanfp.tex) (a-f6coin.tex) ( ) l = Nσ.5 024 N cm (25) (a-f6a.tex) N N l N = (R/l) 2 l/c t = l c ( ) 2 R l l.8cm t 9 0 0 sec (2.8 ) 3

5 (a-f7a.tex) T F = mrω 2 = mr T = a 3/2 F m/r 2 ( ) 2 2π T (a-f7b.tex) (a) a = v2 R = Rω2 = 2.7 0 3 m/s 2 ω = 2π/T = 2π/27.3day (b) 2 ( ) 2 R a = 9.8 = 2.7 0 3 m/s 2 R (a) (a-f7d.tex) R = 6.38 0 6 m v 2 /R v 2 R = g g = 9.8m/s 2 v = gr = 9.8 6.38 0 6 8km/s (a-f8a.tex) v = dx = Aω cos ωt Bω sin ωt a = dv = Aω2 sin ωt Bω 2 cos ωt = ω 2 x (a-f8b.tex) t = 0 T 0 T 0 dv = T 0 ( 5 exp t ) 20 ( v(t ) = 4 4 exp T ) 20 T v 4

6 (a-f9a.tex) F x = mẍ = 0, F y = mÿ = mg (a-f9b.tex) dx dy = R sin θ dθ = R cos θ dθ (26) (27) a a x = d2 x 2 = R cos θ a y = d2 y 2 = R sin θ ( ) 2 dθ R sin θ d2 θ 2 (28) ( ) 2 dθ + R cos θ d2 θ 2 (29) F = ma ( ) ( = mrω 2 F x F y cos θ sin θ ) ( + mr dω sin θ cos θ ) (30) ω = dθ/ F r = Rω 2 F θ = Rdω/ (a-f9c.tex) mrω 2ˆr mg cos θ T )ˆr T T = mg cos θ + mrω 2 ˆθ mr θ mg sin θ d 2 θ 2 = g R sin θ (a9kep.tex). (r(t), θ(t)) t t (/2)r rω t ( ) r 2 ω = r 2 θ = (3) dθ/ = θ = ω

7 2. F = ( F x F y ) = ( mẍ mÿ ) (32) x = r cos θ, y = r sin θ θ r 2 θ = l ẋ = ṙ cos θ l sin θ, ẏ = ṙ sin θ + l cos θ r r (33) ẍ = r cos θ l2 cos θ r 3, ÿ = r sin θ l2 sin θ r 3 (34) r = C( + ϵ cos θ), r = Cϵl2 cos θ r 2 (35) F = ( F x F y ) = mcl2 r 2 ( cos θ sin θ ) = mcl2 r 2 ˆr (36) ˆr r 2 (a-f9e.tex) θ = 0 θ = π r = C + e, r 2 = C e, (37) a = r + r 2 2 = C e 2 (38) P F +P F 2 = QF +QF 2 QF = a ()OF = a r OF Q b = QO b 2 = a 2 (a r ) 2 = C 2 ( e 2 ) 2 e2 C 2 ( e 2 ) 2 = C2 e 2 (39) b = C e 2 = a e 2 (40)

8 T πab h 2 = πab T = πa2 e 2 T C h 2 /GM 2 mv2 + ( 2 mv2 2 = GmM ) r r 2 (4) ht over2 = πa 3/2 C (42) v = h/r v 2 = h/r 2 C h 2 C (43) = GM (44) T 2 a 3 = 4 GM (45) (a-f0a.tex) (a-f0b.tex) MV = (M m)(v + U) + m(v V 0 ) U = m M m V 0

9 (a-feqmo.tex) (a-fvec4.tex) d (fa) f(t + t)a(t + t) f(t)a(t) = lim t 0 t f(t + t)a(t + t) f(t + t)a(t) + f(t + t)a(t) f(t)a(t) = lim t 0 t f(t + t)(a(t + t) A(t)) + (f(t + t) f(t))a(t) = lim t 0 t + t) A(t) = lim f(t + t)a(t + lim t 0 t t 0 (46) (47) (48) f(t + t) f(t) A(t) (49) t = f(t) da + df A (50) (a-fvec.tex) m v + m 2 v 2 = (m + m 2 )v v = 3 i + 2j + 7 3 k (a-fvec6.tex) v = vˆv v = v = v v a = dv = d (vˆv) = dv dˆv ˆv + v dˆv/t n a = dv v2 ˆv + R n R ( R ) ˆv ˆv = ˆv dˆv + dˆv dˆv ˆv = 2ˆv = 0 dˆv/

20 dˆv = lim t 0 ˆv(t + t) ˆv(t) t θ θ ( d ) θ R v t θ = v t dˆv/ = (v/r)n R (a-fvec2.tex) ( ) 2 πt v = i + ( + t)j cos k π 2 a = j + (sin π ) 2 t k t = t = 0 t = r 0 i + 3 2 j 4 π k 2 v i + j 2 π k i + 2j v 2 2 + 4 π 2 5 a j j + k 2 mv2 + 2 5 π 2 2 a = j + k = a ˆv + ( ) a = a v/v = 2/ 5 2 5 i + 5 j + k v 2 /R R = 5 30/6

2 (a-fvec3.tex) 2 r = (cos ϕ cos λ, cos ϕ sin λ, sin ϕ ) r 2 = (cos ϕ 2 cos λ 2, cos ϕ 2 sin λ 2, sin ϕ 2 ) l = Rθ R θ θ cos θ = cos ϕ cos λ cos ϕ 2 cos λ 2 + cos ϕ sin λ cos ϕ 2 sin λ 2 + sin ϕ sin ϕ 2 = cos ϕ cos ϕ 2 cos(λ λ 2 ) + sin ϕ sin ϕ 2 (a3b.tex). dt = d ( ) 2 mv v = mv dv = v F (5) dt = d ( ) 2 mv v = mv dv = v F (52) 2. T 2 T = 2 mv 2 2 mv (53) (z ) F ds = mgdz s = xi + yj + zk, ds = dxi + dyj + dzk, (54) 2 F = F x i + F y j + F z k = mgk, (55) z2 F ds = mg dz = mg(z 2 z ) (56) z z z 2 z F ds = 0 2 mv + mgz = 2 mv 2 + mgz 2 (57) h = z 2 = z 3 z + z 4 z 3 +... + z 2 z n

22 (a3b.tex). dt = d ( ) 2 mv v = mv dv = v F (58) dt = d ( ) 2 mv v = mv dv = v F (59) 2. T 2 T = 2 mv 2 2 mv (60) (z ) F ds = mgdz s = xi + yj + zk, ds = dxi + dyj + dzk, (6) 2 F = F x i + F y j + F z k = mgk, (62) z2 F ds = mg dz = mg(z 2 z ) (63) z z z 2 z F ds = 0 2 mv + mgz = 2 mv 2 + mgz 2 (64) h = z 2 = z 3 z + z 4 z 3 +... + z 2 z n (a3c.tex) m dv = GmM r 3 r v, r r = v G v ( ) ( ) d 2 mv2 = G mm r 3 v r = GmM r 3 r ds ds 2 T = mv 2 /2 T = ( ) ( ) 2 mv2 2 2 mv2 = 2 G mm r 3 r ds

23 ( ) e r e ds = dre r + dye e r = 0 2 G mm r 3 r ds = T = 2 2 G mm r 3 r (dre r + dye ) = G mm r 2 dr = GmM 2 [ ] 2 ( = GmM ) r r 2 r ( ) 2 mv2 GmM ( ) = 2 r 2 2 mv2 GmM r U(r) = GmM r G mm r 3 rdr (a3d.tex) U(x) = F ds = x x 0 ksds = 2 kx2 ( ) F = kx A B A B A B A C B A B C (a3e.tex) dt = ( ) d 2 m iv i v i i = ( ) dv i v i m i i () = ( ) v i G m im j r ij r 3 i ij j i (65) (66) (67)

24 dt = Gm m 2 r2 3 v r 2 G m m 3 r3 3 v r 3... G m 2m r 3 2 G m 3m r 3 3 = (i,j) v 2 r 2 G m 2m 3 r23 3 v 2 r 23... v 3 r 3 G m 3m 2 r32 3 v 3 r 32... G m im j rij 3 (v i v j ) r ij dv = = (i,j) (i,j) r 2 ij = r ij r ij ( d G m ) imj r ij G m imj r 2 ij dr ij (68) (69) (70) 2r ij dr ij dv = = 2r ij dr ij (i,j) G m im j r 3 ij = 2r ij (v i v j ) (??) d(t + V )/ = 0 r ij (v i v j ) (7) (a3f.tex) dc z dm dc z = Gdm a r 2 r = Gaσdρdθ r 3 σ dm = σdxdy = σρdρdθ (ρ, θ) (x, y) r 2 = a 2 + ρ 2 rdr = ρdρ C z = 0 2πGaσ r 3 ρdρ = a 2πGaσ r 3 rdr = 2πGρ

25 (a3h.tex) m = f4f f4f (a4ae.tex). P P 2 F F P 2 P 2 W = P F ds = F ds P 2 (72) F W F 2. ( m) m dv = F (73) v v v 2 = v v mv dv ( ) d 2 mv2 = v F (74) = v F (75) t P t 2 P 2 t2 ( ) d t2 t 2 mv2 = v F (76) t [ ] t2 t2 2 mv2 v F (77) t t = v = ds 2 mv2 2 P 2 2 mv2 = F ds (78) P v v 2 t t 2 ( ) O R F U U R U(R) = R m (??) R O F ds (79) 2 mv2 2 O P 2 2 mv2 = F ds + F ds (80) P O P P 2 = F ds + F ds (8) O O = U(P ) U(P 2 ) (82)

26 2 mv2 2 + U(P 2 ) = 2 mv2 + U(P ) (83) 2 mv2 + U = (84) (a4b.tex) E = 2 v2 GmM R = 2 mv v R v ( ) m M (a) E < 0 v E = 0 v = 0 E > 0 E > 0 2GM v > R (b) E < 0 E = GmM/r max r max = GmM E = GM v 2 /2 GM/R (a4c.tex) () C = N i= Gm i ri 3 r i Ψ = C ds = N i= ( Gmi r 3 i ) r i ds Ψ = ) N i= Gm i r i (

27 (a4d.tex) x C = Ψ x 2 + y 2 + z 2 = r 2 C x = Gmx r 3 C x = Ψ x = Gm x 2x = 2r r x r = Gm r r 2 x C = Gm r 3 x y z = Gmr r 3 (a4e.tex) Ψ = G, r, Ψ 2 = G, r 2 Ψ(x, y, z) = Ψ + Ψ 2 (85) r 2 = (x a) 2 + y 2 + z 2 r 2 = (x + a) 2 + y 2 + z 2 { } Ψ = Gm (x a)2 + y 2 + z + 2 (x + a)2 + y 2 + z 2 (a4f.tex) r P Ψ dψ = Gdm l (86) dm l P O P z (r, θ, φ) ρ dm = ρr 2 sin θdθdφ Ψ = 2π 0 dφ π 0 ( ) dθ GρR2 sin θ l φ OP = r l 2 = r 2 + R 2 2rR cos θ θ l ldl = rr sin θdθ ( Ψ = dl 2πGρR ) r (87) (88)

28 P l = r R r + R P l = R r R + r ρ = M/4πR 2 P Ψ = GM r P Ψ = GM R F = Ψ = { GM r 3 r 0 (89) (90) (9) (a8a.tex) v = (l/2)ω T = ( ) 2 l 2 2 ω 2 = ml2 ω 2 4 v = lω T = 2 (lω)2 = ml2 ω 2 2 (a8b.tex) (a8c.tex) R 2 V 2 l = mv R = mv 2 R 2 = E = 2 mv 2 GmM R = 2 mv 2 2 GmM R 2 = x = R /R 2 V 2 = V x x V R 2 2GmM V g = a = V /V g a 2 = a 2 x 2 x R x = (x )(a 2 x + a 2 ) = 0 x = a2 a 2 = V 2 /(2GM/R ) V 2/(2GM/R )

29 (a8d.tex) I = 2ma 2 I = 2mb 2 Iω = I ω ω 2ma 2 ω = 2mb 2 ω (92) ω = (a 2 /b 2 )ω (a8e.tex) I dω = F R dω = F R = ( ) I t ω = F R I t + ω 0 t = 0 ω = 0 ω 0 = 0 ω = (F R/I)t (a9a.tex) (a9b.tex) N i= x CM = m ix i N i= mi xdm dm ( ) D dm = M σ = 2M/πR 2 ( ) == D xσdxdy = D xσrdrdθ = π/2 π/2 dθ R 0 dr σr 2 cos θ = 2 3 σr3 t = sin θ x CM = 4R 3π ( ) y CM = 0

30 (a9c.tex) I = i m i () 2 (93) I = dm(r sin θ) 2 (94) = R dr π dθ 2π 0 0 0 dϕ ρr 4 sin 3 θ (95) ρ = 3M/4πR 3 cos θ = µ sin θdθ = dµ I = R 0 dr dµ 2π 0 dϕ ρr 4 ( µ 2 ) (96) I = 2 5 MR2 (97) (a9d.tex) R CM I I = MR 2 CM + I (98) = MR 2 + R2 + R2 2 M (99) 2 = 3R2 + R2 2 M (00) 2 (a9f.tex) (a) I 0 + mr 2 dr I 0 + m(r + dr) 2 v r = dr/ (I 0 + mr 2 )ω = [I 0 + m(r + dr) 2 ](ω + dω) (0) dω = 2mωr dr I 0 + mr2 (02) dω = 2mωr I 0 + mr 2 v r (03)

3 (b) (c) L = mr 2 ω (τ tab + τ m = 0) τ tab = I 0 dω = 2mI 0ωrv r I 0 + mr 2 (04) τ m = d (mr2 ω) (05) = 2mrω dr dω + mr2 (06) = 2mI 0ωrv r I 0 + mr 2 (07)

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