2 g g = GM R 2 = 980 cm s ;1 M m potential energy E r E = ; GMm r (1.4) potential = E m = ;GM r (1.5) r F E F = ; de dr (1.6) g g = ; d dr (1.7) g g g
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- あけなお たけくま
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1 1 1 (gravitation) 1.1 m F a ma = F (1.1) F a m F 1.1 m F a (1.1) m a F m F a m a F F a m (universal gravitation) (potential) M m gravitational force F r F = ; GMm r 2 (1.2) G = 6:67 10 ;8 dyn cm 2 g ;2 gravitational acceleration g g = F m = ;GM r 2 (1.3) 1.1 M M =5: g r R = 6378km
2 2 g g = GM R 2 = 980 cm s ;1 M m potential energy E r E = ; GMm r (1.4) potential = E m = ;GM r (1.5) r F E F = ; de dr (1.6) g g = ; d dr (1.7) g g g = ;r (1.8) r (x y z) @! (r ' z) (1.10) 1.2 M m 0-1 LA T EX M g
3 3 2 z = ; 1 2 gt2 + v 0 t + z 0 (2.3) g g = 980 cm s ;2 t = 0 z =0 v =0 dz = v = ;gt (2.4) dt z = ; 1 2 gt2 (2.5) x y z m d2 z dt 2 = ;mg d2 z = ;g (2.1) dt2 m g 2.2 z 0 v 0 dz dt = v = ;gt + v 0 (2.2) 2.3 v z 2.2 z m dv dt = ;mg ; kv (2.6) m g k v z
4 4 2.4 t =0 z =0 v =0 2.6 dv v + mg=k = ; k dt (2.7) m ln v + mg k = ;kt m + C 1 (2.8) v = ; mg k + C 2e ;kt=m (2.9) t =0 v =0 C 2 = mg=k z v = mg k (e;kt=m ; 1) (2.10) t =0 z =0 z = ; mg k t + m k (e;kt=m ; 1) (2.11) t! v! v1 = ; mg k (2.12) z!; mg k t (2.13) 2.12 v1 terminal speed 2.6 ;mg ;kv 0 1mm 7m/s km 1km km 10 km
5 5 3 r t v 2 2 ; GM r = E = v2 0 2 ; GM r 0 (3.3) E 3.3 v 2 =2 ;GM=r E s dr dt = v = ; 2E + 2GM (3.4) r M r g = ;GM=r 2 m m d2 r dt 2 = ;GMm r 2 d 2 r dt 2 = ;GM r 2 (3.1) t = t 0 r = r 0 v = v 0 v 0 0 E = ;GM=r 0 s dr dt = v = ; 2GM r ; 2GM r 0 (3.5) v = dr=dt dr d 2 r dt dt + dr GM =0 (3.2) 2 dt r r = r 0 r =0 t r r 3 0 t = (3.6) 8GM 3.5 r = r 0 cos 2 3.2
6 6 3.2 r = R r = 0 T v T = u t R2 2 GM (3.12) T =21 (3.13) r M r M r = 4 3 r2 (3.7) 3.4 1pc 10 5 M 1M =1: g r g = ; GM r 2 = ; 4G r (3.8) 3 r d 2 r dt 2 = ;4G r (3.9) 3 t =0 r = R v =0 v G 3 r 2 = E = 2G R 2 3 (3.10) M =(4=3)R 3 v s v GM u v = t r 2 1 ; (3.11) R R 2
7 7 4 Kepler's laws 1 v v = r 4.2 s GM = (4.2) r 3 v s GM v = r = r (4.3) M m r M m M m r M m m GMm r 2 1 = mr 2 = m v2 r (4.1) T 2 a 3 T 2 =a 3 P s P = 2r v = 2 =2 r 3 GM (4.4) 4.3 v km 4.3
8 8 4.2 V R g V < p gr V = p gr p gr <V < p 2gR V = p 2gR V > p 2gR V 2 = p 2g R = 11:2kms ;1 escape velocity
9 9 5 binary binary system M 1 M 2 separation center of mass orbital plane inclination angle orbital period mass ratio primary star companion star 5.2 M 1 M 2 a 1 a 2 a distant binary close binary visual binary spectroscopic binary eclipse occultation eclipsing binary i
10 M 1 2 M a 1 a 2 a a 1 +a 2 = a a 1 : a 2 : a = M 2 : M 1 :(M 1 + M 2 ) (5.1) T 2 =a 3 = 5.2 A A B P 50.0 a A B a 1 : a 2 M 1 : M 2 A B GM 1 M 2 a 2 = M 1 a 1 2 (5.2) a 1 a 1 G P (= 2=) 2 2 GM 2 = a 1 a 2 (5.3) P GM 1 = a 2 a 2 (5.4) P a = a 1 + a 2 M 1 + M 2 = a3 G 2 2 (5.5) 3 P M 2 P T
11 11 6 star cluster 6.1 open cluster galactic cluster association OB OB OB association 6.1 M50 NASA globular cluster M80 NASA/STScI M b 10 5 M < M < 10 6 M (6.1) 50 < b < 500 (6.2) 6.2 ;GM=r 1 1
12 12 Plummer Plummer M r = ; p GM (6.3) r2 + b 2 b 0 r b 6.5 d 2 r dt 2 = ; GMr (r 2 + b 2 ) 3=2 = ; GMr b 3 (1 + r2 b 2 ) 3=2 ; GM b 3 r (6.7)! s GM! = (6.8) b 3 A B r = A sin!t + B cos!t (6.9) -GM/b 6.3 r ; d dr = ; GMr (6.4) (r 2 + b 2 ) 3=2! P =2=! 6.1 M 10 5 M b 50 M 10 6 M b 500 N d 2 r dt 2 = ;d dr = ; GMr (6.5) (r 2 + b 2 ) 3=2 1 2 v2 ; p GM = E() (6.6) r2 + z
13 13 7 galaxy The Galaxy (dark matter) 10 Hubble classication M elliptical galaxy disk galaxy irregular galaxy peculiar galaxy spiral galaxy barred spiral galaxy 220km/s 2 rotation curve V [/] r []
14 r m V mv 2 =r r M(r) GM(r)m=r 2 GM(r)m=r 2 GM(r)m=r 2 GM(r)m r 2 = mv 2 r 2 (7.1) M(r) M(r) = rv 2 (7.2) G r V M(r) V [/] r [] 7.4 M31
15 15 8 group of galaxies cluster of galaxies 8.1 NAO M km 500 dark matter
16 16 virial theorem T 2K +=0 (8.1) M R V T NASA/STScI K 1 2 MV 2 (8.2) ; GM 2 (8.3) R 8.1 M RV 2 G (8.4) z =0: km s ;1 2.9 H =75km s ;
17 17 9 3K reball big bang Big Bang Universe H z =(H=c)r (9.1) v v = cz v = Hr (9.2) Hubble's law z r c H Hubble constant 1Mpc [km/s] H =71 4km=s=Mpc (9.3)
18 18 lambda term cosmological constant expanding universe model 3 closed universe at universe open universe 9.3 =0 9.3 M a M a t d 2 a dt 2 = ;GM a 2 + c2 3 a (9.4) c dark energy ! 2 da ; GM dt a ; c2 6 a2 = E = ; 1 2 kc2 (9.5)
19 E 0 E = ;kc 2 =2 k k =1 E k =0 E 0 k = ;1 E 9.5 6= 0 k da=dt 9.4 6= E = ;kc 2 =2 9.4 =0 k da=dt
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1 22 22.1 atomic line spectrum emission line absorption line atom proton neutronnuclei electron Z atomic number A mass number neutral atom ion energy level ground stateexcited state ionized state 22.2
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
12-7 12-7 12-7 12-7 12-8 12-10 12-10 12-10 12-11 12-12 12-12 12-14 12-15 12-17 12-18 10 12-19 12-20 12-20 12-21 12-22 12-22 12-23 12-25 12-26 12-26 12-29 12-30 12-30 12-31 12-33 12-34 12-3 12-35 12-36
1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0
A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
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51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
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I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =
1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
:30 100m 4 3 (2 ) 5 (3 ) 7 (1 ) 16 OB OG m 200m (2 ) m (1 ) 4 (3 ) 2m m 10:35 100m 3 (3 ) 5 OB OG (1
2003 10 1 1 1.1............ 1 1.2............ 2 1.3.......... 2 1.4.............. 2 1.5.............. 7 1.6.............. 10 2 11 100m,200m 200m 21 2.1............ 11 2m A 1 2.2.... 11 1500m 3 1 14m15
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
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II. T = 1 m!! U = mg!(1 cos!) E = T + U! E U = T E U! m U,E mg! U = mg!(1! cos)! < E < mg! mg! < E! L = T!U = 1 m!! mg!(1! cos) d L! L = L = L m!, =!mg!sin m! + mg!sin = d =! g! sin & g! d =! sin ! = v
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
2 3 4 mdv/dt = F cos(-)-mg sin- D -T- B cos mv d/dt = F sin(-)-mg cos+ L- B sin I d 2 /dt 2 = Ms + Md+ Mn FMsMd MnBTm DLg 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Hm H h
5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 3 版 1 刷発行時のものです.
最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/052093 このサンプルページの内容は, 第 3 版 1 刷発行時のものです. i 3 10 3 2000 2007 26 8 2 SI SI 20 1996 2000 SI 15 3 ii 1 56 6
2 X-ray 6 gamma-ray 7 1 17.1 0:38m 0:77m nm 17.2 Hz Hz 1 E p E E = h = ch= (17.2) p = E=c = h=c = h= (17.3) continuum continuous spectrum line spectru
1 17 object 1 observation 17.1 X electromagnetic wave photon 1 = c (17.1) c =3 10 8 ms ;1 m mm = 10 ;3 m m =10 ;6 m nm = 10 ;9 m 1 Hz 17.1 spectrum radio 2 infrared 3 visual light optical light 4 ultraviolet
sec13.dvi
13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
QMI_10.dvi
... black body radiation black body black body radiation Gustav Kirchhoff 859 895 W. Wien O.R. Lummer cavity radiation ν ν +dν f T (ν) f T (ν)dν = 8πν2 c 3 kt dν (Rayleigh Jeans) (.) f T (ν) spectral energy
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() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
77
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
215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4
030801調査結果速報版.PDF
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24.15章.微分方程式
m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt
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Title ブラックホールと重力波天文学 Author(s) 長峯, 健太郎 Citation 高大連携物理教育セミナー報告書. 28 Issue Date Text Version publisher URL DO
Title ブラックホールと重力波天文学 Author(s) 長峯, 健太郎 Citation 高大連携物理教育セミナー報告書. 28 Issue Date 2017-03 Text Version publisher URL http://hdl.handle.net/11094/60516 DOI rights Osaka University Knowledge Archive : OUKA
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Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
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レジャー産業と顧客満足の課題
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( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
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" dt = "r(t "T s dt = "r(t "T s T T s dt T "T s = "r ln(t "T s = "rt + rt 0 T = T s + Ae "rt T(0 = T 0 T(0 = T s + A A = T 0 "T s T(t = T s + (T 0 "T s e "rt dy dx = f (x, y (Euler dy dx = f (x, y y y(x
1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N
1 1 1 1-1 1 1-9 1-3 1-1 13-17 -3 6-4 6 3 3-1 35 3-37 3-3 38 4 4-1 39 4- Fe C TEM 41 4-3 C TEM 44 4-4 Fe TEM 46 4-5 5 4-6 5 5 51 6 5 1 1-1 1991 1,1 multiwall nanotube 1993 singlewall nanotube ( 1,) sp 7.4eV
3 6 6.1: ALMA 6.1 galaxy, galaxies the Galaxy, our Galaxy, Milky Way Galaxy G. Galilei W. Herschel cm J.C. Kapteyn H. Sharpley 30 E.P. Hubble 6.2 6.2.1 b l 6.2 b = 0 6.2: l = 0 6.2.2 6.1 6.3 ( 60-100µm)
