1 1 ϕ ϕ ϕ S F F = ϕ (1) S 1: F 1
1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l
: l r δr θ πrδr δf (1) (5) δf = ϕ πrδr = rθ l πrδr = πθr δr l δl δl = rδf = πθr3 δr l a θ L L = = a 0 a 0 dl = πa4 θ l (8) πθr 3 dr l (6) (7) (8) κ = πa4 l (9) 3
T I T = π κ Il = π (10) πa 4 = 8πIl (11) a 4 T (11) I T 3 I 0 I 0 3: M b c d I 1 I I 0 + I 1 I 0 + I ( 4 ) T 1 T T 1 T = 4π I0 + I 1 κ = 4π I0 + I κ (1) (13) (1) (13) T 1 T = 4π I1 I κ = 8πl I1 I a 4 (14) 4
4: = 8πl a I 1 I 4 T (15) 1 T I 0 I 1, I I 1 = M b + c [ b + c I = M 4 ] + d 1 (16) (17) (15) (16) (17) = πl 3a 4 3b + 3c d T 1 T M (18) 4 1. b c d a l b c d a 5 l 5. 3. ( ) 5
4. ( 90 ) ( 1 )10 90 5. 6. (18) 7. M 3.10 kg M 0.01 kg 5 1. T 1 1: 50 50 ( ) ( ) ( ) 0 50 10 60 0 70 30 80 40 90. T : 50 50 ( ) ( ) ( ) 0 50 10 60 0 70 30 80 40 90 6
6 1. 10. 10 1/ 3. 4. 5. 7 1.. 3. 4. 5. I 1, I (16) (17) 8 9 (57) a a 1, a, a 3, a 4, a 5 ā a ā = 1 5 a i (19) 5 a = 1 1 5 (a i ā) 5 4 (0) 7
(1) = πl 3a 4 3b + 3c d T 1 T M (1) α = 3b + 3c d () β = T 1 T (3) = πl 3a 4 α β M (4) α, β, ᾱ = 3 b + 3 c d (5) β = T 1 T (6) = π l 3ā ᾱ 4 β M (7) (68) ( α) = (6 b) ( b) + (6 c) ( c) + ( d) ( d) (8) ( β) = ( T 1 ) ( T 1 ) + (T ) ( T ) (9) (7) ( ) = ( l l ) ( ) a + 4 + ā ( ) α + ᾱ ( ) ( β M + β M ) (30) M = 3.10 kg M = 0.01 kg 9 9.1 precise 8
accurate 9. X X x x + dx f(x)dx f(x) f(x)dx = 1 (31) µ µ = xf(x)dx (3) σ σ = (x µ) f(x)dx = x f(x)dx µ (33) X F (X) E[F (X)] F (X) E[F (X)] = F (x)f(x)dx (34) µ = E[X] (35) σ = E[(X E[X]) ] = E[X ] (E[X]) (36) F (X) V [F (X)] V [F (X)] = E[(F (X) E[F (X)]) ] (37) σ = V [X] (38) 9
9.3 x s x = 1 x i (39) s 1 = (x i x) (40) 1 i s X x x j (j =,..., ) X x j x j + x f(x j ) x x = 1 x i x j f(x j ) x xf(x)dx = µ j= (41) s = 1 (x i x) 1 (x j x) f(x j ) x 1 j= (x µ) f(x)dx = σ (4) i j E[ X] = E [ 1 ] X i = 1 E[X i ] = 1 µ = µ (43) S = = = 1 1 1 1 1 1 (X i X) (44) { (Xi µ) ( X µ) } (45) (X i µ) 1 ( X µ) (46) 10
E[(X i µ) ] = σ (47) ( ) E[( X µ) ] = E 1 X i µ (48) = E 1 (X i µ) (X j µ) (49) j=1 = 1 [ ] E (X i µ) + 1 E i j(x i µ)(x j µ) (50) 0 E[S ] = 1 1 E[( X µ) ] = 1 [ ] E (X i µ) E[(X i µ) ] 1 E[( X µ) ] = = σ 1 σ (51) σ 1 = σ (5) 1 s σ σ 9.4 F (X) Err[F (X)] = (51) Err[ X] = V [F (X)] (53) V [ X] (54) V [ X] = E[( X µ) ] = σ (55) 11
Err[ X] = σ (56) σ s Err[ X] = s (57) 9.5 Z X, Y Z µ z ẑ Z = ϕ(x, Y ) (58) ẑ = ϕ( x, ȳ) (59) E[Ẑ] = E[ϕ( X, Ȳ )] ϕ(e[ X], E[Ȳ ]) = ϕ(µ x, µ y ) = µ z (60) ẑ ϕ Z Ẑ Err[Ẑ] Err[Ẑ] = V [Ẑ] = E[(Ẑ µ z) ] (61) ϕ Ẑ = ϕ( X, Ȳ ) ϕ(µ x, µ y ) + ( ) ϕ (µ x, µ y )( X µ x ) + ( ) ϕ (µ x, µ y )(Ȳ y µ y) (6) Ẑ µ z ( ) ϕ (µ x, µ y )( X µ x ) + ( ) ϕ (µ x, µ y )(Ȳ y µ y) (63) 1
V [Ẑ] = E[(Ẑ µ z) ] (64) {( ) ( ) } E ϕ ( X ϕ µ x ) + (Ȳ y µ y) (65) ( ) ( ) ϕ = E[( X ϕ µ x ) ] + E[(Ȳ y µ x) ] ( ) ( ) ϕ ϕ + E[( y X µ x )(Ȳ µ y)] (66) X Y E[( X µ x )(Ȳ µ y)] = 0 ( ) ϕ V [Ẑ] V [ X] + (Err[Ẑ]) ( ) ϕ (Err[ X]) + ( ) ϕ V [Ȳ y ] (67) ( ) ϕ (Err[Ȳ y ]) (68) X Y X Y Ẑ µ z ErrMax[F (X)] = max X E[X] (69) ( ) ϕ (µ x, µ y )( X µ x ) + ( ) ϕ (µ x, µ y )(Ȳ y µ y) (70) ( ) ( ) Ẑ µ ϕ z X ϕ µ x + Ȳ y µ y (71) ( ) ϕ ErrMax[Ẑ] ErrMax[ X] + ( ) ϕ y ErrMax[Ȳ ] (7) (68) 13