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. (, , ) .Copyright 2009 Mavromatis Books & Stationery LtdISBN 978-9963-9609-0-3 : 121, 2064 .. 25042, 1306 , : 00357 77777650, : +357 22873037E-mail: [email protected] . 3 , , , , .3 . , . . , - . . : , . - . . Matlab LaTEX. . . , . - . . ,, Email: [email protected]: 994583624 . 1 1.1 1. - 1 109: : - : . m - v :
P = mv . , - . , - . :
P = m1v + m2v + ... + mv
P = (m1 + m2 + ... + m) v2. - 2 109: ;5 1. : :F = Pt , , . .3. - 3 109: ;: - .F = Pt , - . , , .4. - 4 109: ;: , - . .6 . 1.1. 75. - 5 109: , ;: !!!6. - 6 109: - , ;: , -. !7. - 7 109: ; -.: , . . . : , . ( ) .8. - 8 109: .: - :E = 12 m v2(1.1)P = m v (1.2)7 . 1. 1.2 :P2= m2 v2(1.3) 1.1 m:E m = 12 m2 v2(1.4) 1.3 1.4 :E m = 12 P2E = P22m9. - 10 109: m1, m2 . m2 < m1 ;: 8 :E = P22mE1 = E2 P212m1= P222m2P21P22= m1m2 P21P22> 1 |P1| > |P2| .10. - 11 109: m1, m2 . E1E2 .: 8 :E = P22mP1 = P2E1 2m1 = E2 2m2 8 . 1.1. 9E1E2= m2m111. - 12 109: ;: .12. - 13 109: P = f(v) E = f(P): , : P = mv E = P22m. - . 1.1 10 5 .0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 505101520253035404550v (m/s)P (kg m/s)P=f(v) m1=10 kgm2=5 kg0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5020406080100120140v (m/s)E (J)E=f(v) m1=10kgm2=5kg 1.1: P = f(v), E = f(p)9 . 1. 13. - 14 109: F t;: t . . , . = F t = P.14. - 17 109: :() ;() ;() ;: .() , , - . , .() .() - .10 . 1.2. 111.2 1. - 11 7: 1, 2 m/s. ( ). 50 g .: F = Pt . . - ( ) . v = 1.2 m/s v = 0 m/s . :F = Pt = m (vv)t F = 0.05 (0 1.2)1 = 0.06 N .2. - 12 7: , 5 Kg. F . v 1.2.() F = f(t), F = f(x) x .() P E ; - 0 20s 20 30s.() P = f(t) E = f(t)11 . 1. 0 5 10 15 20 25 30 35051015202530354045t (s)v (m/s) 1.2: v = f(t): - . : 2 , 0 20s 20 30s . . :a = vt = 40 020 0 = 2 m/s :F = m a = 5 2 = 10 N12 . 1.2. 13 :a = vt = 0 4030 20 = 4 m/s :F = m a = 5 (4) = 20 N - . :x = 12 a t2= 12 2 202= 400 m 40m/s. - :x30s = x20s + v20s t 12 a t2= 400 + 40 10 200x30s = 600 m 1.3. P E .P = m vE = 12 m v2 v = f(t) 0 30s. - t 20sP = 5 40 = 200 m/sE = 12 5 402= 4000 J P = f(t) E = f(t) 1.4.13 . 1. 0 5 10 15 20 25 30 35252015105051015t (s)F (N)0 100 200 300 400 500 600252015105051015x (m)F (N) 1.3: F = f(t) F = f(x)3. - 14 8: 0, 2 kg 3.2 m . 103s .: . - . 14 . 1.2. 150 5 10 15 20 25 30 35020406080100120140160180200220t (s)P (kg.m.s1)0 5 10 15 20 25 30 35050010001500200025003000350040004500t (s)Ek (J) 1.4: P = f(t) E = f(t) .h = 12 g t2t =2 hg = 0, 8 sv = g t = 8 m/s F = Pt = m (vv)t F = 0.2 (8 (8))103 = 3200 N15 . 1. :F = Pt + m g = 3202 N4. - 15 8: v = 20 m/s, - . 75 kg , 0, 2 s. .: ,v = 20 m/s, v =0 m/s. :F = Pt = m (vv)t F = 75 (0 20)0.2 = 7500 N - :E = EE = 12 m v2 12 m v2 E = 15000 J5. - 18 9: 20 g 30 g - 30m/s 10m/s ( 1.5). . ) 16 . 1.2. 17 1.5: ) .: - . . .P = P m1 v1 + m2 v2 = mo vk vk = m1 v1 + m2 v2movk = 0.02 30 + 0.03 (10)0.05 vk = 6 m/s :E = EE == 12 mo vk2 12 m1 v12+ 12 m2 v22=17 . 1. = 12 0.05 62 12 0.02 302+ 12 0.03 102=E = 9.6 J6. - 19 9: M1 M2 2 kg 3 kg 6m/s 2m/s ( 1.6) -. 0.02 s.() () P1 P2 Po ( ).() F1 = f(t) F2 = f(t) F1 F2 ( ) M1 M2 . 1.6: : - - .P = P 18 . 1.2. 19m1 v1 + m2 v2 = m1 v
1 + m2 v
2 m1 v1m1 v
1 = m2 v
2m2 v2 m1 (v1v
1) = m2 (v
2v2) (1.5) (- ) :E = E 12 m1 v12+ 12 m2 v22= 12 m1 v
12+ 12 m2 v
22m1 (v12v
12) = m2 (v22v
22) m1 (v1v
1) (v1 + v
1) = m2 (v2v
2) (v2 + v
2) (1.6) 1.5 1.6 1.7 :v1 + v
1 = v2 + v
2 (1.7) 1.5 1.7. , - . . - !m1 (v1v
1) = m2 (v
2v2) 12 2 v
1 = 3 v
2 + 6 v
2 = 6 2 v
13 (1.8)19 . 1. v1 + v
1 = v2 + v
2 6 + v
1 = 2 + v
2 v
2 = 8 + v
1 (1.9) 1.8 1.9:6 2 v
13 = 8 + v
1 24 + 3 v
1 = 6 2 v
1 v
1 = 3.6 m/s 1.9:v
2 = 8 + v
1 v
2 = 4.4 m/s P1 P2 Po.P1 = m1 v1 = 12m/sP1 = m1 v1
= 7.2m/sP2 = m2 v2 = 6m/sP2 = m2 v2
= 13.2m/sF1 = Pt = 7.2 120.02 = 960NF2 = Pt = 13.2 + 60.02 = 960N 1.8 :20 . 1.2. 210 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04105051015t (s)P (kg.m.s1) P1P2Ptotal 1.7: 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04100080060040020002004006008001000t (s)F (N) 1.8: 7. - 20 10: P1 1 kg, - m 1.9.21 . 1. m.0 10 20 30 40 50 60 70 80 90 10005101520253035t (s)P (kg.m.s1) 1.9: : . 1 P = 30 kg m s1. - P = 10 kg m s1. 1 :P = 10 kg m s1m1 v1 = 10 kg m s1v1 = 10 m/s , . - m2 .P = P 22 . 1.3. 2330 = (m1 + m2) vo 3010 = 1 + m2 m2 = 2 kg1.3 1. - 26 12: M1 4kg 10 m/s M2 16 kg 4 m/s. 0.02 s. .() , . .() P1 P2 Po .() F1 = f(t) F2 = f(t), F1 F2 ( ) M1 M2 .: . :P = P m1 v1 + m2 v2 = (m1 + m2) vo vo = m1 v1 + m2 v2m1 + m223 . 1. vo = 4 10 + 16 44 + 16 vo = 5.2 m/s 1 2 P1 = f(t), P2 = f(t) Po = f(t).P1 = m1 v1 = 4 10 P1 = 40 kg m s1P1 = m1 vo = 4 5.2 P1 = 20.8 kg m s1P2 = m2 v2 = 16 4 P2 = 64 kg m s1P2 = m2 vo = 16 5.2 P2 = 83.2 kg m s1Po = P1 + P2 Po = 104 kg m s1Po = P1 + P2 Po = 104 kg m s1 - . - - .F1 = Pt = 20.8 400.02 F1 = 960 N24 . 1.3. 25F2 = Pt = 83.2 640.02 F2 = 960 N 1.100102030405060708090100110t (s)P (kg.m.s1) P1P2Ptotal100080060040020002004006008001000t (s)F (N) F1F2 1.10: P = f(t) E = f(t)25 . 1. .P = P m1 v1 + m2 v2 = (m1 + m2) vo vo = m1 v1m2 v2m1 + m2vo = 4 10 16 44 + 16 vo = 1.2 m/s 1 2 P1 = f(t), P2 = f(t) Po = f(t).P1 = m1 v1 = 4 10 P1 = 40 kg m s1P1 = m1 vo = 4 (1.2) P1 = 4.8 kg m s1P2 = m2 v2 = 16 (4) P2 = 64 kg m s1P2 = m2 vo = 16 (1.2) P2 = 19.2 kg m s1Po = P1 + P2 Po = 24 kg m s1Po = P1 + P2 Po = 24 kg m s1 - . -26 . 1.3. 27 - .F1 = Pt = 4.8 400.02 F1 = 2240 NF2 = Pt = 19.2 (64)0.02 F2 = 2240 N 1.118060402002040t (s)P (kg.m.s1) PtotalP1P2250020001500100050005001000150020002500t (s)F (N) F1F2 1.11: P = f(t) E = f(t)27 . 1. 2. - 28 13: m = 3 kg . h1 = 40 m . mA = m3 mB = 2m3 . h2 = 20 m ( 1.12). :() .() .() .() ; 102s. 1.12: 28 . 1.3. 29: , 20 m. - :vA = g t (1.10)h = vA t 12 g t2(1.11)(1.12) 1.10 t 1.11 :t = vAgh = vA2g 12 vA2g h = 12 vA2g vA =_2 g h = 20 m/s h1 = 40 m - :E = 12 m v2= 0 v2= 0 v = 0 - :P = P 0 = mA vA + mB vB 0 = m3 20 + 2m3 vB vB = 10m/s29 . 1. - :E = 12 mA vA2+ 12 mB vB2E = 12 1 202+ 12 2 102E = 300 J . . . . .EB0m = EB40m 12 mB vB
2= mB g h1 + 12 mB vB2vB
2= 2 g h1 + vB2vB
= 30 m/s . : !F = Pt + m g F = mB (vB
vB)t + mB g 30 . 1.3. 31F = 2 (30 (30))102 + 2 10 F = 12020 N31 . 1. 32 . 2 2.1 1. - 10 28: 2,4 . , 6/10 . ;: - :L = L I = I I = 610 I 2f = 106 2 2, 4 f = 4 /s2.2 1. - 11 28: - 1 Hz 33 2. . 8 g 0.5 m . 0.8 Hz.() () . . 2.1: : -:I =n
i=1mi ri2(2.1)L = L (2.2)E = 12 I 2(2.3) = 2 (2.4) 2.2 ( 2.1), :34 . 2.2. 35L = L I = I I = (I + m r2) I I = m r2 I = m r2 I = 0.008 0.52 5.0246.28 5.024 = 8 103kg m2 . :I = I + m r2= 0.01 kg m2 2.3:E = 12 8 103 6.282= 0.1578 JE = 12 0.01 5.0242= 0.1262 J: . - . , , - !2. - 12 28: - 35 . 2. . 2 kg m2. 1.5 rad/s. 1.2 kg m2() () -. ; 1.8 kg m2. , .: ( - ), :L = L I = I = I I= 2.5 rad/s - 2.3:E = 12 1.2 1.52= 2.25 JE = 12 1.2 2.52= 3.75 JE = EE = 1.5 J 1.8 kg m2. - , - :36 . 2.2. 37
= I
I
= 1.2 2.51.8 = 1.67 rad/sE
= 12 1.8 1.672= 2.5 J3. - 13 28: 120 kg 1 m, - . 60kg - 10 m/s. - , - ;(: - . : I = 12 m r2): - - : = vr = 101 = 10 rad/s 2.1 - :I = I + m r2I = 12 m r2+ m r2I = 12 120 1 + 60 12= 120 kg m237 . 2. :I
= I + m r
2I
= 12 m r2+ m r
2I
= 12 120 1 + 60 02= 60 kg m2 - . :L = L I = I
= I I
= 120 1060 = 20 rad/s = = 10 rad/sE = 12 120 102= 6000 JE = 12 60 202= 12000 JE = EE = 6000 J4. 14 29: , 60 kg, - 2 m. 360 kg m2. 1.5 m/s .38 . 2.2. 39() ;() . ;: . - , - - . :L = L 0 = I I I = I = I I= m r2 I = 60 22 0.75360 = 0.5 rad/s : = vr = 1.52 = 0.75 rad/s -. . .L = L 0 = I I = I I= 0 0360 = 0 rad/s39 . 2. 2.3 1. 15 29: I1 0 . - , I2 , .() I1, I2 0.() .() 0 ;: , . , , . :L = L I1 0 = (I1 + I2) = I1 0I1 + I2 . :E = 12 I 240 . 2.3. 41 :EE=12 (I1 + I2) ( I10I1+I2)212 I1 02 EE= I1I1 + I22. - 1 135: l = 0.4 m . IK = ml212 . :() .() . 2.2: .: . . :41 . 2. Steiner . 2.5. IK , IA , m x .IA = IK + m x2(2.5) , 2.2 :I0 = IK + m x2= m l212 + m x2I0 = m 0.4212 + m 0.22I0 = 0.0533 m - . , 120. :I0 = m l23 = 0.0533 m . . !!!Eo = E 42 . 2.3. 43m g l = m g l2 + 12 IO 2m 10 0.4 = m 10 0.42 + 12 0.0533 m 24 = 2 + 0.0266 2 = 4 20.0266 = 8.67 rad/s :v = r r . :vK = r = 8.67 0.2 = 1.734 m/sv = r = 8.67 0.4 = 3.468 m/s3. - 2 135: 0.2 kg R = 0.1 m IK = 25 m R2 - 1.75 m , 30o 2.3. :() () () -.: - , . .43 . 2. 2.3: . , . . - , , 2.3 , ! . - : , .I0 = IK + m x2= 25 m R2+ m x2I0 = 2 0.2 0.125 + 0.2 0.12I0 = 2.8 103kg m2EA = E 44 . 2.3. 45m g h = 12 I0 2 =2 m g hI0= 50 rad/s , , :vK = R = 5 m/s :E = 12 m vK2= 2.5 JE = 12 I0 2= 3.5 J4. - 4 135: - . 2.5 kg m2. = 2 rad/s. , 0.5 kg m2. :() () .: , , :45 . 2. L = L I = I
= I I
= 2.5 20.5 = 10 rad/s - :E = 12 I 2= 5 JE = 12 I
2= 25 J2.4 1. - 2007-2008: m1 m2 I. - 2.4. , - h :v =_2 (m2m1) g hm1 + m2 + IR2: . 1 2. 2 h 46 . 2.4. 47 2.4: . 1, 2 .E = E m2 g h = m1 g h + 12 m1 v2+ 12 m2 v2+ 12 I 2m2 g h = m1 g h + v2 (12 m1 + 12 m2 + 12 I 1R2) v2= 2 (m2 g h m1 g h)m1 + m2 + IR2v =_2 (m2 g h m1 g h)m1 + m2 + IR22. - 1986: 1. 47 . 2. 2, .: . , , . :I =2
i=1mi ri2= m1 r2+ m2 r2I =2
i=1mi ri2= m1 (r2)2+ m2 (r2)2I = 4 I :L = L I 1 = I 2 I 1 = 14 I 2 2 = 4 13. - 1993: m1 r2 r. m = 2 m1 1. - . 2 = 10 Hz. . 1 . I = 12 m r248 . 2.4. 49 2.5: - .: -, 2.5 .I = m1 (r2)2+ I = m1 r2+ 12 m r2I = 54 m1 r2I = m1 r2+ I = m1 r2+ 12 m r2I = 2 m1 r2 :L = L I 1 = I 2 54m1r2 21 = 2m1r2 22 1 = 85 2 = 16 Hz49 . 2. 4. - 1996: R = 20 cm M = 0.5 kg - 1 = 120cyclesmin . m = 100 g r = 10 cm -. , .: :I = moo r2+ I = moo r2+ 12 m R2I = 103+ 0.01 = 0.011I = I = 0.01 :L = L I 1 = I 2 0.011 21 = 0.01 22 2 = 0.0110.01 1 2 = 132 cyclesmin = 132 cycles60 s 2 = 2.2 Hz5. - 1997: m M r. 1 K. v2. 2. :50 . 2.4. 51() - () :M = 2kg, r = 1m, 1 = 10rad/s, 2 = 15rad/s, m =0.1kg, to = 0.1s, I = 12 m r2i. v2ii. iii. t = 0s t = 0.5s, t = 0.2s 2.6: - , ) ) .: . :I = m r2+ I = m r2+ 12 M r2L = I 1 L = (m r2+ 12 M r2) 1 . :51 . 2. . . ( v = r).L = I 2I v2r L = 12 M r2 2m r2 v2r L = 12 M r2 2m r2 v2r L = 12 M r2 2m v2 r . - . v2 :L = L (m r2+ 12 M r2) 1 = 12 M r2 2m v2 r v2 =12 M r2 2(m r2+ 12 M r2) 1m r v2 = 0.5 2 12 15 (0.1 12+ 0.5 2 12) 100.1 1 v2 = 40 m/s . . :E = 12 I 12= 12 (m r2+ 12 M r2) 1252 . 2.4. 53E = 0.5 (0.1 12+ 0.5 2 12) 102E = 55 JE = 12 I 22+ 12 m v22E = 12 (12 M r2) 22+ 12 m v22E = 0.5 (0.5 2 12) 152+ 0.5 0.1 402E = 192.5 JE = EE = 192.5 55 E = 137.5 J - . :L = I 1 = 10 kg m2 rad/sL = I 1 = 1 kg m2 rad/s -:L = I 2 = 15 kg m2 rad/sL = I v2r = 4 kg m2 rad/s53 . 2. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5420246810121416t (s)L (kg.m2.rad/s) LdiscLparticle 2.7: () (): - . 2.7 . .M = I a = I t == I ()t= LLt M = Lt54 . 2.4. 55 , .55 . 2. 56 . 33.1 1. - 3 41: . a = f(t) - 3.1. t = 0.5 s 2= 10.0 1 2 3 4 5 6 7 81086420246810t (s)a (cm/s2) 3.1: ..57 3. : t =5s. : ( ) T = 4s a = 10cm/s2 y0. = 2T = 2 104 = 1.58a0 = 2 y0 y0 = a02 = 10(24 )2 y0 = 4 cm :y = y0 sin(t + ) :y = y0 sin(t + ) (3.1)dydt = v = v0 cos(t + ) (3.2)dvdt = a = y0 2 sin(t + ) (3.3) .58 . 3.1. 59 t = 1s a = 10cms2 . 3.3 :a = y0 2 sin(t + ) 10 = 4 (24 )2 sin(24 1 + ) 1 = sin(2 + ) = 3.1 :y = y0 sin(t + ) y = 4 sin(24 0.5 + ) = 4 (22 )y = 222. - 4 41: 3.2 y t ... :() t = 4s() t = 2s() t = 8s() t = 8s() t = 4s: - .() t = 4s . . .59 . 3. 0 1 2 3 4 5 6 7 8 9 101086420246810t (s)y (m) 3.2: ..() t = 2s y = 0. .() t = 8s .() t = 8s . 3.1 3.3y = y0 sin(t + )a = y0 2 sin(t + )a = 2 y() t = 4s . . y ymax :E = 12 k y260 . 3.1. 613. - 5 42: l0 m 3.3.() - ; .() .15 16 17 18 19 20 21 22 23 24 2502468101214 3.3: : 3.3 ( MatLab) l0 = 20cm x0 = 5cm. . - - (x = 15 x = 25). - 61 . 3. . . - . -, , . 3.4 .15 16 17 18 19 20 21 22 23 24 2502468101214 3.4: , 4. - 6 42:() . , ;() 30o. ;() - - .62 . 3.1. 63: 3.5. 3.5: . 3.5, . : W S , .Sx = S sinSy = S cos , 6o sin = tan = rad . 63 . 3. l. y y:Sy = W Sy = m g S cos = m g S = m gcos x :___Sx = S sinS = mgcos___ Sx = m g tan , , ly l. :tan = xly= xlSx = m g xl (3.4) , 3.4. . - , 3.4 ._ Sx = m g xlF = k x_k = mgl64 . 3.1. 65k = m 2k = mglm 2= mgl =_gl 2T =_gl T = 2 lg: - l, . - g, .() . . . : , . - !!!() : 6o sin = tan = rad. .() . . l ly = cos l.65 . 3. 5. - 9 43: - - x0. :() () , () : . :E = 12 k x02= 12 m v02 :E = 12 m v02E = 12 2m v02= m v02 . . . k :k = m 2k = 2m 2 .66 . 3.2. 676. - 10 43: , ;: : - . - ( ) l = g2, . , , .3.2 1. - 12 43: ... ( ) - 52cm.s2. 4cm , 3cm/s. , .:, :() ...:x = x0 sin(t + )v = v0 cos(t + )a = a0 sin(t + )67 . 3. () a0 = 52cm.s2() x = 4cm v = 3cm/s :4 = x0 sin(t + )3 = x0 cos(t + ) sin cos. - , , - :4 = x0 sin(t + )3 = x0 cos(t + )162= 2 x02 sin2(t + )92= 2 x02 cos2(t + )162+ 92= 2 x02 [sin2(t + ) + cos2(t + )] 162+ 92= 2 x02 :2 x0 = 52 , - :(x0216)2= 922 x0 = 52(x0216)52x0= 92x0216 = 95x068 . 3.2. 69 - :x02 95x016 = 0x01,2 =95 _(95)2+ 4 1 162 1 :x01 = 5cm x02 = 3.2cm. . :2 x0 = 522= 2 = T = 2 = 2 = 2s :v0 = x0 = 5cm/s2. - 13 44: 3.6 , ... ..: - ... :69 . 3. 6 4 2 0 2 4 615105051015X: 6Y: 12x (m)a (m/s2)X: 3Y: 6X: 3Y: 6X: 6Y: 12 3.6: ...x = x0 sin(t + )a = x0 2 sin(t + )a = 2 x (3.5) y = x + = 2 = 0. 2. ... , . = 12 06 0 = 22= 2 =2 2T =2 T = 2270 . 3.2. 71T = 4.44s3. - 14 44: 2kg ... 0.5m. D = 200N/m. t1 - v1 = 2.5m/s. : t1 : m = 2kg D = 200N/m t1 v1 = 2.5m/s D = m2 =Dm =2002 = 10rad/s t1. (x0, D ) - (v1, m ). 1 x1:Eo = 12 D x02= 25J71 . 3. E = 12 m v12= 6.25JE1 = 12 D x12= 25 6.25 x1 =2 18.75200 = 0.433m :a1 = 2 x1 = 100 0.433 a1 = 43.3m/s24. - 15 44: 0.05kg x = 5sin(40t + 2) x cm, t s. 2= 10. :() () () - t = 0.5s() t = 0s t = 0.5s() , - : ... :x = x0 sin(t + )x = 5 sin(40t + 2) :72 . 3.2. 73 x0 = 5cm = 40 T = 0.05s :v0 = x0 = 0.05 40 = 210Emax = 12 m v20 = 12 0.05 (210)2= 1J t = 0.5s ...:_ F = m aa = 2 x0 sin(40t + 2)_F = m 2 x0 sin(40t + 2) F = 0.05 (40)2 0.05 1 = 40N t = 0s t =0.5s ( ). T = 0.05s. , , t = 0 x0 = 0.05m. :C = tT = 0.50.05 = 10 4 , d1o = 4 0.05m = 0.2m :Soo = C d1o = 2m73 . 3. 5. - 17 45: m h . - y. k . kg; 3.7: h: h . y. 1 , .m g (h + y) = 12 k y2m = k y22g (h + y)1 + y74 . 3.2. 756. - 18 45: 3.8 .. T = 2s , , x0 20cm. . 3.8: : . . 3.9. - . F . , . : 3.9: 75 . 3. T = N = m gFmax = m a0 = m 2 x0T = Fmax m g = m 2 x0 = 2 x0g = 42 x0T2 g = 0.27. - 19 45: 3.10 . K1 = 120N/m K2 = 80N/m. 2kg . , ... . 3.10: : ... , . x , 1 K1 2 K2 , 3.11. 76 . 3.2. 77 1 2 2. 3.11: (Hooke) :F1 = k1 xF2 = k2 x :F = F1F2 F = k1 x k2 x F = (k1k2) x (3.6) 3.6 - ... k1k2:2 1 . 2 77 . 3. k1k2 = m 242T2 = k1k2m T =m 42k1k2= 2 42120 80 T = 0.28. - 20 46: (Body Measuring Device - BMMD), - . . -. K = 600N/m. 0.9s 2.1s . 2= 10.: - , k . k :k = m 2 - 0.9s 2.1s.k = m 2 78 . 3.2. 79m = k2= 600420.92= 12.15kgk = mo 2o = (m + m) 2om 2 = (m + m) 2o m =m 22om m =12.15 42T242T2o12.15 = 12.15 T2oT212.15 m = 54kg9. - 25 48: 1 m1 = 1kg k = 400N/m x0 = 0.1m . 1 - 2 m2 = 3kg. . :() () : 1 2. 1 . : x0 = 0.1m k = 400N/m m1 = 100kg79 . 3. 3.12: - m2 = 3kg :k = m1 2= 400 2= km1= 4001 =400 = 20rad/s v0 = x0 = 20 0.1 v0 = 2m/s . :E = E 80 . 3.2. 8112 m1 v21 = 12 m1 v1
2+ 12 m2 v22 12 1 22= 12 1 v1
2+ 12 3 v22 3 v22 + v1
2= 4 (3.7)P = P m1 v1 = m1 v1
+ m2 v2 1 2 = 1 v1
+ 3 v2 v1
= 2 3 v2 (3.8) 3.7 3.8 :3 v22 + v1
2= 42 3 v2 = v1
3 v22 + (2 3 v2)2= 4 3 v22 + 4 12 v2 + 9 v22 = 4 12 v22 = 12 v2 v2 = 1m/s :v1
= 1m/s 1 v1
= 1m/s. :x0 = v1
= 0.05m , . :81 . 3. P = P m1 v1 = (m1 + m2) V V = m1 v1m1 + m2= 1 21 + 3 V = 0.5m/s .k = (m1 + m2) 2 =4004 = 10rad/s :x0 = V = 0.510 = 0.05m/s3.3 1. - 27 48: , , 6 , 12 6 . 0oC , 10oC - 0oC . - ... :() . 6:00 . t = 0s.() 9 .82 . 3.3. 83() - 5oC, ;: ... . .. : T4 = 6h t = 0s = 0oC 6h 0 = 10oC . .. : = 0 sin(t + ) : = 10 sin( 24 6t + )t = 0s = 0oC0 = 10 sin( 12 0 + ) sin = 0 = 0 : = 10 sin( 12t) 3.1383 . 3. 0 2 4 6 8 10 12012345678910t (hours)o C 3.13: , 9.. . t = 0s 6 .. 9 6 = 3. = 10 sin( 12t), t = 3 = 10 sin( 12 3) = 10 sin4 = 7oC 5oC. : = 10 sin( 12t), 5oC84 . 3.3. 855 = 10 sin( 12t) 0.5 = sin( 12t) 12t = 6 12t = 56 t = 2h t = 10h 8:00. 16:00, 8 !!!2. - 29 49: M = 2kg ..., x0 = 10cm T, K = 800N/m, 3.14. m = 500g M v = 4m/s M . . :() T () T
() () 3.14: 85 . 3. : T :k = M 2 = kM T = 2 Mk T = 2 2800 = 0.314s . , mo = 2.5kgT
= 2 _mok = 2 2.5800 T
= 0.351s : M ( ) v2 = x0 = 2T x0 = 20.314 0.1 v2 = 2m/sP = P m v1 + M v2 = (m + M) V V = m v1 + M v2m + M V = 0.5 4 + 2 22.5 = 2.4m/s86 . 3.3. 87 . ( - ):V = x0
x0
= V = V2T
x0
= 0.134m3. - 30 49: M = 1kg - k = 200N/m . m = 0.2kg - v = 2m/s. :() () () : . - , . 0 ( ), :P = P 0 = m v + M v 0 = 0.2 2 + 1 v v = 0.4 0.4m/s. 87 . 3. . , , Hooke:F = k x x = Fk = m gk = 0.2 10200 x = 0.01m :Eo = E + E Eo = 12 M v2 + 12 k x2= 12 1 0.42+ 12 200 0.012Eo = 0.09JEo = 12 m x20 x0 =2 Eok =2 0.09200 x0 = 0.03m4. - 32 50: k = 100N/m l0 = lm . m = 1kg .88 . 3.3. 89() ... t =0s .() () () , .() . ; ; 3.15: ) ) ) 89 . 3. : ... . 0.F = 0 F.B = 0 F. = B F. = m g 3.15. x :F = B F
F
= k (x + x) = F. + k x (3.9)F. = B = m g (3.10) 3.9, 3.10:F
= k x ... . :y = y0 sin(t + )k = m 290 . 3.3. 91 =km = 10rad/sT = 2 _mk = 0.628sF = k y0 y0 = m gk = 0.1m t = 0s x0 :0.1 = 0.1 sin(10t + ) 1 = sin() = 32 :E = 12 k (2 y0)2= 2J , :E = 12 k y20 = 0.5JE = m g h = m g y0 = 1JE = 12 m v20 = 12 m ( y0)2= 0.5J . .91 . 3. 3.4 1. 2003: l0 = 10cm k = 100N/cm. . () 10cm 15cm() 15cm 20cm: S.I.. k =104N/m, l0 = 0.1m. - :E0.15 = 12 k x2= 12 104 0.052= 12.5J - 0.15m 0.2m, - 0.2m 0.15m:E0.2 = 12 k x2= 12 104 0.12= 50JE = E0.2E0.15 = 50 12.5 E = 37.5J2. 2000: x .92 . 3.4. 93: ..:x = x0 sin(t + )a = d2xdt2 = 2x0 sin(t + ) a = 2 x 2 .xa 3.16: a = f(x)93 . 3. 3. 1999: .. 3.17. , .0 0.5 1 1.5 2 2.5 3 3.5 454321012345t (s)x (cm) 3.17: x = f(t): : x0 = 5cm T = 2s f = 1T = 0.5Hz = 2T = - , (t =0, x = x0 = 5cm) :94 . 3.4. 95x = x0 sin(t + )5 = 5 sin( 0 + ) sin = 1 = 2 :x = 5 sin(t + 2)4. 1999: x = x04 , x0 .: . , Emax = 12 k x20. x04 :E = EoE E = 12 k x20 12 k (x04 )2E = 12 k (x20 x2016) = k (15x2032 )EE=15x2032x2032= 1595 . 3. 5. 2001: 3.18 v x ... .0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.23210123X: 0Y: 3.142X: 0.2Y: 0 3.18: v = f(x): . - , .x = x0 sin(t + ) (3.11)v = x0 cos(t + ) (3.12) 3.40 , :96 . 3.4. 972 x2+ v2= 2 x20 sin2(t + ) + 2 x20 cos2(t + )2 x2+ v2= 2 x20 (sin2(t + ) + cos2(t + )) 2 x2+ v2= 2 x20 x2x20+ v22 x20= 1 (3.13) 3.42 x0 = 0.2m v0 = x0 = . : x0 = 2T 0.2 = T = 0.2 2 = 2.5s6. 2000: ... x0 T. - - ( 3.19). 3.19: - ..: - 97 . 3. . . :x = x0 sin( tOB) 0.5 x0 = x0 sin( tOB) tOB = 6 tOB = 6 = T 12 = T12 . , , :x = x0 sin( tOA) 0.707 x0 = x0 sin( tOA) tOA = 4 tOA = 4 = T 8 = T8tAB = tOB + tOA = T8 + T12 tAB = 524 T7. 2003: , y0 2y0. :() () () 98 . 3.4. 99: :T1 = 2 _mkT2 = 2 _mkT1T2= 1 k . :E1 = 12 k x20E2 = 12 k (2x0)2E1E2=12 k x2012 k 4x20= 14 :a1 = 2x0a2 = 2 2x0a1a2= 2x02 2x0= 128. 2002: - .. 3.20. - .99 . 3. 5 4 3 2 1 0 1 2 3 4 50.50.40.30.20.100.10.20.30.40.5X: 5Y: 0.4935x (cm)a (m/s2)X: 5Y: 0.4935 3.20: a = f(x): - : a0 = 0.493m/s2 x0 = 5cm. :a0 = 2 x0 =a0x0 =0.4930.05 = 100 . 3.4. 101T = 2 = 2s - 3.210 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 254321012345t (s)x (cm) 3.21: x = f(t)9. 2000: (x) (x0) .: : x = x02 () , 101 . 3. E = EoE E = 12 k x20 12 k x0222E = 12 k (x20 + x204 ) E = 12 k 34 x20EE=12 k 34 x2012 k x0222= 310. 2001: . 3.23. :() (t > 0) :i. ii. () a0Aa0B, a0A, a0B .: : : TA = 2s y0A = 10cm : TB = 4s y0B = 5cm .. . T4 = 0.5s T4 = 1s. 102 . 3.4. 103 3.22: x = f(t) . T2 = 1s T2 = 2s. :a0Aa0B= y0A 2Ay0B 2B= 0.1 (22 )20.05 (24 )2 = 811. 2001: m = 0.5kg . - F x.() - k() m .i. ;ii. ;iii. ;() 1.3J - .103 . 3. 0 2 4 6 8 10 12 14 16 18 2000.511.522.533.544.55x (mm)F (N) 3.23: F = f(x): . Hooke F = k x F = f(x) . :k = FFxxk = 5 00.02 0 = 250 N/m . , . 104 . 3.4. 105 - . E = 1.3J. . :E = 12 k x20 = 1.3 J x0 =2 1.3250 = 0.1 m12. 2002: - . m , t, - . : g 0 10 20 30 40 50 s 4 5,6 6,9 8 8,9 9,8() - .() . ;() ;: . k - :105 . 3. k = m 2 :k = m 2 =km 2T =kmT = 2 _mk T2= 42k m - T2= f(m) 42k . , k : 20 . - 20 - . T2= f(m) 42k k. 3.24 :42k = yyxx42k = 0.2401 0.040.050 0 = 4 k = 424 = 9.87 N/m106 . 3.4. 107 g s T T20 4 0,2 0,0410 5,6 0,28 0,078420 6,9 0,345 0,11902530 8 0,4 0,1640 8,9 0,445 0,19802550 9,8 0,49 0,24010 5 10 15 20 25 30 35 40 45 5000.050.10.150.20.25m (g)T2 3.24: T2= f(m) k :k = m 2m = k2 = k T242 = 4 0.0442 m = 4 103kg = 4 g107 . 3. . . - , . 20 -. . .13. 2003: :() () , k, - . m, - l0 0. .i. .ii. .:() - . -: F = k : .108 . 3.4. 109 : , . 3.25: ) ) ) () ... - . 0.F = 0 F.B = 0 F. = B F. = m g 3.25. x :109 . 3. F = B F
(3.14)F
= k (l0 + x) = F. + k x (3.15)F. = B = m g (3.16) 3.14, 3.15 3.16:F = k x ... . : = 2T (3.17)k = m 2(3.18) 3.17, 3.22 :k = m 2= m (2T )22T =km T = 2_mk .14. 2002: :() - . .110 . 3.4. 111() - . - 60kg f =3.2Hz. - f
= 2.2Hz. - .() . f
=2.2Hz. () , .:() . .() - . k = m 2(3.19) . : = 2f (3.20) 3.22, 3.20 :k = m (2f)2(3.21) k () 111 . 3. ! -. mchair - , mtot = mchair + mastr. :(mchair + mastr) (2f
)2= mchair (2f)2(mchair + mastr) f
2= mchair f2mastr f
2= mchair f2mchair f
2mastr = mchair (f2f
2)f
2 mastr = 60 3.222.222.22 mastr = 66.94kg() . - f
f
.mastr = mchair (f2f
2)f
2 mastr = 64.67kg 2.27kg !!!() . . ,112 . 3.4. 113 - , .15. 1998: , m, l, g. , .() .() .() - . 3.26: : . : .113 . 3. .. - . 6 . 3.33 . 3.27: , , l . 3.33 . - 0.Fy = 0 SyB = 0 Sy = B Sy = mg (3.22) S :Sy = S cos (3.23)Sx = S sin (3.24)114 . 3.4. 115 3.32 3.33 S cos = mg (3.25)S = mgcos (3.26) 3.36 3.34 :Sx = mgcos sin (3.27) 3.33 l :sincos = tan = xl (3.28) 3.38 3.37 :Sx = mgl x (3.29) Sx, . . - . - D ( - ). - D = m2 . : D = mgl D = m2 = 2T115 . 3. m2= mgl 2= gl :(2T )2= gl 2T =_gl T = 2 lg - l g. . g. : 20 ( n) 3 . - 20 20 . T2= f(l) (T2= 42g l)3 20 n . 20 .116 . 3.4. 117 , , = 42g g = 4216. 1999: ; :() ;() ;() ;: . - .T = 2 lg (3.30) 3.30. . :T = 2 lgT
= 2 _l4g = 2 l4 g = 12 (2 lg) T
= 12 T . (g = g6 ), :117 . 3. T = 2 lgT
= 2 _lg6= 2 6 lg =6 (2 lg) T
=6 T17. 1999: T m - k. - . / X0 kg k N/m Ts1 0,10 0,2 20 0,632 0,10 0,3 20 0,773 0,10 0,2 30 0,514 0,15 0,2 20 0,635 0,10 0,4 20 0,896 0,10 0,2 40 0,447 0,20 0,2 20 0,63 3.1: m, x0, k :() .() T :i. X0ii. miii. k() - T = f(X0), T = f(m), T = f(k). ;118 . 3.4. 119() - T .: . : n ( -) n . ( ) . - . y = f(x1, x2, ..., xn) n . y x1 , x2 ... ... x1 x1 . , y, , . :i X0 ; -, m k X0. :119 . 3. / X0 kg k N/m Ts1 0,10 0,2 20 0,634 0,15 0,2 20 0,637 0,20 0,2 20 0,63 3.2: X0. m, k . .ii m ; , X0 k m. : / X0 kg k N/m Ts1 0,10 0,2 20 0,632 0,10 0,3 20 0,775 0,10 0,4 20 0,89 3.3: m. X0, k .iii k ; k , X0 m - k. : / X0 kg k N/m Ts1 0,10 0,2 20 0,633 0,10 0,2 30 0,516 0,10 0,2 40 0,44 3.4: k. m, X0 k, .120 . 3.4. 121 .0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20.500.511.520 (m)T (s) 3.28: . 3.28 .0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.40.650.70.750.80.850.90.95m (kg)T (s) 3.29: 3.29 . 3.30 - .121 . 3. 20 22 24 26 28 30 32 34 36 38 400.40.450.50.550.60.65k (N/m)T (s) 3.30: 18. 2004: : / l(m) 10T(s) T 1 0,6 152 0,8 183 1,0 204 1,2 225 1,4 24 3.5: g() .() ;: 3.5 -: - , T T2. 10 , 10 - . 3.6122 . 3.4. 123 T2= f(l) (T2= 42g l) , , = 42g g = 42 / l(m) 10T(s) T T21 0,6 15 1,5 2,252 0,8 18 1,8 3,243 1 20 2 44 1,2 22 2,2 4,845 1,4 24 2,4 5,76 3.6: g 3.31:0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.422.533.544.555.56l (m)T2 (s2) y = 4.3*x 0.29data 1 linear 3.31: g 3.6 , . .123 . 3. - 3.31 ( , x), = 4.3 g. = 42g g = 42 g = 424.3 g = 9.18 ms2 , - -. , l. .19. 2002: - 3.7.() - ;() (); ;() .() T .124 . 3.4. 125 / 100T(s) l(m) m(kg) X0(cm) gms21 199 1 0,05 5 10.02 199 1 0,05 10 10.03 199 1 0,02 5 10.04 162 1 0,05 5 15.05 140 1 0,05 5 20.06 126 1 0,05 5 25.07 115 1 0,05 5 30.08 140 0,5 0,05 5 10.0 3.7: g: . - 17 . ( ) , .i ; 3.7 . 3.8. / 100T(s) l(m) m(kg) X0(cm) gms21 199 1 0,05 5 10.03 199 1 0,02 5 10.0 3.8: ( ) . .ii ; . 3.9.125 . 3. / 100T(s) l(m) m(kg) X0(cm) gms21 199 1 0,05 5 10.02 199 1 0,05 10 10.0 3.9: 3.9 . . . - 3.10. / 100T(s) l(m) m(kg) X0(cm) gms21 199 1 0,05 5 10.08 140 0,5 0,05 5 10.0 3.10: . . ( T =2 _lg ) . -. 3.32 . a . - a .F = m a S B = m a S = m a + m g 126 . 3.4. 127S = m (a + g) (3.31) 3.32: 3.31 () g a . 15 .20. 2004: l m. .() .() .() .127 . 3. () , , (E = f(x2), E = f(x2) E =f(x2)).: . : . .. - . 6 . 3.33 . 3.33: , , l . 3.33 . - 0.128 . 3.4. 129Fy = 0 SyB = 0 Sy = B Sy = mg (3.32) S :Sy = S cos (3.33)Sx = S sin (3.34) 3.32 3.33 S cos = mg (3.35)S = mgcos (3.36) 3.36 3.34 :Sx = mgcos sin (3.37) 3.33 l :sincos = tan = xl (3.38) 3.38 3.37 :Sx = mgl x (3.39) Sx, . .129 . 3. . 1s. .T = 2 lg T242 = lg l = g T242 l = 10 1242 = 0.253 m 0.253 m. - . , .x = x0 sin(t + ) (3.40)v = x0 cos(t + ) (3.41) 3.40 , :2 x2+ v2= 2 x20 sin2(t + ) + 2 x20 cos2(t + )2 x2+ v2= 2 x20 (sin2(t + ) + cos2(t + )) 2 x2+ v2= 2 x20 x2x20+ v22 x20= 1 (3.42)130 . 3.4. 131x2E EdynEmechEkin 3.34: - - x2 E =f(x2), E = f(x2) E = f(x2).21. 2001: m = 0, 25kg - k = 50Nm. m = 0, 25kg 3.35. 20cm t = 0.() () .() , .() , . :i. ii. :131 . 3. 3.35: , . : F = k 0.F = 0 F.(B + B) = 0 F. = B + B F. = (m + m) g 3.25. x :132 . 3.4. 133 3.36: ) ) ) F = B + BF
(3.43)F
= k (l0 + x) = F. + k x (3.44)F. = B + B = (m + m) g (3.45) 3.43, 3.44 3.45:F = k x - ...133 . 3. . - : 0 :k = m 2 =km =500.5 = 10rads - 20cm . t = 0 s 20cm:x = x0 sin(t + 0) 0.2 = 0.2 sin(10 0 + 0) sin(0) = 1 0 = 2 :x = 0.2 sin(10t 2) .134 . 3.4. 135 . . :dxdt = v = 0.2 10 cos(10t 2) v = 2 cos(10t 2)dvdt = a = 2 10 sin(10t 2) a = 20 sin(10t 2) 3.37 3.38 - .0 0.1 0.2 0.3 0.4 0.5 0.6 0.721.510.500.511.52t (s)v (m/s)v=f(t) 3.37: . 3.39 m = 0.25kg , : , N = 0. 135 . 3. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7201510505101520t (s)a (m/s2)a=f(t) 3.38: 3.39: . ... , :F = N BN B = k xN = 0 :136 . 3.4. 137B = k x m g = kx x = m gk = 0.25 1050 x = 0.05m 20cm 5cm. 5cm :x = x0 sin(10t 2) 0.05 = 0.2sin(10t 2) sin(10t 2) = 0.25 10t 2 = sin1(0.25) t = sin1(0.25) + 210 = 0.182s t = 0.182s22. 1999: 1 2 m1 = 150g m2 = 50g , k 3.40.() 20cm :i. kii. - g = 10ms2.137 . 3. () 1 2 2 1 . :i. x0 ii. .() 1 t0, .() - 1 f, 1; 1 ; 3.40: .: . 0.2m. Hooke - . Hooke :F = k x k = Fx (3.46)138 . 3.4. 139: F . 1 2. k x 3.46 - :k = Fx = B1 + B2x k = m1 g + m2 gx k = 0.15 10 + 0.05 100.2 = 10Nm - , :k = m2k = m (2T )2 km = (2T )2 2T =km T = 2 _mk (3.47) 3.47 - :T = 2 m1 + m2k T = 2 0.210 = 0.89s139 . 3. , m2 = 0.05kg 1 . , X0 . ( -) 2 - 1. Hooke .F = k x x = Fk = B1k x = m2 gk x = 0.05 1010 = 0.05m 1 3.47 1.T = 2 _m1k T = 2 0.0510 = 0.44s 1 3.48:x = x0 sin(t + 0) (3.48)140 . 3.4. 141 x0 = 0.05m =_ km1=_ 100.15 = 8.16rads 0: 3.48 t = 0s0.05 = 0.05 sin(8.16 0 + 0) sin(0) = 1 0 = 2 : x = 0.05 sin(8.16t + 2) - f 1, . f , f0. : = 2f0 f0 = 2 f0 = 1.298Hz23. 1998: m = 0.5kg . v = 20 sin(5t + 23 ), v cms , t s cm.() y = f(t) .() - 3.41, - .() .141 . 3. 3.41: - : y = f(t) . :v = 20 sin(5t + 23 ) :v = y0 cos(t + 0) :v = 20 sin(5t + 23 ) v = 20 sin(5t + 23 2 + 2) v = 20 sin(5t + 46 36 + 2) v = 20 sin(5t + 6 + 2) v = 20 cos(5t + 6) 142 . 3.4. 143 :sin( + 2) = cos() : : , = 5rads 0: 0 = 6 y0:y0 = 20 y0 = 20 = 205 y0 = 4cm (y cm) :y = 4 sin(5t + 6) . t = 0s. 0 :y = 4 sin(5 0 + 6) y = 4 sin(6) y = 4 0.5 y = 2cm143 . 3. 3.42: , . y = 2cm y = 4cm. t = 0s , y = 4cm:y = 4 sin(5tA + 6) 4 = 4 sin(5tA + 6) sin(5tA + 6) = 1 5tA + 6 = 2 5tA = 2 6 tA =35 tA = 0.209s24. 2000: m = 150g , 3.43 , f = 10 Hz. - F = 12N. :() 144 . 3.4. 145() () () , :i. x0ii. 3.43: .: :k = m2k = 0.15 (2 f)2 k = 0.15 (2 10 )2 k = 60Nm . Hooke k :145 . 3. F = k x x = Fk = 1260 x = 0.2m :EA = 12 k x2 EA = 12 60 0.22 EA = 1.2J . - - , . - (x0), .Eo = Emax EA + EA = 12 k x20 EA + 3 EA = 12 k x20 1.2 + 3 1.2 = 12 k x20 1.2 + 3 1.2 = 12 60 x20 146 . 3.4. 147 x20 = 2 (1.2 + 3 1.2)60 x0 =2 (1.2 + 3 1.2)60 x0 = 0.4m , EA =3.6J:EA = 12 m v2 12 m v2= 3.6 v =2 3.6m =2 3.60.15 v = 6.928ms25. 2004: M T = 2s. 3.44 E x . - :() () k() () M() 147 . 3. 0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.0500.511.522.5x 103X: 0Y: 0.002406x (m)E (J)Ekin=f(x) 3.44: M .() E x : 3.44 : 2.4 103J. M x0 = 0.05m. T = 2s - . k. - . :Emax = Emax 12 k x20 = 2.4 103 k = 2 (2.4 103)x20= 2 (2.4 103)0.052 148 . 3.4. 149 k = 1.92 Nm :f = 1T = 12 f = 0.5Hz :k = M 2 M = k2 = k(2T )2 M = 1.92(22 )2 = 0.195 kg - :12 M v20 = Emax 12 0.195 v20 = 2.4 103 v20 = 2 (2.4 103)0.195 v0 =2 (2.4 103)0.195 = 0.157 ms . 3.45 :149 . 3. E = 12 k x2E = 12 k (x0 sin(t))20.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.0500.511.522.5 x 103x (m)E (J)E=f(x) 3.45: M .26. 1999: - 3.46. , m =50g, . f0 = 10Hz.f (Hz) 2 4 6 8 9 10 11 12 14 16 18f (Hz) 2,1 3,9 6 8,1 8,9 10 11,1 11,9 14,1 16 17,9 a (cm) 0,7 0,9 1,3 2,2 3,2 4 3,1 2,3 1,2 0,8 0,6 3.11: () - () :i. f = f(f)ii. a = f(f)150 . 3.4. 151 3.46: . m =50g, m
= 200g - (ii);151 . 3. : 3.46, . . . 3.11. f = f(f) 3.47 a = f(f) 3.48.2 4 6 8 10 12 14 16 1824681012141618f (Hz)f (Hz)f = f(f) data 1 linear 3.47: a. : . 3.47. . , 152 . 3.4. 1532 4 6 8 10 12 14 16 180.511.522.533.544.5f (Hz)a (cm)a=f( f ) data 1 spline 3.48: a. . - . (f0 = 10Hz), . 3.48. m = 50g, m
= 200g , - . k - :k = m 2= m (2f)2 k = 197.39 Nmk = m
2= m
(2f
)2 2f
= km
f
= 12 km
f
= 5 Hz153 . 3. 0 5Hz . f =5Hz . .27. 2004: 3.49, k = 4 Nm m =0.15kg. . = 4rads x0. 0.20kg, 0.25kg, 0.30kg, 0.35kg 0.40kg.() () x0, m x0 = f(m) m 3.49: .154 . 3.4. 155: - . 3.49 = 4rads , . ( ), . -. - f = 12_km k = m(2f)2. . . :f = 12 km : m f = 2 _km0,15 0,8220,2 0,7120,25 0,6370,3 0,5810,35 0,5380,4 0,503 3.12: . : = 2f f = 2 = 0.637Hz155 . 3. - . . x0 = f(m), 3.50.0.15 0.2 0.25 0.3 0.35 0.4mx0x0=(m) 3.50: .28. 2005: ( ) 12cm Hooke . 4cm F = 0.2N. 14cm 16cm.: 14cm 16cm : 14cm 16cm - 16cm 156 . 3.4. 15714cm. k :F = k x k = Fx k = 0.20.04 = 5NmE1216cm = 12 k x2 E1216cm = 12 5 0.042 E1216cm = 4 103JE1214cm = 12 k x2 E1214cm = 12 5 0.022 E1214cm = 1 103JE1416cm = E1216cmE1214cm E1416cm = 4 1031 103 E1416cm = 3 103J29. 2005: k = 20Nm 0, 1J - x. 3x.: - . .157 . 3. E = 12 k x2= 0.1JE
= 12 k (3x)2 E
= 12 k 9 (x)2 E
= 9 12 k (x)2 E
= 9 E = 0.9J30. 2005: l . . = l4, l g. 3.51. 3.51: .: l, , 3l4, , , - l. 158 . 3.4. 159 - l 3l4.T = T12 + T22 T =2 _lg2 +2 _3l4g2 T = lg + 3l4g T = lg +32 lg T = (1 +32 ) lg T = 5.86 lg159 . 3. 160 . 44.1 1. - 1 244: . - ;: . - . , - ( ) .2. - 2 244: ; ;: . : f y0 161 4. v 3. - 3 244: .: :() : . .() : . .4. - 4 244: () ;() ;: () . , :x = = 1, 2, ...() 162 . 4.1. 163 yA = yB ,vA = vB . - , :x = 2 = 1, 2, ...5. - 8 244: ;: (), .6. - 9 244: - ;: - . 4.1 = 2 ( tT x) (4.1) , :A = 2 ( tT xA ) B = 2 ( tT xB )163 . 4. = AB = 2 ( tT xA ) 2 ( tT xB ) = 2 tT 2 xA 2 tT + 2 xB = 2 xB 2 xA = 2 xBxA = 2 x .7. - 10 244: 74 ;: - . : = 2 x = 2 74 = 2 74 = 7 28. - 11 244: ;164 . 4.1. 165: . .9. - 12 15 244: ; ;: - . , - . . .: . . x t = xv. t = xv. :y1 = y0 sin_2( tT x)_ y2 = y0 sin_2( tT + x)_ : = y1 + y2 =165 . 4. = y0 sin_2( tT x)_+ y0 sin_2( tT + x)_== y0_sin_2( tT x)_+ sin_2( tT + x)__ 4.7 .sinA + sinB = 2 cos(A B2 ) sin(A + B2 ) (4.2) = y0_sin_2( tT x)_+ sin_2( tT + x)__== 2y0_cos(2 tT x tT x2 ) sin(2 tT x + tT + x2 )_== 2y0_cos(2 2x2 ) sin(2 2tT2 )_ = 2y0 cos(2 x ) sin(2 tT ) cos(x) = cos(x) : = 2y0 cos(2 x) sin(2 tT ) (4.3) 4.3 .: 2y0cos(2x) . - . sin(2 tT ), .166 . 4.1. 167 , , , .: cos(2 x) , :cos(2 x) = 1 2 x = x = 2 = 0, 1, 2, ... (4.4) 4.4 - . , . cos(2 x) 0.cos(2 x) = 0 2 x = (2 + 1) 2 x = (2 + 1) 4 = 0, 1, 2, ... (4.5) 4.5 - . 15 . 12 . .167 . 4. x1o = (2 + 1) 4, x1o = 2 = 0x = 2 4 x = 410. - 13 244: y = y0 sin_2 ( tT x)_. x = 54 ;: , . 0. , = 2 ( tT x). x = 54 . = 0 0 = 2 ( tT 54 ) tT = 54 t = 54 T11. - 14 244: 6 3 .() ;() ;: . 168 . 4.1. 169 : 2y0 cos(2 x) . . : - - 2y0 cos(2 x) . :cos(2 xA ) = cos(2 6) = cos(26 ) = 0.5 :cos(2 xB ) = cos(2 3) = cos(23 ) = 0.5 . : :A = 2y0 0.5 A = y0 :B = 2y0 (0.5) B = y0 .169 . 4. 4.2 1. - 1 247: . - . y0 = 0.2m. 0.5m. 5 5sec. t = 0 x = 0 x, , .: , : y0 :y = y0 sin_2(t f x)_ 0.2m y0 = 0.2m 0.5m. . 5 t = 5sec.f = cyclest = 55 = 1Hz , ().170 . 4.2. 171 :y = y0 sin_2(t f x)_== 0.2 sin_2(t 1 x0.5)_y = 0.2 sin[2(t 2x)]2. - 2 248: y = 2sin_(8t x16)_.( cm s).() , , - .() , 60cm .() , 13cm .: . - :y = y0 sin_2( tT x)_ (4.6) 2:y = 2sin_22 (8t x16)_= 2sin_2(82 t x2 16)_= 2sin_2(4 t x32)_171 . 4. - 4.6 : y0 = 2cm 4t = tT T = t4t T = 0.25s x = x32 = 32xx = 32cm v = 1T = 320.25 v = 128cms - . :t = xv = 60128 t = 0.47s t = 0s t = 0.47s . t = 0.47s yM = 2sin_2(4 t x32)_0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 121.510.500.511.52t (s)yM (cm)y=y0 sin[(8tx/16)] 4.1: . t = 13s. .172 . 4.2. 173x = v t = 128 13 = 1664cm t = 13s x 0 1664cm. - 1600cm 1664cm.1600 1610 1620 1630 1640 1650 1660 167021.510.500.511.52x (cm)y (cm) 4.2: t = 13s.3. - 3 248: O1 O2 v = 3ms . M d1 = 5m d2 = 7m . f = 2Hz y0 = 2cm. M t1 = 25s t2 = 3.125s.173 . 4. 4.3: M.: - . - M - M . . :y1 = y0 sin_2( tT x1 )_ y2 = y0 sin_2( tT x2 )_ . x1 x2 - . - : = y1 + y2 == y0 sin_2( tT x1 )_+ y0 sin_2( tT x2 )_174 . 4.2. 175= y0_sin_2( tT x1 )_+ sin_2( tT x2 )__ 4.7 .sinA + sinB = 2 cos(A B2 ) sin(A + B2 ) (4.7) = y0_sin_2( tT x1 )_+ sin_2( tT x2 )__= 2y0_cos_2( tT x1 ) 2( tT x2 )2_ sin_2( tT x1 ) + 2( tT x2 )2__== 2y0_cos_2tT 2x1 2tT + 2x22_ sin_2tT 2x1 + 2tT 2x22__== 2y0__cos(2 (x2x1)2 ) sin(4tT 2(x2+x1)2 )__ = 2y0 cos_2 (x2x12 )_ sin_2 ( tT x1 + x22 )_ (4.8) 4.8 , x1 x2 . , - 4.8 M . x1 x2 . ( ). , - . :v = f = vf = 1.5m175 . 4. :T = 1f T = 12 T = 0.5sM = 2y0 cos_2 (x2x12 )_ sin_2 ( tT x1 + x22 )_M = 2 0.02 cos_2 ( 7 52 1.5)_ sin_2 ( t0.5 5 + 72 1.5)_M = 0.04 cos(4.189) sin[2 (2t 4)] (4.9) : t1 = 25sM = 0.04 cos(4.189) sin[2 (2 25 4)] = 0m t1 = 3.125sM = 0.04 cos(4.189) sin[2 (2 3.125 4)] = 0.02m M .4. - 5 248: y = 16 cos(x30) sin(80t) x, y cm t s. :() () 67.5cm() - 8cm.176 . 4.2. 177: 9 153, : = 2y0 cos(2 x) sin(2 tT ) - : 2 y0 = 16 y0 = 8cm x30 = 2x = 2x30x = 60cm 2 tT = 80t T = 2t80t T = 0.025s . , 9 153.x = (2 + 1) 4 = 0, 1, 2, ... = 0 = 1: :x = (2 + 1) 4 x = (2 0 + 1) 604 = 15cm :x = (2 + 1) 4 x = (2 1 + 1) 604 = 45cm177 . 4. x = 67.5cm. 2y0 cos(2 x), :0(67.5) = 2y0 cos(2 x) 0(67.5) = 16 cos(2 67.560 ) 0(67.5) = 11.31cm y = 8cm. 2y0 cos(2 x). :2y0 cos(2 x) = 8 16 cos(2 x60) = 8 16 cos(2 x60) = 8 cos(2 x60) = 12 cos(2 x60) = 12 2 x60 = 3 x = 603 2 = 10cm5. - 6 248: y =4 sin(15.7t), (y cm, t s). M1 x1 = 2m , t = 3s. M2 x2 = 5m t1 = 10.25s t2 = 12.5s.: M2 :178 . 4.2. 179 M2 - , , . :y = y0 sin(2T t) :y = 4 sin(15.7 t) : y0 = 4cm 2T = 15.7 T = 215.7 = 0.4s t = 3s M1 x1 = 2m :v = x1t = 0.67msv = 1T = v T = 0.267m179 . 4. :y = y0 sin_2( tT x)_y = 4 sin_2( t0.4 x0.267)_y = 4 sin[2(2.5 t 3.745 x)] M2 - x x2 , t1 t2. t1 = 10.25syM2 = 4 sin[2(2.5 t 3.745 x2)]yM2 = 4 sin[2(2.5 10.25 3.745 5)]yM2 = 2.35cm t1 = 12.5syM2 = 4 sin[2(2.5 t 3.745 x2)]yM2 = 4 sin[2(2.5 12.5 3.745 5)]yM2 = 0.625cm6. - 7 249: , 1cm, 21.6cm t = 1.2s. :() () 8cm 12.180 . 4.2. 181: - , 12. , = 1cm. :v = xt = 21.61.2 v = 18cms :v = f f = v f = 18Hz 12 - t. . :B = 2(t f xB ) B2 = t f xB t =B2 + xBf =122 + 8118 t = 0.778s t
= 1.2s . .: , 181 . 4. :A = 2(t f xA )7. - 8 249: y = y0sin[2(2 t 0.01 x)]. x, y cm t s. :() , , () - : :y = y0 sin_2( tT x)_ y0 = 10cm = 0.1m tT = 2t T = 0.5s x = 0.01 x = 100cm v = 1T = 100 10.5 v = 200cms = 2ms . , , . - . = dydt = ddt_y0 sin_2( tT x)__ = 2T y0 cos_2( tT x)_ (4.10)0 = 2T y0182 . 4.2. 183a = ddt = ddt_2T y0 cos_2( tT x)__ a = (2T )2y0 sin_2( tT x)_ (4.11)a0 = (2T )2y0 0 = 2T y0 0 = 0.4ms a0 = (2T )2y0 a0 = 1.6 2 ms2: 4.10 4.11 - , .8. - 9 249: y0 = 10m, f = 10Hz v = 1ms .: :y = y0 sin_2(f t x)_ . y0 = 10m f = 10Hz v = f = vf = 110 = 0.1m :y = 10 sin_2(10 t x0.1)_183 . 4. 4.3 -1. - 1 272: l = 1m, m = 5g . f = 130.5Hz. ;: , . . , . , -. 2. :v =F (4.12)v = f (4.13) 4.12, F = ml . . : : l = 2. 4.12,4.13 . F : f =F ( f)2= F F = ( f)2 l = 2 = 2 l184 . 4.3. 185F = (2 l f)2= ml (2 l f)2 F = 0.0051 (2 1 130.5)2 F = 340.6N2. - 2 272: l1 = 33cm f1 = 440Hz. f2 f2f1= 32; f2;: f1 = 440Hz. l, 2. - :l = 12 1 = 2 l = 66cm f2 f2f1= 32:f2f1= 32v = 1 f1 = 2 f2 :185 . 4. f2f1= 12f2f1= 322 = 23 1 = 44cm - f2. , f2 l
:l
= 22 l
= 442 = 22cm l
= 22cm. 11cm . f2:f2f1= 32f2 = 32 f1 = 32 440Hz f2 = 660Hz3. - 3 272: 68cm. v = 340ms .() ;() f2 f2f1= 32;186 . 4.3. 187: , , . . - , 4 = l, l .( 4.4). 4.4: . v = f :4 = l = 4 l = 4 0.68 = 2.72m :187 . 4. v = f f = v f = 3402.72 = 125Hz : f2 f1f2= 32 v =
f2
= vf , l
=
4 f2:f1f2= 32 f2 = 23 f1 = 23 125 f2 = 83.33Hz :v =
f2
= vf
= 34083.33
= 4.08m :l
=
4 l
= 4.084 l
= 1.02m188 . 4.3. 1894. - 7 273: l = 0.75m . v = 1500ms . :() () 4 .: 4 . 4.5. 4.5: , 4 . l = 3 2. 2.l = 3 2 = 2 l3 = 2 0.753 = 0.5m . :189 . 4. v = f f = v f = 15000.5 = 3000Hz5. - 8 273: y = 10 cos( x6 ) sin(50t) x, y cm t s. :() 1,2 () : . , ( ). - .y = 2y0 cos(2 x ) sin(2f t) y0: 2y0 = 10 y0 = 5cm f: 2f = 50 f = 502 = 25Hz : 2x = x6 = 12cm - 2. x. :x. = 2 = 122 x. = 6cm190 . 4.4. YOUNG 1914.4 Young4.4.1 1. ; ;: - . Huygens : . 4.6: ) ) - .191 . 4. 2. - 1 283: ;: . , 4.6.3. - 2 283: ;: - , . - - .4. - 3 283: ;: .5. - 4 283: Young - . .: Young 4.7. 1 2. 1 2 - 2 . 192 . 4.4. YOUNG 193 4.7: Young , - . -. . - , , . x = k (4.14) x = (2k + 1) 2 (4.15)193 . 4. x , k . 2, x = O2. 4.7 2. 2 - - 1 :sin = O2a O2 = sin a D :sin tan :tan = ykD :x = ykD a (4.16) : 4.14 4.16x = ykD ax = k ykD a = k (4.17)194 . 4.4. YOUNG 195 4.15 4.16x = ykD ax = (2k + 1) 2ykD a = (2k + 1) 2 (4.18) - .6. - 5 285: - () ;() ;: 5, . 4.14 yk:ykD a = k yk = D k a (4.19) : - 4.19 . k .195 . 4. : . , a - a 4.19 yk . (k ) .: .4.4.2 1. - 1 285: Young -. a = 0.5 103m D = 2m x = 10 103m() - () ;: -. , :yk = D k a . 196 . 4.4. YOUNG 197 x 5 . k = 5.yk = D k a = yk aD k = 10 103 0.5 1032 5 = 5 107m . a
= 2a yk = Dka .2. - 2 285: Young = 5.89 107m 2m . 1.178cm.() ;() ( =4.5 107m), ;() () .: a. D = 2m y10 = 1.178cm k = 10 = 5.89 107m197 . 4. yk = Dka a :yk = D k a a = D k yk= 2 10 5.89 1070.01178 a = 1 103m = 1mm - , =4.5 107m. - yk yk+1. o = yk+1yk:o = yk+1yk == D (k + 1) a D k a == D k a + D 1 a D k a o = D a :o = D a = 2 4.5 1071 103 o = 9 104m = 0.9mm :y5 = D 5 a = 2 5 4.5 1071 103 y5 = 4.5 103m = 4.5mm198 . 4.5. 1994.5 -1. 2005: 4.8 4.9 .0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0810.80.60.40.200.20.40.60.81x (m)y(m) 4.8: .0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.610.80.60.40.200.20.40.60.81t (s)y(m) 4.9: . :() 199 . 4. () () : 4.8 . - 2 . = 0.04m 4.9, , . . . :T = 0.8s v = T:v = T = 0.040.8 v = 0.05ms2. 1993: 4.10 y = f(x) t = t0. 50Hz 0 . :() t0200 . 4.5. 201() () () , dAB = 0.3m, t = t +0.01s. xA = 0.9m xB = 1.2m() , = f(x), t = t0.0 0,3 0,6 0,9 1,20.40.30.20.100.10.20.30.4x (m)y (m) 4.10: .: - 4.10. . : = 0.6m f = 50Hz, - :v = f = 0.6 50 v = 30ms201 . 4. 4.10 0.9m t0 . v = xt t0:v = xt t0 = xv = 0.930 t0 = 0.03s :y = y0 sin_2(t f x)_ y0 y0 = 0.4m. - :y = y0 sin_2(t f x)_ y = 0.4 sin_2(50t x0.6)_ t = t + 0.01s. : yB x t = t + 0.01s = 0.04syB = 0.4 sin_2(50t xB0.6)_ yB = 0.4 sin_2(50 0.04 1.20.6)_ yB = 0m202 . 4.5. 203 () t = t +0.01s = 0.04s :B = ddt_0.4 sin_2(50t xB0.6)__ B = 2 50 0.4 cos_2(50t xB0.6)_ B = 2 50 0.4 cos_2(50 0.04 1.20.6)_ B = 40ms =(x) t = t0. t0 : = 2(t0 f x) = 2(50 0.03 x0.6) t0 = 0.03s x = 0.9m:0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9012345678910x (m) (rad) 4.11: t = t0.203 . 4. 3. 1992: x - y = 0.02sin(20t 0.5x). . . :() () t = 0.2s .() x, t = 0.3s, .() y = 0.015m .() , .: . - .y = y0 sin_2( tT x)_ y = y0 sin(2T t 2v T x) . :y = 0.02sin(20t 0.5x)y = y0 sin(2T t 2v T x)204 . 4.5. 205 y0 = 0.02m 2T = 20 T = 0.1s 2v T = 0.5 v = 20.5 T v = 40ms t = 0.2s . v = 40ms : x = v t = 40 0.2 = 8m0 1 2 3 4 5 6 7 80.020.0150.010.00500.0050.010.0150.02x (m)y (m) 4.12: t = 0.2s. t = 0.3s. = 20t 0.5x t = 0.3 .x = v t = 40 0.3 = 12m - y = 0.015m - .205 . 4. 0 2 4 6 8 10 1202468101214161820x (m) (rad) 4.13: t = 0.3s.y = 0.02sin(20t 0.5x)v = dydt = 20 0.02cos(20t 0.5x)a = d2ydt2 = dvdt = (20)2 0.02sin(20t 0.5x) sin(20t0.5x) . . :y = 0.02sin(20t 0.5x)0.015 = 0.02sin(20t 0.5x) sin(20t 0.5x) = 34 sin2 + cos2 = 1 :sin2(20t 0.5x) + cos2(20t 0.5x) = 1 206 . 4.5. 207 cos2(20t 0.5x) = 1 sin2(20t 0.5x) cos2(20t 0.5x) = 1 (34)2 cos(20t 0.5x) =1 (34)2 cos(20t 0.5x) = 0.6614 cos(20t 0.5x) = 0.6614 sin(20t 0.5x) = 0.75. - - :v = 20 0.02cos(20t 0.5x) v = 20 0.02 0.6614 = 0.83msa = (20)2 0.02sin(20t 0.5x) a = (20)2 0.02 0.75 = 59.2ms2 , . y0 = 0.01m,f = 20Hz + .: . v = f = vf = 4020 = 2my = 0.01sin_2(f t + x)_y = 0.01sin_2(20 t + x2)_207 . 4. 4. 2003: 4.14 t0, f =0.25Hz. :() xA = 0.5cm xB = 2cm() t = t0 + 1s0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 53210123x (cm)y (cm) 4.14: t = t0.: . = 2 ( tT x) + 0. . t0. = 2cm, 4.14 v = f = 2 0.25 = 0.5cms t0 4cm v = 0.5cms .v = xt0t0 = xv = 40.5 t0 = 8s208 . 4.5. 209 : = AB == 2 ( tT xA ) + 02 ( tT xB ) 0 == 2 (84 0.52 ) 2 (84 22) == 2 1.75 2 1 = 1.5 - t = t0 + 1s. 1s T4 T = 4s. t = t0 = 8s. 1s 0. - - t = 9s , xA =0.5cm.y = 2sin_2(f t + x)_ y = 2sin_2(0.25 9 + 0.52 )_ y = 05. 1991: Young = 6 107m. S1 = 1mm. D = 2m, S2 = 3mm. .209 . 4. : :o = yk+1yk == D (k + 1) a D k a == D k a + D 1 a D k a o = D a . D - D
= D + D .o = D a (4.20)
o = D
a
o = (D + D) a (4.21) a - 4.20 D 4.21.D = a o
o = (ao + D) a (4.22) 4.22 a - . :210 . 4.5. 211 = 6 107m S1 = o = 1mm S2 =
o = 3mm D = 2m
o = (ao + D) a a
o = a o + D a (
oo) = D a = D
oo= 2 6 1070.003 0.001 a = 6 104m6. 2005: - f = 2Hz, y0 = 0.01m v = 0.1ms . . t0 = 0s . x1 = 0.4m 1 x2 = 0.6m 1 = 2. :() 1 1 t0 = 0s .() .() , , 0 t 8s.211 . 4. 4.15: .: 1 , : y0 = 0.01m f = 2Hz v = 0.1ms = vf = 0.12 = 0.05m :y = y0sin_2(f t x)_ :212 . 4.5. 213y = y0sin_2(f t x)_ y = 0.01sin_2(2t x0.05)_ , . . 3 161 : = 2y0 cos_2 (x2x12 )_ sin_2 ( tT x1 + x22 )_ () -1 1 0 -. :cos_2 (x2x12 )_ = cos_2 (0.6 0.42 0.05 )_= cos_2 (0.20.1)_= cos(4) = 1 - , ., 0 t 8s. .: , , - . .213 . 4. - . - . 1 :v = x1t1t1 = x1v = 0.40.1 t1 = 4s, 2 :v = x2t2t2 = x2v = 0.60.1 t = 6s : 0 t 4s 4 t 6s 1 y = 0.01sin_2(2t x0.05)_ 6 t 8s = 2y0 cos_2 (x2x12 )_ sin_2 ( tT x1 + x22 )_ 4.16.214 . 4.5. 2150 1 2 3 4 5 6 7 80.020.0150.010.00500.0050.010.0150.02 t (s)yo ( m ) 4.16: t.215 . 4. 216 . 5 5.1 1. II - 1 24: , , 5.1. .t1 t2 t3Iot II=f(t) 5.1: t.: 217 5. . 5.1E = LdIdt (5.1) 5.1 - . . 5.1 () : 0 I0 0 t1. :E = LdIdt = LIt E = L(I00)(t10) E = LI0t1() : I = I0 t1 t2. :E = LdIdt = LIt E = L(I0I0)(t2t1) E = 0() : I0 0 t2 t3. - :218 . 5.1. 219E = LdIdt = LIt E = L(0 I0)(t3t2) E = L I0t3t2 5.2.t1 t2 t3L*I_0\t105L*I_0\(t3t2)tE 5.2: t.2. II - 6 26: 5.3 - . .: - Faraday : :219 . 5. 0 0.5 1 1.5 2 2.5 3 3.5 4201510505101520t (s) (Wb)=f(t) 5.3: t.E = t (5.2) - = 1. (), 5.3 ( ) . 5.2.() : = 0 =20Wb t = 0 t = 2s. :E = t = 20 02 0 E = 202 = 10 V() : = 20 =20Wb t = 2 t = 4s. :220 . 5.1. 221E = t = 20 (20)4 2 E = 402 = 20 V t 5.4.0 0.5 1 1.5 2 2.5 3 3.5 4252015105051015t (s)V (v)V=f(t) 5.4: t.3. II - 1 27: 5.5 . SA, SB = 2 SA S = 3 SA . :() ;() ; .: . - Weber Wb. :221 . 5. 5.5: . = B S = BScos (5.3) B S 1S B. 5.6 . 5.3 5.6 :() :A = B SA = BSAcos A = BSAcos900= 0 Wb() :B = B SB = BSBcos B = BSBcos00= 2BSA Wb1 , .222 . 5.1. 223 5.6: .() : = B S = BScos = 3BSAcos600= 1.5BSA Wb - .4. II - 2 27: 1 T, 15 , -, 20cm. . , , 0.1s.: d = 20cm. :S = r2= (d2)2= (0.22 )2 S = 0.0314m2 . 223 . 5. = 00 . , B = B2 Faraday :E = t E = BScos BScost = 150.5 0.0314 1 1 0.0314 10.1 E = 2.353 V5. II - 3 27: , , v = 360km/h. l = 30m. B = 0.3 104T.: , . - . Faraday. S.I :v = 360kmh = 3601000m3600s = 100ms . . Faraday :224 . 5.1. 225E = t = t(BS) E = B t(l x) = B lxt E = B l v = 0.3 104 30 100 E = 0.09 V6. II - 4 27: 0.2m - - . 0.1m2s , 18mV . ;: . - - , . St =0.1m2s . , - . :E = t = t(BS) E = B St B = E 1St= 18 103 10.1 225 . 5. B = 18 102T7. II - 5 28: , a = 0.4m, . Bz - 5.7. E = f(t) , .0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160.0500.050.10.150.20.250.30.350.40.45t(ms)Bz (T)Bz=f(t) 5.7: Bz t.: Fara-day. . B, Bz. :E = t = t(BS) E = S Bzt (5.4)226 . 5.1. 227 5.4 5.7 7 .: 5.7 Bzt . S = a a = 0.16m2 1:E = S Bzt = 0.16 0.4 0(3 0) 103 E = 21.33 V 2:E = S Bzt = 0.16 0.4 0.4(5 3) 103 E = 0 V 3:E = S Bzt = 0.16 0 0.4(7 5) 103 E = 32 V 4:E = S Bzt = 0.4 0 0(9 7) 103 E = 0 V 5:E = S Bzt = 0.4 0.4 0(12 9) 103 E = 21.33 V227 . 5. 6:E = S Bzt = 0.16 0.4 0.4(14 12) 103 E = 0 V 7:E = S Bzt = 0.16 0 0.4(16 14) 103 E = 32 V 5.80 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16302010010203040t ( ms )E ( v )E=f( t ) 5.8: E t.8. II - 6 28: 5.9 , l = 1m = 20rads , B = 0.5T. 228 . 5.1. 229 5.9: . . .: . - , . - , : - , , . Faraday . : = 2T T = 2 = 220 T = 10s r = l = 1m :S = r2= 12229 . 5. S = m2 :St = STT = 10 St = 10 m2s Faraday .: . 1800 B 5.10.: :E = t(B S) = BSt 5.10: S B.E = t = t(B S) 230 . 5.1. 231 E = BSt cos = 0.5 10 cos1800= 0.5 10 (1) E = 5 V9. II - 7 28: B = 0.2T - 5.11. v = 15ms t = 0 . xA = 0.5m. :() , () ; 5.11: .: 5.11 v = 15ms . . 5.12 . :231 . 5. 5.12: .t = B St t = B (l x)t l = . t = B lxt = B l v t = 0.2 0.5 15 = 1.5 Wb - . Faraday = 1 :E = t = 1.5 V10. II - 8 28: 1m, 103 30cm2 . . , 232 . 5.1. 233 = 1950;: 5.5:E = L It (5.5) . 5.5 Faraday. Faraday .: , . B = 0 N Il (5.6)E = ddt = ddt(B S) E = dBdt S dSdt B dSdt = 0 E = d(0NIl )dt S E = 0 Nl S dIdt 5.5, :L = 0 N Sl 233 . 5. 0 = 4 107 N = 103 l = 1m S = 30 104m2L = 4 107 103 30 104 1031 L = 3.77 mH . 5.5 = 1950 :L = 0 N Sl L = 1950 3.77 L = 7.35 H11. - - 23: A1y1 A2y2 1m . A1 A2 , , - .. E = 20V r = 2. , l = 1m, m = 0.3kg 8, , A1y1 A2y2 . B = 1T, A1y1 A2y2. . t = 0s . :() .234 . 5.1. 235() -() - vo , v = vo.() I v , I = f(v) .() v = vo, :i. ii. iii. . 5.13: .235 . 5. : . , - Faraday .. .() - . r = 2 RK = 8. , , :Ro = r + RK = 2 + 8 Ro = 10 - Ohm :I = VRo= 2010 I = 2 A() . F = m a . : B = mg = 0.3 10 = 3 N Laplace FL = B I L = 1 2 1 = 2 NF = m a a = Fm = B FLm a = 3 20.3 = 3.33ms2236 . 5.1. 237 5.14: .() . , Faraday ... ... I = ERo LaplaceFL = B I L. FL a = 0ms2 . - F = 0 .. Faraday.F = 0 FLB = 0 FL = B B Io L = mg Io = mgBL = 3 A . Faraday . 237 . 5. I = ER = 2010 = 2 A .. ..:I = IoI = 3 2 = 1 AE = I Ro = 1 10 = 10V Faraday :E = t = B St E = B (l y)t l = . lt = 0 E = B lyt = B l vo vo = EB l = 101 1 vo = 10ms vo =10ms() . :I = I + I I = ERo+tRo I = 2 + B v lRo I = 2 + v10238 . 5.1. 239 5.15.0 2 4 6 8 10 12 14 16 18 2000.511.522.533.54I=f(v)v (m/s)I (A) 5.15: , I, v.() :i. :P = I V = 3 20 P = 60 Wii. :P = I Vo = 3 30 P = 90 Wiii. :P = I Vo = PP = 90 60 P = 30 W12. 2004: OX1 OX2 -, O, 239 . 5. X1OX2, . yy
v X1OX2 . . B, 5.16. t0 = 0s yy
O t(t > 0s) OX1 OX2 . : 5.16: .() Faraday() t.() t, ... E = (2Bv2tan)t() - OX1 OX2 yy
r.() .() -, 240 . 5.1. 241 . .:() Faraday : -() . 5.16 2 . x v t v = xt .tan = KMx (5.7)x = v t (5.8) 5.7, 5.8 K = 2KM :KMtan = v t KM = tan v tK = 2KM = 2 tan v t , E = o2 , 5.16 - :S = K x2 241 . 5. S = (2 tan v t) (v t)2 S = tan v2 t2() Faraday :E = ddt = ddt(B S) E = BdSdt E = Bd(tan v2 t2)dt E = 2Btan v2 t() :RK = r K RK = 2r tan v t() . .. , Ohm, :I = ERo I = 2Btan v2 t2r tan v t I = B vr242 . 5.1. 243() - Laplace FL. 0 ( ), vo. :F = FL F = B I L F = B (B vr ) (2 tan v t) F = B2 v2r 2tan t13. : - . 5.17: Biot-Savart.: Biot-Savart 5.9:243 . 5. d B = 0I ds sin4R2 (5.9) I ds Biot-Savart = 0 = 2. : = 2. ds ds = Rd :d B = 0I ds sin4R2 B =_ 200I ds sin4R2 B =_ 200I R d sin(2)4R2 B = 0I4R_ 20d = 0I4R []20 B = 0I2R244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 51.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 . . . . . . . . . . . . . . . . . . . . . 111.3 . . . . . . . . . . . . . . . . . . . . . . 232 332.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 . . . . . 463 573.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.4 . . . . 924 1614.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.3 . . . . . . . . . . 1844.4 Young . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 1914.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.5 . . . . . . . . 1995 2175.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217245246 . 1.1 P = f(v), E = f(p) . . . . . . . . . . . . . . . . 91.2 v = f(t) . . . . . . . . . . . . . . . . . . . . . . . 121.3 F = f(t) F = f(x) . . . . . . . . . . . . . . . 141.4 P = f(t) E = f(t) . . . . . . . . . . . . . 151.5 . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 . . . . . . . . . . . . . . . . . . . . . . . . . 181.7 . . . . . . . . . . . . 211.8 . . . . . . . . . . . . 211.9 . . . . . . . . . . . . 221.10 P = f(t) E = f(t) . . . . . . . . . . . . . 251.11 P = f(t) E = f(t) . . . . . . . . . . . . . 271.12 . . . . . . . . . . . . . . . . . . . . . . . . 282.1 . . . . . . . . . . . . 342.2 . . . . . . . . . . . . 412.3 . . . . . . . . . . . . . . 442.4 . . . . . . . . . . . 472.5 - . . . . . . . . . . . . . . . . . . . . 492.6 - , ) ) . . . . . . . . . . . 512.7 () () . . . . . . . . 543.1 .. . . . . . . . . 573.2 .. . . . . . . . 603.3 . . . . . 613.4 , . . . . . . . 623.5 . . . . . . . . . . . . . . . . . . . . . . . . 633.6 ... . . . . . . . . 703.7 h . . . . . . . . . . . . . . . . . 743.8 . . . . . . . . . . . . . . . . . . . . . 753.9 . . . . . . . 753.10 . . . . . . . . . . . . . . . . . . . 76247 3.11 . . . 773.12 - . . . . . . . . . . . . . . . . . . 803.13 . . . . . . . . . . . . . . . . . . . . 843.14 . . . . . . . . .