1
1
1. ,
1.
,
, x, y, z , - .
,
S(t) , .
2.
.
.
.
3. , t1 t2
.
4.
.
5.
.
6.
,
, , .
7.
.
8.
.
9.
,
J , L = Jw , M .
10.
.
1. R m | 0.4mR2 | |
2. R m | 0.5mR2 | |
3. R m | mR2 | |
4. L m | mL2/12 | |
5. L m | mL2/3 |
,
J m ;
J0 , ; a .
11.
,
- .
|
|
12.
13. ,
,
m , vc , J , , w - .
14. .
15. : , .
16. h: ,
r - .
17.
,
P , V .
18. h r V
.
19. :
,
t - , ; - , , v, .
20. :
,
l0 - , l - , , v.
21. , m0:
.
22. :
.
23. :
.
24. :
.
1. ( ) = + t + t3, = 4 , = 2 /, = - 0,5 /2. t1 = 2 : 1) 1 ; 2) V1; 3) 1.
. , , t t1:
1 = + t1 + t13; 1 = 4 .
V t , :
V = dx/dt = B + 3Ct2.
:
V1 = B + 3Ct21; V1 = - 4 /.
, t1 = 2 .
, :
a = d2x/dt2 = 6Ct.
: a1 = 6Ct1; a1 = - 6 /c2.
, .
2. , φ = 10 + 20 t - 2 t2 (. 1). , R = 0,1 , t1 = 4 .
:
φ=10+20t-2t2;
R=0,1 ;
t1=4 c;
a -? α -?
. . :
|
|
a = (a2t + a2n)1/2. (1)
:
t = εR; (2)
an = ω2R, (3)
ω - ; ε - ; R - .
t n (1) :
a = R(ε2 + ω4)1/2. (4)
ω = dφ/dt = 20 4t.
t = 4 ω = 4 -1.
: ε = dω/dt = - 4 c-2.
(4) : a = 1,65 /c2.
, , :
cos α = at/a. (5)
(2) (3) t an:
at = - 0,4 /c2; an = 1,6 /c2.
(5), :
cos α = 0,242; α = 760.
3. m = 60 . : 1)
1 = 3 /2; 2) ; 3) 3 = 9,8 /2.
:
m=60 ;
1=3 /2;
v2=const, a2=0;
3=9,8 /;
F1-? F2 -? F3 -?
. , : m g N. :
m a = m g + N (6)
:
N = - F N = F (7)
1. . 2 (6)
ma1 = N1 mg
(7) (8)
F1 = N1 = m (g + a1), F1 = 783 H.
2. 2 = 0 , , e : F2 = N2 = mg.
3. , ma3 = mg N3.
: F3 = N3 = =m(g a3).
, 3 = g F3 = 0. e, .
4. , , (R = 6400 ).
: h = R = 6370 ;
-?
.
= 2π(R + h)/V = 4πR/V.
, Ft, :
Ft = Fn; GmM/(R+h)2 = mV2/(R+h),
G , m , M .
V = (GM/(R + h)1/2 = (GM/2R)1/2.
,
GmM/R2 = mg,
g ,
V = (gR/2)1/2.
, ,
= 4 (2R/g)1/2 = 14360 c = 3 59 .
5. oo S= 3 2 , F = 4 . 104 y L1 = 2 . F1 = 104 . =2 . 1011 .
:
= 21011 ;
S= 3 2 =310-6 2;
L1 = 2 ;
F = 4104 ;
F1 =1,0104 ;
ΔL2 -?
. , L.
|
|
F = εE = E(L1 L)S/L
L = EL1S/(F +ES).
F1
F + F1 = EΔL2S/L.
ΔL2 = (F + F1)L/ES.
L , ,
ΔL2 = (F + F1)L1/(F + ES).
, : ΔL2 = 0, 16 .
6. , m = 5 r = 20 , , n = 720 -1. Δt = 20 . N, .
:
m = 5
r = 20 =0,20
n =720 -1 = 12 -1
Δt =20
-? N -?
. , , :
J∆ ω = M ∆t, (9)
∆ ω = ω ω0 - ∆t; .
N , , .
(9)
J∆ω = M∆t, (10)
∆ω, M - .
,
∆ω = |ω ω0 | = ω0 =2πn (11)
,
J = mr2 (12)
(11), (12) (10)
mr22πn = M∆t.
M = 2πnmr2/Δt = 0,75 H.
M, Δ ω ω0.
,
φ = ω0∆t ε∆t2/2. (13)
, ω = ωo - ε∆t = 0 (13)
φ = ω0∆t/2.
φ = 2πN, ω =2πn,
N - , ,
N = nt/2 = 120 .
7. m =10 . l 1 = 50 . n1 = 1,0 -1. A , , l 2 = 20 . J =2,5 2. .
:
m = 10 ;
l 1=50 = 0,5 ;
n1 =1,0 -1;
l 2 =20 =0,2 ;
J = 2,5 2.
n2 -? -?
. , . , , . . ( .) , :
L1 = L2; J1 ω1 = J2 ω2, (14)
J1 ω 1, J2 ω 2 - .
(14) :
|
|
J1ω1 = J2ω2. (15)
J1 = J0 + 2m l 12.
J2 = J0 + 2m l 22,
m - .
ω = 2πn (15)
(J0 + 2m l 12)n1 = (J0 + 2m l 22)n2.
n2 = n1(J0 + 2ml12)/(J0 + 2ml22) = 2,3 c-1.
, , . , :
A = W2 - W1 = J2ω22 /2 J1ω12/2.
, ω2 = J1ω1/J2, , :
A = J1(J1 J2)ω12/2J2 = (J0 + 2m l 12) 2π2 n12(l 12 l 22)/(J0 + 2m l 22) = 190 .
8. m = 2000 α = 0,1, S = 100 v = 36 /. μ = 0,05. .
:
m =2000 ;
S=100 ;
a=0,1 /2;
μ=0,05;
v0 =0;
v =36/ = 10/;
-? max -?
. , . , , (. 3).
: F =m g, N, F F. :
m a = N + m g + F + FTP.
ma = F mg sina - FTP,
0 = N mg cosa,
FTP = μ N.
F
F = mg sina + μmg cosa + ma.
a = (vk 2 - v02)/(2s) = vk2/(2s).
:
A = Fs cos α,
α F s, .
F,
A = [mg sinα + μmg cosα + mvk2/(2s)]s.
PCP = A/t, t = (vk v0)/a = 2s/vk2,
PCP = m[g sin α + μg cos α + vk2/(2s)](vk/2).
, :
Pmax = Fvk,
P = 27104 , Pmax = 47104 .
9. =6,0 l=2,0 , (. 4). m = 10 , V0 =1,0103 /, . .
:
= 6,0 ;
l = 2,0 ;
m = 10 = 1,0102;
v0 = 1,0 . 103 /;
W -?
. : . , . , . . .
:
L1 =L2 (16),
L1 =mv0l ;
L2 = Jω ;
J = J1 + J2,
J1 = Ml2/3 ; J2 = ml2 .
(16),
mv0l = (M/3 + m)l2ω
m << M, ,
mV0l = Ml2ω /3,
ω = 3mv0/Ml.
W = Jω2/2 = 3 m2V20/2M = 25 .
10. , V=0,9 ( ).
:
V = 0,9 c;
m0 = 9,110-31 ;
-? W -?
. = m0V/ (1 V2/c2)1/2 = =5,610-22 /.
W W0
W = W W0 = mc2 m0c2 = (m m0)c2
W = m0c2 = 0,66 .
Q23 = i (p2 V2 p V2) = - 1050 .
100. 30 v0=10 /. , t=2,5 ?
|
|
101. x1 = A1 + B1t2 + C1t3 x2 = A2 + B2t2 + C2t3, B1 = 4 /2; C1 = - 3 /3; B2 = - 2 /2; 2 = 1 /3. , .
102. : = 8 t2 + 4, (); y = 6 t2 3, (); z = 0. t = 10 . .
103.
x = At + Bt3,
=6 /, =0,125 /3. = 2 = 6 .
104. V0 = 10 / a=600 . R .
105. , , , , 30 .
106. , n = 955 /, t = 10 . , . ?
107. n=600 ./. , , N = 125 . t ?
108. e = t3, = 0,02 /3. 60 ?
109. j = t t3, = 6,0 /, = 2,0 /3. . .
110. m = 2 N = 80 v = 24 /. , . , .
111. , m1=1,5 , m1=2,5 , m2=1,5 , , . m1.
112. , 1 2 ? m1=2 m2=1 , k =0,05. .
113. F = 120 a=60 , . , b=30 ? m = 25 .
114. , , m = 240 . m , t = 4 c S = 160 c?
115. m1=10 v1=300 /. m2=200 , k=25 /. ? (, .)
116. v=4/ . 30. . .
117. , V =100 /, H =40 . t = 1 . V2 .
118. =60 , , m = 1 V = 10 /. , m=0,01?
119. <F> , m = 10 . V= 300 /. N = 300 .
120. 990 . 10 . 700 / . ? 0.05.
121. - m = 40 v =27 /, μ = 0,1.
122. h = 5 m1 = 10 , m2 = 0,20 . A ?
123. , . 0,2 , 0,1 . , 4,5 , . , : ) , ) ?
124. m = 1,0 h=20 . , , h, 10.
125. m1 = 100 , m2 = 400 . H =5 h = 5 . F .
126. 4,5.103 8 . 8,832 . ?
127. , , . , v1 v2. , m1 = 1 , m2 = 3 , W = 3 .
128. 50 12 / 30 . , . , , 5% .
129. = 2540 , , m= 12 . v0=800 /. , a=50, k = 0,12?
130. = 3,751026 . ?
131. 0,8 ?
132. v , k = 5%?
133. , 30- 20 . . 35 .
134. 100 . .
135. , m = 0,5 , R= 5 .
136. , m = 10 r = 10 , , , ν = 6 -1. t = 5 . M.
137. m1= 1 m2=1,5 , k=0,2 m1 , m = 0,5 , .
m2 , m1, 5 .
138. . J = 63,6 .2, = 31,4 /. t = 20 . .
139. R = 20 , J = 0.1 .2 , m= 0,5 . h = 1 . . .
140. , , , , ω1 = 5,0 / ω2 = 10 /. ω , . ρ1 = 2,6103 /3, ρ2 = 8,6103 /3.
141. m1 = 60 m2=100 . ω , R1 =5 ? v = 3,6 / . R2= 10 . , .
142. m = 80 R = 1 =20 /. . , , , J1 = 2,94 J2 = 0.98 2? .
143. R = 1 = 6 --1. , m = 80 . , . J=120 .3.
144. , . , , , =10 -1. R = 20 c, m = 3 . , f= 180? J= 6 .2. .
145. v = 18 /. L , ( h/L) α = 0,10.
146. ε = 0,5 /2 t1 = 20 W = 500 . L t2 = 15 ?
147. S , α = 300, v0 = 7,0 /, ?
148. N , , R = 1,0 m = 1000 , t = 1 ω = 31,4 /. .
149. d= 5 D = 5 m = 0,4 . , . , . .
150. d= 1 . D = 0,5 . v h. h = 0,2 .
151. , 4 . , , ?
152. d = 1 v =0,185 / , . . rc=8600 /3, - r = 900 /3.
153. = 2 /. , D = 6,65 . r = 800 /3.
154. d1 = 20 c. v1 = 1 / , d2 = 2 . v2 ?. ?
155. , , d = 12 . , 9g. σ =500 .
156. 1 2 . , 0,5 ? = 200 .
157. d , m= 10 ? σ = 500 , k =6.
158. , m= 2. L = 5 , S = 4 2. , ∆ L ε , = 21011 .
159. . h ? . σ = 500 , ρ = 1030 /3, ρ = 7800 /3.
2
,
i.
1.
:
Q = = const.
2. :
Q = ne,
n = 1, 2...; = 1,6 . 10-19
3.
:
,
:
,
F12 - () ; r - ; ε0 = 8,8510-12 Է/ - .
4. :
. .
5. :
. .
6. :
. .
7. :
, ,
- , Q0, .
8. :
, ,
W - Q0; A∞ - .
9. :
, , :
1 φi - , Qi.
10. :
,
A12 1 2.
11. :
,
= ,
- .
12. :
; Δφ = Ed.
13. :
,
l - E dl. .
14. , 1 2:
,
A12 = Q0 Q0 .
15. Q Q0:
.
16. :
A12 = QEl cosα.
17. :
dN ,
dN = EdScos α = EndS,
- , dS, ; En = Ecos α -
18. :
NE = = .
19. :
NE= = ,
NE = ,
ε0 , Qi - ; n- ; ρ - .
20. .
Q | E = Q/4πε0r2 | |
σ | E = σ/2ε0 | |
, d | r ≤ 0, r ≥ d: E= 0 0 < r < d: E = σ/ε0 | |
R | 0 < r < R: E = 0 r = R: E = Q/4πε0R2 r > R: E = Q/4πε0r2 | 0 < r ≤ R: r > R: |
, R | 0 < r < R: E = Qr/4πε0R3 r = R: E = O/4πε0R2 r > R: E = Q/4πε0r2 | 0 < r < R: r = R: φ = Q/4πε0R r > R: φ = Q/4πε0r |
R () τ | r < R: E = 0 r = R: E = τ/2πε0R; r > R: E = τ/2πε0r | r < R: r > R: |
21. :
, ,
Q, , φ - .
, :
C = εC0.
22. :
C = 4πε0εR,
R ; ε - .
23. :
C = ,
Q , ; ∆φ - .
24. :
,
S - ; d .
25. :
:
1/ C = 1/ Ci;
:
C = Ci,
Ci - i- , n - .
26. :
,
- , , i , Qi.
27. :
,
Q ; C ,
28. :
,
- .
29. ( ):
,
S ; V = Sd -
30. :
w = ; .
II.
1. :
, , ,
dQ , dt.
2. R G :
, , ,
ρ ; ℓ - ; γ ; S .
3. :
) - ,
) - ,
Ri i- .
4. :
)