ΛΥΣΕΙΣ ΤΩΝ ΑΣΚΗΣΕΩΝ
1.
∞. ·)
n2 = 0,75 n1 P1V1 = n1 RT1 P2V1 = n2 RT1
}
⇒ P2 = 3?105 N/m2.
‚) ∞ÊÔ‡ ∆ = ÛÙ·ı ⇒ Urms = 2,5?103 m/s. B. ·) Uµ = 3 U∞ 3RTB =3 n2 ∞ → µ:
3RT∞ ⇒ ∆µ = 9∆∞ n2
V1 V = 2 ⇒V2=9V1=18?10 -3 m3 T A TB
WAB=P2 (V2-V1)=4800 J.
‚) Uµ = 3 U∞ U° =U∞ ¢Uµ° = U° - Uµ = U∞ - 3U∞= -2 U∞= -5?103 m/s.
P 2.
n=
6 R
PA
VA = 10?10-3 m3
A
T∞ = 273 + ı1 = 300 ∫ ƒ∞V∞ = nRT∞ ⇒ƒ∞= 1,8 ? 105 N/m2 A→B : T∞= ÛÙ·ı.
}
ƒ∞V ∞ = P BV B PB = 1 ƒ∞ 32
° ⇒VB=32 V∞
CP = 5 CV 3
P BV ° Á = P ∞V ∞ Á PB = 1 ƒ∞ 32 µ→° : ƒµ = ÛÙ·ı. Vµ V = ° ∆A ∆A VB= 32 V∞ V °= 8 V ∞
}
}
⇒ V° = 8 V∞
⇒T° =
T∞ 4
TA T°
O
°→A : CP = CV + R = 5R 2 Á=
B
PB
VA
V°
VB
V
πÛfiıÂÚÌË :
¢T∞µ = 0 ¢V∞µ = Vµ - V∞ =31 V∞ = 310?10-3 m3 31 ¢ƒ∞µ = ƒµ - ƒ∞ = ƒ = -1,74?105 N/m2 32 ∞ W∞µ = nR∆Aln
Vµ = 9000ln2 J = 6210 J VA
¢U∞µ = 0 Q∞µ =W∞µ =6210 J πÛÔ‚·Ú‹˜:
Tµ =T∞ 3T ¢∆µ° = ∆° - ∆µ = - ∞ = -225∫ 4 ¢Vµ° = V° - Vµ = 8V∞ - 32 V∞= - 24 V∞= - 240?10-3 m3 ¢ƒµ° = 0 Wµ° = PB?¢Vµ° = -13500 J ¢Uµ° = ncV¢∆µ° = -2025 J Qµ° =Wµ° + ¢Uµ° = -15525 J
A‰È·‚·ÙÈ΋:
¢∆°∞ = ∆∞ - ∆° = +
3T∞ = 225∫ 4
¢V°∞ = V∞ - V° = - 7V∞= -70?10-3 m3 31 ¢ƒ°∞ = ƒ∞ - ƒ° = ƒ = 1,74?105 N/m2 32 ∞ ¢U°∞ = ncV¢∆°∞ = 2025 J W°∞ = - ¢U°∞ = -2025 J Q°∞ = 0
3.
⇒ ∆∞ = 400 ∫
∞. ·) P∞V∞ = n RT∞ VA V = µ ∆∞ ∆µ
ƒµ ƒ = ° ∆µ ∆°
V° V = ¢ ∆° ∆¢
∞ ƒ∞= 6?105 N/m2 VA = 10-3 m3
µ ƒB= 6?105 N/m2 VB = 3?10-3 m3
° ƒ°= 2?105 N/m2 V° = 3?10-3 m3
∆∞ = 400 ∫
∆B = 1200 ∫
∆° = 400 ∫
ƒ(103¡/m2) 6 A
2 O
B
3
°
1 ¢
°
¢ 1
V(10-3m3) 3
V(10-3m3)
‚) ¢U∞° = ncV(∆° - ∆∞) = 0 Á) W∞µ°¢∞ =∂∞µ°¢∞ = 1200 J
400/3
¢ ƒ¢= 2?105 N/m2 V¢ = 10-3 m3 ∆¢ = 400 ∫ 3 B
A 400
1200
T(K)
B. ·)
·=
WÔÏ W∞µ°¢∞ = QÚÔÛÊ Q¢∞+QAµ
Q¢∞= ncV (∆∞ - ∆¢) = 1200 J QAµ = ncƒ (∆µ - ∆∞) ⇒ QAµ = 6000 J 5R Cƒ = CV + R = 2
}
ÕÚ· ‚)
·=
1200 J = 0,1667 1200 J+6000 J
∆max =∆B = 1200 K ∆min =∆¢ = 400 K 3 ·c = 1 -
∆min = 0,889 ∆max
P 4.
n = 10 moles PA = 2?105 N/m2 TA = 273 + 127 = 400 K cV =
PA
A
B
cp = 11,94 J/(mole?K) Á
°
A → B (ÈÛÔ‚·Ú‹˜)
¢
VA 2V∞ = ⇒ ∆µ= 800 ∫ ∆∞ ∆µ
VA
ƒµ = P∞ = 2?105 N/m2. ¢UAµ = ncV(∆µ - ∆∞) = 47760 J A → ° (ÈÛfiıÂÚÌË) ƒ∞V∞ = P°2VA ⇒ P° = 1?105 N/m2 Q∞° = W∞° = nR∆Aln A → ¢ (·‰È·‚·ÙÈ΋)
V° = 22908 J VA
P∞V∞ Á = P¢(2V∞)Á ⇒ P¢ = P∞V∞ = n RT∞ P¢V¢ = n RT¢
u
TA
⇒ ∆¢ =
V
2VA
QAµ = ncƒ(∆µ - ∆∞) = 81200 J
T° = ∆∞ = 400 ∫ ¢U∞° = 0
P∞ = 0,61?105 N/m2 21,7
2∆∞ = 246 ∫ 21,7
Q∞¢ = 0 ¢U∞¢ = ncV(∆A - ∆¢) = 18387,6 J. P 5.
¢
°
A
B
∂›Ó·È: WAµ° = 50 J Q∞µ° = 80 J W∞µ°¢∞ = -40 J WAµ° = WAµ + Wµ° = WAµ ⇒ WAµ = 50 J ¢U∞µ° = Q∞µ° - WAµ° = 30 J ·) ¢U∞° = ¢U∞µ° = 30 J
V
‚) W∞µ°¢∞ = WAµ + Wµ° +W°¢ + W¢∞ Wµ° = 0 WA¢ = 0 W°¢∞ = W°¢ + W¢∞ = -90 J ¢U°¢∞ = ¢U°∞ = - ¢U∞° = -30 J Q°¢∞ = W°¢∞ + ¢U°¢∞ = -120 J ∂ W∞µ°¢ Á) W∞µ°¢∞ = ∞µ°¢∞ = = -20 J 2 2
⇒ W°¢ = -90 J
W∞µ°∞ = WAµ + Wµ° +W°A ⇒ W°A = -40 J ⇒ WA° = 40 J
6.
Q1 )Q ) = 2 ⇒ ∆2= 0,75T1 ⇒ ∆2=450 K ∆1 ∆2
∞. ·) Q2 = 0,75Q1
‚) · = 1 c Á) ·c =
µ.
∆2 ⇒ ·c = 0,25 ∆1
P A
W Q1 ⇒ W = 90 J
∆ 19 = T 1 +
B
20 T = 1,20T1 = 720 ∫ 100 1
T1
¢ °
20 ∆ 29 = T 2 T = 0,80T2 = 360 ∫ 100 2 · c9 = 1 -
7.
∆ 29 ∆ 19
T2 V
= 0,50
∞) ∏ ·fi‰ÔÛË Ù˘ Ì˯·Ó‹˜ Carnot ›ӷÈ: e = 1-
∆c ∆h
ÁÈ· e = 0,4 Î·È ∆c = 250 ∫
‚Ú›ÛÎÔ˘ÌÂ: ∆h = 416,6 ∫. ŸÙ·Ó Ë ·fi‰ÔÛË Á›ÓÂÈ ec = 0,5 ÙfiÙ ·fi ÙË Û¯¤ÛË ∆c ec = 1‚Ú›ÛÎÔ˘ÌÂ: ∆h9 = 500 ∫. ∆h ∏ ·‡ÍËÛË Ù˘ ıÂÚÌÔÎÚ·Û›·˜ Â›Ó·È ¢∆ = 500∫ - 416,6∫ = 83,4∫ µ) ŸÙ·Ó Ë ·fi‰ÔÛË Ù˘ Ì˯·Ó‹˜ Carnot Â›Ó·È e = 0,4 ‚ڋηÌ ∆h = 416,6 ∫. ªÂÙ·‚¿ÏÏÔ˘Ì ÙË ıÂÚÌÎÚ·Û›· ÒÛÙÂ Ë ·fi‰ÔÛË Ó· Á›ÓÂÈ ec = 0,5 Ì ∆h = 416,6 ∫, ∆9 ÙfiÙ ·fi ÙË Û¯¤ÛË ec = 1- c ‚Ú›ÛÎÔ˘ÌÂ: ∆c9 = 208,3 ∫. ∏ ÂÏ¿ÙÙˆÛË Ù˘ ∆h ıÂÚÌÔÎÚ·Û›·˜ Â›Ó·È ¢∆ = 208,3∫ - 250∫ = -41,7∫. P
8.
·) MÂÙ·‚ÔÏ‹ A→B : 3P0 P P P P Œ¯Ô˘Ì A = B ⇒ 0 = B ⇒Pµ= 3P0. V∞ Vµ V0 3V0 MÂÙ·‚ÔÏ‹ B →° : P P 3P P Œ¯Ô˘Ì B = ° ⇒ 0 = 0 ⇒Tµ= 9T0. ∆B ∆° ∆B 3∆0 ™¯Â‰È¿˙Ô˘Ì ÙÔ ‰È¿ÁÚ·ÌÌ· P-V.
P0
B
A
°
V0
3V0
V
P0 + 3P0 Afi ÙÔ ‰È¿ÁÚ·ÌÌ· P-V Ì ÙÔ ÂÌ‚·‰fiÓ ˘ÔÏÔÁ›˙Ô˘Ì ÙÔ W∞µ = ? V0 ⇒ 2 W∞µ = 4P0? V0 . E›Ó·È ·fi ÙÔ 1Ô £ÂÚÌÔ‰˘Ó·ÌÈÎfi ÓfiÌÔ: Q∞B = W∞B + ¢U∞B ⇒ 16P0? V0 = 4P0? V0 + ncV(9∆0 - ∆0) ⇒ 12P0? V0 = = n R ? 8∆0 ⇒ Á - 1 = 2 ⇒ Á = 5 Á-1 3 3 ‚) E›Ó·È WÔÏ = 1 ?2P0? 2V0 ⇒ WÔÏ = 2P0? V0 2 ™˘ÓÙÂÏÂÛÙ‹˜ ·fi‰ÔÛ˘ e = QB° = Q°A
Á) ∂›Ó·È
9.
2P0? V0 WÔÏ = = 1 16P0? V0 Q∞B 8
ncV(3∆0 - 9∆0) ncP(∆0 - 3∆0)
= 3 = 9 Á 5
A. ·) VÔÏ(¢) = 0 kËÏ
¢ñ
QB QA Q + kËÏ + kËÏ ° = 0 ⇒ Q° = -4ÌC. ·ü · · ·
‚) U= UAµ + UA° + Uµ° = kËÏ µ. ·)
ñ°
Q AQ B Q Q Q Q + kËÏ A ° + kËÏ B ° ⇒ U = -0,408J · · ·ü
˘A
A
Añ
B
·
˘B ñ r IÛ¯‡Ô˘Ó: 0 = mA? ˘A - mB? ˘B (·Ú¯‹ ‰È·Ù‹ÚËÛ˘ ÔÚÌ‹˜) k
ñ
Q AQ B Q Q =k A B + 1 mA? ˘A2 + 1 mB? ˘B2 (·Ú¯‹ ‰È·Ù‹ÚËÛ˘ Ù˘ ÂÓ¤ÚÁÂÈ·˜) · r 2 2
Afi ÙȘ ÂÍÈÛÒÛÂȘ ·˘Ù¤˜ ÚÔ·ÙÂÈ ÙÂÏÈο: ˘A= 34 m/s, ‚) Afi 0 = mA ? ˘A - mB ? ˘B ÚÔ·ÙÂÈ ˘A=2˘B
˘B = 17 m/s
AÓ Â›Ó·È ˘A= 60m/s ⇒ ˘B= 30m/s k
Q AQ B = Ux + 1 mA ? ˘A2 + 1 mB ? ˘B2 ⇒ Ux = 0,141 J. · 2 2
Á) TfiÙ ÈÛ¯‡ÂÈ: 0 = mA? ˘A - mB ? ˘B k
Q AQ B = 1 mA ? ˘A + 2 1 · 2 2
⇒ mB ? ˘B2
˘A= 78,4 m/s ˘B =39,2 m/s
ñB
10.
A. ·) F∞= B ?˘0 ? qA = 10 Ø 10-4 ¡ F° = B ?˘0 ? )q°) = 10 Ø 10-4 ¡
→
A RA
2m∞ = Ø 10-4 s ‚) ∆∞= BqA 5
O
2m° 2 ∆° = B)q ) = Ø 10-4 s 5 °
R
m ∞ ˘0 Á) R∞= Bq = 10 cm < d A R° =
^B
° d
m ° ˘0 = 20 cm < d B)q°)
‰) (∞°) = 2R∞ + 2R° = 60 cm
R9A R9A
d
Ø10-4N
B. ·) F∞ = BUo9qA = 40 F° = BUo9q° = 40 Ø 10-4N ‚) ∆∞ Î·È ∆° ‰ÂÓ ÂËÚÚ¿˙ÔÓÙ·È ·fi ÙËÓ Ù·¯‡ÙËÙ· m ∞ ˘0 = 40 cm > d Á) R∞ = BqA Rr9=
d
A
^B
XA
O X°
R9r
d
°
m∞˘09 = 80 cm > d B)q°)
R9r
‰) XA = RA9- RA92 - d2 = 4 cm X° = R°9 - R°92 - d2 = 6 cm (∞°) = XA + X° = 10 cm
11.
A
K
x
∞. ∂ÈÙ·¯˘ÓfiÌÂÓË (ÌË ÔÌ·Ï‹) ΛÓËÛË, ± ± ± B^ FL F ̤¯ÚÈ Ó· ·ÔÎÙ‹ÛÂÈ ÔÚȷ΋ Ù·¯‡ÙËÙ·. R1 ™ÙË Û˘Ó¤¯ÂÈ· ¢ı‡ÁÚ·ÌÌË ÔÌ·Ï‹. ± µ. ¶·Ú¤¯ÂÈ ¤ÚÁÔ Ë F ÙÔ ÔÔ›Ô ·˘Í¿ÓÂÈ ° „ ÙËÓ ÎÈÓËÙÈ΋ ÂÓ¤ÚÁÂÈ· Ù˘ Ú¿‚‰Ô˘ § ± (¢∫ = W™F) Î·È Ù·˘Ùfi¯ÚÔÓ· ̤ۈ ÙÔ˘ ¤ÚÁÔ˘ Ù˘ FL ıÂÚÌfiÙËÙ·˜ ÛÙȘ R1, R2 (Ê·ÈÓfiÌÂÓÔ Joule). ∞fi ÙË ÛÙÈÁÌ‹ Ô˘ Ô ·ÁˆÁfi˜ ·ÔÎÙ‹ÛÂÈ ÔÚȷ΋ Ù·¯‡ÙËÙ· ÙÔ ± ¤ÚÁÔ Ù˘ F ÌÂÙ·ÙÚ¤ÂÙ·È Û ıÂÚÌfiÙËÙ·. I = E = 2A. °. ·) ∂ = µu, = 10V R1+R2 ‚) FL = BI, = 2N Á) V∫§ = VA° = πR1 = 4V. ±
™F = F - FL = 8N
¢. ·) ŸÙ·Ó ˘ = ÔÚȷ΋ ⇒ ™F = 0 ⇒ FL= F = 10N I=
E ⇒ ∂ = 50V R1+R2
‚) V∫§= V = IR1 = 20V Á) ƒ1 = π2 R1 = 200W ‰)
¢WÂÍ = I Ø EÂ = 500 J/s. ¢t
™F = m· ⇒ = 4 m/s2
FL = BI, ⇒ I = 10A
∂ = µ˘ÔÚ, ⇒ ˘ÔÚ= 50 m/s
12.
+Q(∞)
FL
-q
°-q
uo
u=0
r1 r2
∞) To ÊÔÚÙ›Ô -q ÌÂÙ·ÎÈÓÂ›Ù·È ·fi ÙËÓ ·Ú¯È΋ ÙÔ˘ ı¤ÛË µ ˆ˜ ÙË ı¤ÛË ° Ô˘ ·ÎÈÓËÙÔÔÈ›ٷÈ. ∂Ê·ÚÌfi˙ÔÓÙ·˜ ∞.¢.ª.∂ ·›ÚÓÔ˘ÌÂ: 1 muo2+ kc -q?Q = kc -q?Q ·ÓÙÈηıÈÛÙÒÓÙ·˜ ÙȘ ‰Â‰Ô̤Ó˜ ∂ª(µ) = ∂ª(°) ⇒ 2 r1 r2 ÙÈ̤˜ ÙˆÓ Ê˘ÛÈÎÒÓ ÌÂÁÂıÒÓ ·›ÚÓÔ˘ÌÂ: r2 = 0,36m B) AÓ ÙÔ ÊÔÚÙ›Ô -q ÂÎÙÔÍ¢ı› Ì ٷ¯‡ÙËÙ· uo Êı¿ÓÂÈ ÛÙÔ ¿ÂÈÚÔ Î·È ÛÙ·Ì·Ù¿. ∂Ê·ÚÌfi˙ÔÓÙ·˜ ∞.¢.ª.∂ ·›ÚÓÔ˘ÌÂ: 2kc?Qq 1 ∂ª(µ)=∂M(`) ⇒ mu9o2+ kc -q?Q = 0 ⇒ uo= . ∞ÓÙÈηıÈÛÙÒÓÙ·˜ ÙȘ mr1 2 r1 ‰Â‰Ô̤Ó˜ ÙÈ̤˜ ÙˆÓ Ê˘ÛÈÎÒÓ ÌÂÁÂıÒÓ ·›ÚÓÔ˘ÌÂ: uo= 20 3 m/sec . u 3 BÚ›ÛÎÔ˘Ì ÙÔ ÏfiÁÔ ÙˆÓ Ù·¯˘Ù‹ÙˆÓ: o = . 3 uo
13.
i(A) Ī =
2
ü
V™ ⇒ R = 40ø R
0
∞. ŒÛÙˆ i = I ËÌ (ˆt+Ê0)
∞fi ÙË ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË ÚÔ·ÙÂÈ: π=üA
t(ms) 10
20
-ü
Ê0 = 0 T = 20 ?10-3s ⇒ ˆ = 2 = 314 rad/s. ∆ ÕÚ· i= 2ËÌ(314 t) ÛÙÔ S.I. Î·È ·fi u=VË̈t, V=IR ÚÔ·ÙÂÈ : u=40ü ËÌ(314 t) B. ·) πÂÓ = π
ü
‚) ¢t = 1 min = 60s, Q = °.
VÂÓ = V = 40 V.
= 1 ∞.
ü
IÂÓ2R¢t
= 2400 J.
°È· t1 = 2,5 ms = 2,5 ? 10-3s, ›ӷÈ: ·) Ê1 = 314 t1 = 100 ? 2,5 ? 10-3rad = rad. 4 ‚) i1 = I ? ËÌÊ1 = üËÌ = 1A 4 Á) ƒ1 = i1 ? v1 = 40 W. °È· t2 = 12,5 ms = 12,5 ? 10-3s ›ӷÈ:
v1 = VËÌÊ1 = 40üËÌ = 40 V. 4
·) Ê2 = 100 ? 12,5 ? 10-3 = ( + ) rad. 4 ‚) i2 = I?ËÌÊ2 =üËÌ ( + )=- üËÌ =-1A 4 4 v2= V?ËÌÊ2 = - 40üËÌ (+ )= - 40 V. 4 Á) ƒ2= i2 ? v2 = 40 W.
14.
2 ·) L1 = kÌ 4 ¡1 A1 = ? 10-2 ∏ ,1
‚) ª = kÌ 4
¡1¡2 A2 = ? 10-4 ∏ ,1
Á) ∂·˘Ù(1) = - L1 ¢π = ? V. ¢t ‰) ∂·Ì(2) = - ª ¢π = ? 10-2 V. ¢t
Â) π2 =
∂·Ì(2) = ?10-3 ∞. R
ÛÙ) U = 1 L2 π22 = 23 ? 10-12J. 2
¡22 L2 = kÌ 4 , A2 = 4 ? 10-6 ∏ 2