ÔnÆ I � åÆ�Chap.4 Å�UÅð
Qijin Chen �é>
Department of Physics, Zhejiang University
úô�ÆÔnX!úôC�Ôn¥%
September 19, 2013
Copyright c©2013 by Qijin Chen; all rights reserved.
'uÄþÅðÚÄUÅð�{¤�Ø
1. ÄþÅðkd(k�JÑ René Descartes, 1637
2. Gottfried Leibniz → ¹å vis viva Åð (During 1676-1689)
living force∑imiv
2i Åð
Thomas Young, 1807 : 1�gr vis viva ��Uþ(energy)¦^
1703c§Huygens �[�ã�5-E¯K↓
��LãÄUÅð
mv2 Ú m~v Åð��§±Y100õc⇓
Two independent laws, both are right
19VÐϧ^Å�õÿ/¹å0§rN/õ0�Vg
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
Å�U£Åð¤
ØʱÏ{Ä�¨{ −→ �3,«ØCþ
1(7���±$ħ~v C§� |~v| ØC1
2mv2 ØC = const. −→ ÄU´Åðþ
®¥þeEaÄ®¥� �9�Ý÷v
v =√
2g(h0 − z)§þ,!eüÑ�(1
2mv2Ø2Åð§�
1
2mv2 +mgz = mgh0=const.
↑å³U
ÄU ↔ ³U �p=�
Å�U£Åð¤ (cont'd)
Am
mW
z
O
0hXmã§ÔWlh0?.X31wS¡þ�ÔNA�å$Ä"
Щµt = 0 �§W uz = h0, v0 = 0
=⇒ v =m
m+mAgt, z = h0−
1
2
m
m+mAgt2, (a =
mg
m+mA)
1
2mv2 +mgz = mgh0 −
1
2mAa
2t2 6= const.
�A�öħ1
2mAv
2 =1
2mAa
2t2
∴1
2mv2 +mgz +
1
2mAv
2 = mgh0 = const.
� ÔÔÔNNNA���WmmmUUUþþþ���===£££§§§���oooUUUØØØCCC
Å�U£Åð¤ (cont'd)
½½½ÂµµµT =
1
2mv2 � Kinetic energe, KE
3å|¥§å³U V = mgz
ÔNNX�ÄU�³U�Ú¡�Å�U
ÅÅÅ���UUUÅÅÅððð:
E = T + V = const.
Uþþjµ[E] = ML2T−2
ü µ1 J = 1 kg ·m2/s2 = 1 N ·m
Å�U£Åð¤ (cont'd)
ÅÅÅ���UUUÅÅÅððð333éééõõõ���ÿÿÿØØؤ¤¤ááá
~µ1 k�Þ�§~f ‖ −~v −→ Uþ���9U9UáSU§3�*��f!©f�ÄU
T + V +SU = const.
2 zÆUX�¿�A
3 ØU!>^U�=�3Å�U�Ù¦/ª�Uþ�p=z�§T + VØÅð
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
Ù¦/ª�³U
ÄU�k�«/ª§³Ukéõ«
m
0x
���555³³³UUUµ���x0§\Ô����x0 + h
Hook's law: f = −k (x− x0)
mg − k (x− x0) = mdv
dt= m
dv
dx
dx
dt= m
dv
dxv =
1
2
d
dx
(mv2
)∫ x0+h
x0
:1
2mv2
∣∣∣∣v=0
v=0
=
[mgx− 1
2k (x− x0)2
]∣∣∣∣x0+h
x0
= mgh−1
2kh2
∴ mgh = 12kh
2 å³U��5³Um=�↑
�5³UEp =
1
2kh2 =
1
2k (x− x0)2
�áX �«/ªUþ�oÚÅð ←− �m²£ØC5Question: Big bang§t = 0 �UþÅðíº
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
õ Work
½Â�:ÄU�CzL§�£å3¤�õ
A = T2 − T1 = W
�¦�:ÄU±T1C�T2§I�õW
T2 = W + T1
�� £2¤UþÅð§W´Uþ��«
W > 0§��õ¶W < 0§�Kõ
²þõÇ 〈P 〉 =T2 − T1
t2 − t1=
∆T
∆t
]�õÇ P =dT
dt= lim
∆t→0
∆T
∆t
õÚõÇ: ���/
�Ä����$ħ�:3å F �^e\�
mdv
dt= F
v× :1
2m
dv2
dt= Fv =
dT
dt= P�� ��P = Fv
W =
∫ t2
t1
Pdt =
∫ t2
t1
Fdx
dtdt =
∫ x2
x1
Fdx
W =
∫Pdt =
∫Fdx
O 1x 2x 3x
=⇒ �õ�9�L§ =⇒ õ´ÝþL§�Ônþ§ØU½Â,��]�½,�:�õ
XÔN3å��^ed x1 � x3 2� x2§ØU�È© x1 � x2
õÚõÇ: 3D �/
ííí222��� 3D case
md~v
dt= ~F
m~v · d~v
dt= ~F · ~v
d
dt
(1
2m~v2
)= ~F · ~v =
d
dtT = P�� ��P = ~F · ~v
W =
∫ t2
t1
Pdt =
∫ t2
t1
~F · ~vdt =
∫ ~x2
~x1
~F · d~x
(l)←÷;,È©
∆W = ~F ·∆~r = F∆s cos θ θ =(~F ,∆~r
)Y�
R�u$Ä��Ø�õ~Fcor = −2m~ω × ~v′ ⊥ ~v′, Ø�õ
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�:ÄU½n
�^uÔNþ�Üå¤��õ�udL§¥ÄU�Oþ��
� W =
∫ ~r
~r0
~F · d~r = EK − EK0 = T − T0
~F · d~r � �õ
P =dT
dt=
dW
dt= ~F · ~v
õ�ü J: 1 J �u 1 N �åí£ 1 m�´§¤��õ§õÇ1 W = 1 J/s
>þ ݵ103 W · h = Z�� = 3.6× 106 J
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�:XÄU½n
mi~ri = ~Fi +∑j 6=i
~fji
∫ t
t0
·~vidt :
∫ t
t0
mi~vi · ~vidt =
∫ t
t0
~Fi · ~vidt+∑j 6=i
∫ t
t0
~fji · ~vidt
=
∫ t
t0
1
2d(mi~v
2i
)
=⇒ Ti (t)− Ti (t0) = Wi +∑j 6=i
Wji
Wi =
∫ t
t0
~Fi · ~vidt =
∫ xi
xi0
~Fi · d~xi
Wji =
∫ t
t0
~fji · ~vidt
�:XÄU½n (cont'd)
∑i
[Ti (t)− Ti (t0)] =∑iWi +
∑i,j 6=i
Wji
=⇒ T − T (t0) = W +WS �� �:XÄU½n
T =∑iTi, W =
∑iWi, WS =
∑i,j 6=i
Wji
5¿WS�Ñy§��:Äþ½nØÓd?Så�ØUCoÄþ§�O\�õ�ϵÄþ�¥þ§ÄU£õ¤´Iþå�Så�õ�Ú�u�:XÄU�Oþ�:XÄþ½nÚõU½n�pÕá
~~~4.1 m = 10�§v0 = 200 m/s§�\7¬§²þ{åF = 5× 103 N, ¦\��Ý.
E1 =1
2mv0
2 = Fs =⇒ s =mv0
2
2F= 0.04 m
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
Úå³U
å³U mgz
M 1rrm
1 3
2
5
4�kÚå ~F = −GmM
r2r
1→ 2 : W = A = ~F ·∆~r = −GmMr1
2∆r
�� ���Ã'
1→ 3 : A = ~F · d~r =GmM
r12
∆r
cos θcos (π − θ) = −GmM
r12
∆r
4→ 5 �Óu 1→ 2
�kÚå�õ§��Ð!"�ålk'§���Ã'=�´»Ã'
a→ b : A =
∫(l)
~F · d~r =
∫ rb
ra
~F · d~r = −∫ rb
ra
GmM
r2dr
=GmM
rb− GmM
ra= Tb − Ta
Úå³U
Ta +
(−GmM
ra
)= Tb +
(−GmM
rb
)= const.
V = −GmMr
��Úå³U
Úå�k%å§4Ü´»�õ�". ← é ∀ k%å¤á
k%å ~F (~r) = F (r) r
´»Lþ?��� ds = KM , k
dA = ~F ·−−→KM = F (rk) ds cos θ
= F (rk′)K ′M ′ →ÝK�,�»���þ
A =
∫ Q
PdA =
∫ Q′
PF (r) dr =
∫ rQ
rP
F (r) dr, ���»k',�´»Ã'
=⇒∮
~F · d~r = 0
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�Åå
Úå9k%å�õ�´»Ã'§��Þ�´»k'y
O
1y
2y2 3
1
x
Xmã§1→ 2 gdáN
åõµAG1→2 = mg (y1 − y2)
FrictionµAf1→2 = 0
1→ 3 : AG1→3 = mg
y1 − y2
cos θ· cos θ = mg (y1 − y2)
Af1→3 = µmg sin θ
y1 − y2
cos θcosπ
= −µmg (y1 − y2) tan θ
3→ 2 : AG3→2 = 0
Af3→2 = (y1 − y2) tan θ · µmg cosπ = −µmg (y1 − y2) tan θ
=⇒ AG1→3 +AG
3→2 = AG1→2
Af1→3 +Af
3→2 = −2µmg (y1 − y2) tan θ 6= Af1→2
�Åå (cont'd)
XJå��õ�´»Ã'§��Ð": �k' −→ �Åå
é�Åå§�½Â¼ê
V (rb) = Va −Aa→b, Va�T¼ê3a??À�ê�
V (r) ¡�³U¼ê=⇒ Aa→b = Va − Vb∵ Aa→b = Tb − Ta
=⇒ Ta + Va = Tb + Vb = const.
∴ �ÅåX¥§Å�UÅð
�L5If Ta + Va = Tb + Vb = const., (∀a, b)
=⇒ Tb − Ta = Va − Vb = Aa→b
K Aa→b = Va − Vb �´»Ã' =⇒ �Åå
�Åå (cont'd)
=⇒ Å�UÅð (T + V = const.) ´NX¥��3�Åå�¿�^�§´³UVg·^�¿�^�.[
é��Ååµ~f · d~r ≤ 0, ÑÑå§Å�U~�~f · d~r ≥ 0, Å�UO\
]
V (~r) = −∫ r
r0
~F · d~r + V0
~F (~r) = −~∇V (~r) =
(i∂V
∂x+ j
∂V
∂y+ k
∂V
∂z
)Úå³
~F = −GmMr2
r
� V (r =∞) = 0
V (~r) = −∫ r
∞~F · d~r =
∫ r
+∞GMm
r2dr= −GMm
r
⇐⇒ ~F = −~∇V (r) = −GmMr2
r
�Åå (cont'd)
5¿µ1 Úå³U�m,M�ö�k§�3Äþ¥%X¥
m~v +M~V = 0 =⇒ 1
2mv2 � 1
2MV 2
ÄUÌ��m¤k§T + V Åð�¦ M �Czé�2
A (~r0 − ~r) = V (r0)− V (r) = Tr − T0
= − [V (r)− V (r0)]
�Åå�õ¦³U~�
3 XJNX¥�k�Åå§K��ÅNX4 ØÓ�ÅåÚå�³U��\
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
³U�
å´¥þ§³U´Iþ§�´(½§��B��³U¼ê=�³U�
地表重力 引力，库仑力
V V
O z
mgz cr
r
2
cr
rF F
mg
O
O O
³U�
弹性势能 双原子分子
V V
O
O
OO
r
r
FF
x
x
212
kx
kx
0r
0r
1 ~F = −~∇V → d³¦å2 V (r) → ¦²ï �!½5©Û
~: /¥Úå��<º�Ý
0E
0E
V
maxr
V引
rE =
1
2mv2 − GmM
r= const.
�E < 0, rmax =GmM
−E, åP$Ä
�E > 0, v > 0, ∀r, � gd$Ä
1
2mv2−GmM
r= 0 =⇒ v =
√2GME
R=√
2gR ' 11.2 km/s2
(g =
GME
R2
)
~: �E&�
�EÅ�UØU�p½�$.
À�Ü·�Uþ§¦�E?\�¥Úå�§2�{~�
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
��$Ä���5� → �Åå�/
V (x1)− V (x2) =
∫ x2
x1
Fdx
F = −dV
dx1
2mv2 + V (x) = E
=⇒ E ≥ V (x)
If E = E2 =⇒ x1 ≤ x ≤ x2
E1 ≤ E ≤ 0�§�åP$ÄE > 0�§��±∞
If E = E3,−∞ ≤ x ≤ x3 or x ≥ x4. (x3 ∼ x4�m�³^)
E ≥ E4,−∞ < x < +∞gd�åP$Ä� E = E1 �, x = x5§·�§4��:§
F (x5) = − dV
dx
∣∣∣∣x=x5
= 0 ØÉå
� x = x6 �§�k F (x6) = 0, ØÉå� E = E4 = V (x6) �§x6 ? v = 0§·�§�ؽ
��$Ä���5�→�Åå�/
dx
dt= ±
√2 (E − V ) /m , −→÷± x��$Ä
t = ±√m
2
∫ x
x0
dx√E − V (x)
, −→�¦$Ä�m
t1→2 =
√m
2
∫ x2
x1
dx√E2 − V (x)
, F (x2) = − dV
dx
∣∣∣∣x2
< 0§$Ä��
t2→1 = −√m
2
∫ x1
x2
dx√E2 − V (x)
w,, t2→1 = t1→2
=⇒ ±Ï T = 2t1→2 =√
2m
∫ x2
x1
dx√E2 − V (x)
~: á��³²¥�$Ä
V (x) =
{0,∞,
−l/2 ≤ x ≤ l/2elsewhere
V x
V V
0V
/ 2l/ 2l x
T =
√2m
E
∫ l/2
−l/2dx =
√2m
El
E =1
2mv2 =⇒ T =
2l
v
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�:XÅ�UÅð½Æ
T − T0 = W +WS → �:XÄU½nWS = W�S +W��S
W�S = V0 − V, V (r)��ÅSå�³¼ê
=⇒ (T + V )− (T0 + V0) = W +W��S
= E − E0 = W +W��S�:XõU½n½õU�n
W = 0 :
1!�áNX2!d~r = 0, → å�^:vk £3!~Fext ⊥ d~r§å�Ù�A�^:� £p�R�
If W = 0,
1 W��S > 0§X�¿2 W��S < 0§X�Þå3 W��S = 0§Å�UÅð
�:XÅ�UÅð½Æ
�.5X¥§X�A^õU½n§LO\.5å9Ù�'³U
5¿1 å¤��õ W �ë�Xk'§ Så¤��õ�ë�
XÃ'2 K.E. �ë�Xk'3 ³U V (r) �ë�XÃ'Å�UÅð3��ë�X¥¤á§3,��¥�Uؤá
~~~4.2 n�:n:���²¡þ§ØO�Þ§¥m�:¼Ð�Ý~v0, ¦ü>�:�����Ç v.
0v
y)))µµµ ÄþÅðµmv0 = 3mvy
UþÅðµ12mv0
2 = 12mv
2y + 2 · 1
2mv2
=⇒ v2 =1
2v0
2 − 1
2
(v0
3
)2=
4
9v0
2
v =2
3v0
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�ÅX��m�üØC5
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
üN¯K
1rCr
2r2m
1mC1r
C2r
X㤫ü�:NX
mc = m1 +m2
~rc =m1~r1 +m2~r2
m1 +m2
m1d2~r1
dt2= ~f21 = −~f12 = −m2
d2~r2
dt2= ~F
=⇒ mcd2~rcdt2
= 0 = mcd~vcdt
=d~Pc
dt
ØÉå�§�%�!�$Ä£�,�k�%ÄþÅð¤
üN¯K (cont'd)
��%X§K ~rc = 0
~rc1 = ~r1 − ~rc = ~r1 −m1~r1 +m2~r2
m1 +m2=
m2
m1 +m2(~r1 − ~r2) =
m2~r
m1 +m2
~rc2 = ~r2 − ~rc = − m1~r
m1 +m2
d? ~r = ~r1 − ~r2 → �é £
~rc1 − ~rc2 = ~r =⇒ ~v =d~r
dt= ~v1 − ~v2 �é�Ý
rc1rc2
=m2
m1or ~rc1m1 +m2~rc2 = 0
�:��%�ål��þ¤�'
üN¯K (cont'd)
XXX m1 � m2§§§���ÄÄÄ m1 ���ééé m2 ���$$$ÄÄÄ� S X§�éu m2 ·��²Äë�X§m2 u�:§~r2 = 0, ~r1 = ~r§SX��.5X
.5X¥µd2~r
dt2=
d2 (~r1 − ~r2)
dt2=
d2~r1
dt2− d2~r2
dt2
=
(1
m1+
1
m2
)~F =
m1 +m2
m1m2
~F =1
µ~F
µ ≡ m1m2
m1 +m2� �z�þ§òÜ�þ
=⇒ µd2~r
dt2= ~F
µ ' m1 −→ @���þ m2 Øħ ^ µ �� m1§KSXCq�.5X
üN¯K (cont'd)
�%X¥µ
~vc1 =m2
m1 +m2~v, ~vc2 =
−m1
m1 +m2~v
E =1
2m1v
2c1 +
1
2m2v
2c2 + V (r) ←³UV
=1
2m1
m22v
2
(m1 +m2)2 +1
2m2
m21v
2
(m1 +m2)2 + V (r)
=1
2
m1m2
m1 +m2v2 + V (r)
E =1
2µv2 + V (r)
��±µ��m1§K�@�SX´.5X��¦�Å�U£�%X¥�¤£�w,ØU¦��%XÄþ¤=|^�z�þ§�òüN¯Kz�üN¯K>
üN¯K (cont'd)
ggg������ÖÖÖKKKµµµ
nN¯K�±�z¤üN¯K½Ù¦{ü¯Kíº
���ÖÖÖ���ááá
Laplace�û½Øg�£ÄuÚîåƤ↓
Poincare�Chaos → �R�A��5XÚ¥§Ð©����Cz¬��XÚ1�4�UC↓
intrinsic stochasticity → S��Å5
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�ZF½n (Konig's Theorem)
�KX→.5X§Kc��%X
~ri = ~rc + ~rci, ~vi = ~vc + ~vci, ~ai = ~ac + ~aci
T =1
2
∑imiv
2i , Tc =
∑i
1
2miv
2ci
=⇒ T =1
2
∑imi (~vc + ~vci)
2 =∑
i
1
2mi
(~v2c + ~v2
ci + 2~vc · ~vci)
=∑
i
1
2mi~v
2c + ~vc ·
∑imi~vci +
∑i
1
2mi~v
2ci
=1
2mc~v
2c +
∑i
1
2mi~v
2ci (
∑imi~vci = 0)
=1
2mc~v
2c + Tc
NXÄU = �%ÄU + NX�éu�%X�ÄU�� ���ZZZFFF½½½nnn£3�.5X¥��¤
£Äþ�/µNXÄþ=�%Äþ¤
²þ�/½n0
∑i
XiWi = X∑i
Wi = WX, Xi = X + δi∑i
X2iWi =
∑i
(X + δi
)2Wi = X2
∑i
Wi +∑i
Wiδ2i + 2X
∑i
δiWi
= WX2 +∑i
Wiδ2i
ëYµW =
∫W (~r) d3r, X =
∫W (~r)Xd3r
W∫W (~r)X2 (~r) d3r =
∫W (~r)
(X + δ
)2d3r
= X2
∫W (~r) d3r +
∫W (~r) δ2 (~r) d3r + 2X
∫W (~r) δ (~r) d3r
= WX2 +
∫W (~r) δ2 (~r) d3r
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�%X¥.5å¤��õ
�Ä ~ac 6= 0
.5å−mi~ac = ~F i
in
õWi =
∫~F iin · d~rci
W. =∑i
∫(−mi~ac) · d~rci
= −~ac ·∫ ∑
i
mid~rci = −~ac ·∫
d (mc~rcc) = 0
(∵ ~rcc = 0)
=⇒ �%X¥ÃI�Ä.5å¤��õ
~~~4.3 ÓEþ�<± v′ = 4 m/s ��é� Ç�cÚ���ÑÔN m = 1 kg§b�E± v0 = 2 m/s��Ç$Ľ·�§ü«�¹e<éÔN©O�õ �õº
)))µdÄþÅð§ Mu+m (v′ − u) = 0or
M (v0 + u) +m(v′ − u+ v0
)= (m+M) v0
=⇒ u = − mv′
m+M' −mv
′
M, m�M
If v0 = 0:
Tm − T 0m =
1
2m(v′ − u
)2 ' 1
2mv′2
TM − T 0M =
1
2Mu2 ≈
(mM
) mv′22
=m
M
(Tm − T 0
m
)�(Tm − T 0
m
)
If v0 6= 0:
TM − T 0M =
1
2M (v0 + u)2 − 1
2Mv0
2
=1
2Mu2 +Mv0u
≈Mv0u = −mv0v′ , |u| � v0
Tm − T 0m =
1
2m(v′ − u+ v0
)2 − 1
2mv0
2
' 1
2m(v′ + v0
)2 − 1
2mv0
2
=1
2mv′2 +mv′v0
ü���é'�v0
v′§Ø�6uM
=⇒ ∆Tm+M '1
2mv′2
XXX^���%%%XXXµv0 = 0 =⇒ ∆T ' 12mv
′2, ÃIO�ÓE�$ÄG�
~~~4.4 1n�»�Ý(ºÑ��X��Ý)
�»��é/¡�Ç� v′§<Ñ/¥Úå�� v§³UV (∞) = 0d?À»�¨/¥�%X
1
2mv2 =
1
2mv′2 −GmME
RE=
1
2mv′2 −mgRE
¿©|^/¥ú=;��Ý1
2m (v + 29.8)2 −GmM�
R�= 0 →»�¨���%X
=⇒ v =
√2GM�R�
− 29.8 = 42.2− 29.8 = 12.4 km/s
v′2 = v2 + 2gRE = 12.42 + 11.22
=⇒ v′ ' 16.7 km/s
XJ�ÜÀ»�¨���%X§K�O9/¥�ÄUCz
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
-E
�5-Eµ-�ÔN©m§ÃUþ��§Å�UÅð�ÄL§4á§:�> → ÄU!ÄþÅð
��5µ��kÅ�U��§=z�-EÔ�m�SÜUþ����5µ-�üÔNÜ3�å$ħÅ�U����
���Ä-EL§4á§åÀþ9�õØO§okÄþÅð
���---£££ééé%%%---EEE¤¤¤µµµhead-on collision
-�ü¥�ÝE÷� ~r1 − ~r2 ��(1) Ø �ã§Ð��ü¥�Ý�� v§Ð� u1, u2
m1u1 +m2u2 = (m1 +m2) v
ÀþIµ�åé m2
m1v −m1u1 = −Im2v −m2u2 = I
}=⇒ (u1 − u2) = I
(1
m1+
1
m2
)I = µ (u1 − u2) , µ =
m1m2
m1 +m2
�-£é%-E¤
(2) ¡E�ã§g¥mü¥�Ý���"�Ý v1, v2
(m1 +m2) v = m1v1 +m2v2
�å� m2 �Àþ J
J = m2v2 −m2v−J = m1v1 −m1v
}=⇒ J = µ (v2 − v1)
½Â e = J : I −→ ¡EXê
v2 − v1 = e (u1 − u2)m1u1 +m2u2 = m1v1 +m2v2
}=⇒
v1 =
m1 − em2
m1 +m2u1 +
(1 + e)m2
m1 +m2u2
v2 =(1 + e)m1
m1 +m2u1 −
em1 −m2
m1 +m2u2
e = 1 :
v1 =
m1 −m2
m1 +m2u1 +
2m2
m1 +m2u2
v2 =2m1
m1 +m2u1 −
m1 −m2
m1 +m2u2
-EL§¥�ÄU��
Tc =1
2µv2
�
=1
2µ (u1 − u2)2
-E�T ′c =
1
2µ (v2 − v1)2 =
1
2µe2 (u1 − u2)2
ÄU��
∆T = Tc − T ′c =1
2
(1− e2
)µ (u1 − u2)2
(1) e = 1µ���5§UÄþ�Åðm1 = m2: v1 = u2, v2 = u1§���Ý
-EL§¥�ÄU��
ifu2 = 0 :
m1 > m2 =⇒ v1 > 0§�¥ØUUC�¥��m1 < m2 =⇒ v1 < 0§�¥��¥��m2 � m1 =⇒ v1 ' −u1, v2 ' 0§�¥���£m2 � m1 =⇒ v1 ' u1, v2 ' 2u1§�¥��c?§
�¥¼2��Ý
T f2
T i1
=12m2v
22
12m1u2
1
=4m1m2
(m1 +m2)2 =4µ
m1 +m2=
4m1/m2
(1 +m1/m2)2
d(T f
2 /Ti1
)d (m1/m2)
= 0 =⇒ m1
m2= 1
=��þ-E§m2 ¼���ÄU§½ö`§� u2 = 0 �§m1
��C m2§m1 ¿��ÄU�õ
(2) e = 0µ����5-E§ÄU���õ
∆T = Tc − T ′c =1
2µ (u1 − u2)2 ,
(T ′c = 0
)-� v1 = v2 = vc =
m1u1 +m2u2
m1 +m2
éuâf-E�A§du E = Tc +1
2mcv
2c �
1
2mcv
2c
=⇒ =¦´����5§Ù�%�ÄU�جUC§Ïd �^�ÄU�´�%X£½Äþ¥%X¤¥�ÄU§½�éÄU
éu��þ m1 E· � m2§µ = 12m1,
Tc = 12µu
21 = 1
4m1u21 = oÄU��.
(3) 0 < e < 1µ���/
If m2 � m1, u2 = 0, K v1 = −eu1, v2 = 0
�¥£��Ý�-c�e�.
�¥�/¡-E�¡EXêÿ½µ
u1 =√
2gh0 , h =v2
1
2g
e =v1
u1=
√h
h0
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�-µ~v1�~u1ز1
�Ä�5-Eµm1~u1 +m2~u2 = m1~v1 +m2~v2
1
2m1u1
2 +1
2m2u2
2 =1
2m1v1
2 +1
2m2v2
2
If u2 = 0µm1~u1 = m1~v1 +m2~v2
1
2m1u1
2 =1
2m1v1
2 +1
2m2v2
2
1u 1v
2vx2m1m
b 12
� ~u1 ���� x ¶§-E¤3¡� x− y ²¡§K
m1u1 = m1v1 cos θ1 +m2v2 cos θ2
0 = m1v1 sin θ1 −m2v2 sin θ2
��ê'�-�õ§UC¤Oål§�UCÑ�� θ1, θ2
?�Ú§�Ä m1 = m2 = m§� u2 = 0:
~u1 = ~v1 + ~v2
u21 = v2
1 + v22
=⇒ ± ~u1 ��>���n�/§~v1 ⊥ ~v2
-E�üâf$Ä��R�§/¤XãÑ��.
ÿ� ~v1 =� ~v2
1v
2v
1u
散射圆
�±^Ñ��5©Û��þ£XP − P¤âf-E´Ä�5
入射质子
靶
detector
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
�%X¥�-E
oÄþ ~P = 01!�- (~u1, ~u2)→ (~v1, ~v2) → LabX¥.
~vc =m1~u1 +m2~u2
m1 +m2
~u1 = ~uc1 + ~vc, ~u2 = ~uc2 + ~vc
~v1 = ~vc1 + ~vc, ~v2 = ~vc2 + ~vc
=⇒ m1uc1 +m2uc2 = m1vc1 +m2vc2 = 0
vc2 − vc1 = e (uc1 − uc2)
=⇒{vc1 = −euc1vc2 = −euc2
�%X¥§-���Ý´Ù-c�Ý�−e�
∆T = Tc − T ′c =(1− e2
) 1
2
(m1u
2c1 +m2u
2c2
)=(1− e2
)T
e = 1 �§ÄUÅðe = 0 �§-�ÄU�Ü��§C�"
2!!!���---
m1~uc1 +m2~uc2 = m1~vc1 +m2~vc2 = 0
m1uc1 +m2uc2 = m1vc1 +m2vc2 = 0
vc2 − vc1 = uc1 − uc2 ���5
=⇒ vc1 = uc1, vc2 = uc2
2v2u
1v
1u
�%X¥§���5üÔN-��ÇØC§�k��UC
1v
2u
1u2m
1m
1m~~~4.5 ���üüü���AAA& ÿ ì m1 � 1 ( m2 �é � � � Ý © O � ~u1, ~u2,
m1 �� m2§m1 7 m2 �� ��Ñ ~v1§¦~v1.
)))µµµ ���5-E§e = 1
v1 =m1 − em2
m1 +m2u1 +
(1 + e)m2
m1 +m2u2 =
m1 −m2
m1 +m2u1 +
2m2
m1 +m2u2
' −u1 + 2u2
Alternatively§m1 ± |u1|+ |u2|��Ý7 m2 �7�±���Ç���Ñ=⇒ m1 �éu����Ç� |u1|+ |u2|+ |u2| = |u1|+ 2 |u2|.
?ë-E
M m m mO
u 3 2 1x
~~~4.6 ?ë-E§Ã�Þ�5§M > m§¦�w¬�ª�Ý
)))µµµ M − 3-Eµ
{Mu = Mu1 +mv11
2Mu2 =
1
2Mu2
1 +1
2mv2
1
=⇒
u1 =
M −mM +m
u
v1 =2M
M +mu
M±u1 < v1��ÝUYc?§3↔ 2 ���Ý=⇒ 2↔ 1 ���ݧm1±v1c?
M − 31�g-
Eµ
u2 =
(M −mM +m
)2
u
v2 =2M
M +mu1 =
2M
M +m
M −mM +m
u < v1
?ë-E (cont'd)
m3±v2c? → 3↔ 2 ���ݧm2±v2c?§á�um1§m3·�.
M − 31ng-Eµ
u3 =
(M −mM +m
)3
u
v3 =2M
M +m
(M −mM +m
)2
u < v2
m3±v3 < v2 < v1��Ýc?M±u3 < v3��Ýc?
N�m�§��Ý©O�µ
uN =
(M −mM +m
)N
u, vN =2M
M +m
(M −mM +m
)N−1
u
· · · · · ·
v2 =2M
M +m
M −mM +m
u, v1 =2M
M +mu
Outline
1 Å�U£Åð¤Å�U£Åð¤Ù¦/ª�³U
2 ÄU½nõ Work
�:ÄU½n�:XÄU½n
3 ³UÚå³U(k%å)
�Åå�³U³U���$Ä���5�
4 Å�UÅð½Æ�:XÅ�UÅð½Æ�ÅX��m�üØC5üN¯K
5 �%X��ZF½n�ZF½n�%X¥.5å¤��õ
6 -E�-�-�%X¥�-E
7 é¡5�Åð½Æé¡5�Åð½Æ
é¡5�Åð½Æ Symmetry and Conservation Laws
���ooo���ééé¡¡¡555ºººXÚ½5Æ3,«ö�eØC§KTXÚ½5ÆäkTé¡5
���mmm²²²£££ØØØCCC555µµµ�áNX�UþØ��mCz → UþÅð
���mmm²²²£££ØØØCCC555µµµ
B
'A As
B 'B
A
s
a b
�ÄA−Bm�^³V§K ∆V = ∆V ′
∆V = −~fB→A ·∆~s, ∆V ′ = −~fA→B · (−∆~s)
A′B = AB′
²£ØC5�¦ V + ∆V = V + ∆V ′
⇐⇒ ∆V = −~fB→A ·∆~s = ∆V ′ = ~fA→B ·∆~s∵ ∆~s ?¿§
=⇒ ~fB→A = −~fA→B
⇐⇒ ÄþÅð
![DENKEN : DIE GROSSEN FRAGEN DER PHILOSOPHIE...DESCARTES ZWEIFEL René Descartes (1596-1650) „Ich nehme […] an, alles, was ich wahrnehme, sei falsch; ich glaube, das nichts von](https://static.unterlagen.site/doc/80x56/609f810ce64fe74e775cd745/denken-die-grossen-fragen-der-philosophie-descartes-zweifel-ren-descartes.jpg)
