Mecànica Fonamental

Anuncio
ƌŽŶŽůŽŐşĂƚĞĐŶŽůſŐŝĐĂ
ϭϲϱϬ
WƌŽƚĂŐŽŶŝƐƚĂƐ
/ŶƐƚƌƵŵĞŶƚŽƐŵĞĐĄŶŝĐŽƐ͕
ŚŝĚƌĄƵůŝĐŽƐ͕ĂŐĂƐ͘͘͘
ƐŝŐŶĂƚƵƌĂƐ
ƌƋşŵĞĚĞƐ͕<ĞƉůĞƌ͕'ĂůŝůĞŽ͕,ŽŽŬĞ͕
EĞǁƚŽŶ͕ƵůĞƌ͕,ƵLJŐĞŶƐ͕Ě͛ůĞŵďĞƌƚ
DĞĐĄŶŝĐĂ
ƉƌŝŵĞƌŽƐƐŝƐƚĞŵĂƐĚĞǀĂƉŽƌ
ϭϳϱϬ
ϭϳϴϬͲϭϴϯϬ͗ϭǐƌĞǀŽůƵĐŝſŶ
ŝŶĚƵƐƚƌŝĂů ;ŵĄƋƵŝŶĂƐĚĞǀĂƉŽƌͿ
ϭϴϱϬ
ϭϵϱϬ
ĂƌŶŽƚ͕:ŽƵůĞ͕<ĞůǀŝŶ͕ůĂƵƐŝƵƐ͕
ŽůƚnjŵĂŶŶ͕,ĞůŵŚŽůƚnj͕'ŝďďƐ
dĞƌŵŽĚŝŶĄŵŝĐĂ
>ĂǀŽŝƐŝĞƌ͕'ĂLJͲ>ƵƐƐĂĐ͕ǀŽŐĂĚƌŽ͕ĂůƚŽŶ͕
ƌƌŚĞŶŝƵƐ͕DĞŶĚĞůĞĞǀ͕<ĞŬƵůĠ͕>ĞǁŝƐ
YƵşŵŝĐĂ
&ƌĂŶŬůŝŶ sŽůƚĂ KŚŵ <ŝƌĐŚŽĨĨ
&ƌĂŶŬůŝŶ͕sŽůƚĂ͕KŚŵ͕<ŝƌĐŚŽĨĨ͕
ůĞĐƚƌŽŵĂŐŶĞƚŝƐŵŽ
ů
ƚ
ƚŝ
ϭϴϳϬͲϭϵϬϬ͗ϮǐƌĞǀŽůƵĐŝſŶ
ϭϴϳϬ
ϭϵϬϬ Ϯǐ
ů ŝſ
ŵƉğƌĞ͕&ĂƌĂĚĂLJ͕DĂdžǁĞůů͕,Ğƌƚnj
ŝŶĚƵƐƚƌŝĂů ;ĞůĞĐƚƌŝĐŝĚĂĚн
ƋƵşŵŝĐĂŝŶĚƵƐƚƌŝĂů;ƉĞƚƌſůĞŽ͕ĨĄƌŵĂĐŽƐ͘͘͘ͿͿ
DĂƚĞƌŝĂůĞƐ
ϭϵϲϬͲϭϵϵϬ͗ϯǐƌĞǀŽůƵĐŝſŶ
ŝŶĚƵƐƚƌŝĂů ;ĞůĞĐƚƌſŶŝĐĂ;ƐŝůŝĐŝŽͿ͕
ŝŶĨŽƌŵĄƚŝĐĂ͕ŽƉƚŽĞůĞĐƚƌſŶŝĐĂͿ
MƉƚŝĐĂĂƉůŝĐĂĚĂ
,ŽLJLJŵĂŹĂŶĂ͗ŐĞŶĠƚŝĐĂ͕ŶĂŶŽƚĞĐŶŽůŽŐşĂ͕ĞŶĞƌŐşĂƐůŝŵƉŝĂƐ
dŝĞŵƉŽ
;ĂŹŽĚ͘͘Ϳ
ƉůŝĐĂĐŝŽŶĞƐĚĞůĂŵĞĐĄŶŝĐĂ͗ŵĄƋƵŝŶĂƐƐŝŵƉůĞƐ;ĂŵƉůŝĨŝĐĂŶůĂĨƵĞƌnjĂŚƵŵĂŶĂͿ͗
ƉůĂŶŽŝŶĐůŝŶĂĚŽ͕ŵĂƌƚŝůůŽ͕ƉŽůĞĂ͕ƉŽůŝƉĂƐƚŽ͕ƉĂůĂŶĐĂ;ŶŽƌŵĂůĞŚŝĚƌĄƵůŝĐĂͿ͕ƚŽƌŶŽ
LJĂĚĞŵĄƐ͗ƌƵĞĚĂƐ͕ƐƵƐƉĞŶƐŝŽŶĞƐ͕ĂŵŽƌƚŝŐƵĂĚŽƌĞƐ͕ƌĞůŽũĞƐĚĞƉĠŶĚƵůŽ͙
DĞĐăŶŝĐĂ&ŽŶĂŵĞŶƚĂů
ŚƚƚƉ͗ͬͬĂƚĞŶĞĂ͘ƵƉĐ͘ĞĚƵ
ƌŽďĞƌƚŽ͘ŵĂĐŽǀĞnjΛƵƉĐ͘ĞĚƵ ;ĚĞƐƉĂĐŚŽϭϭ͘ϰϱ͕ƉůĂŶƚĂϭϭͿ
ŚƚƚƉ͗ͬͬŐĐŵ͘ƵƉĐ͘ĞĚƵͬŵĞŵďĞƌƐͬƌŽďĞƌƚŽͲŵĂĐŽǀĞnj
DĞĐĄŶŝĐĂ͗
^dh/K>
DKs/D/EdK
z ^h^ h^^
z^h^h^^
^ŝƐƚĞŵĂƐƌşŐŝĚŽƐ
;ĐƵĞƌƉŽƐƐſůŝĚŽƐͿ
^ŝƐƚĞŵĂƐďůĂŶĚŽƐ
ŽĞůĄƐƚŝĐŽƐ;ůşƋƵŝĚŽƐ͕
Ͳ ŝŶĞŵĄƚŝĐĂ
Ͳ ƐƚĄƚŝĐĂLJŝŶĄŵŝĐĂ
Ͳ &ůƵŝĚŽĞƐƚĄƚŝĐĂ
Ͳ KƐĐŝůĂĐŝŽŶĞƐLJKŶĚĂƐ
& dp&
F=
dt
ŐĂƐĞƐ͕ŵƵĞůůĞƐͿ
NOTA = 0.6 EXfinal + 0.1MQ + 0.1EvC1 + 0.1EvC2 + 0.1LAB
ϰƉƌƵĞďĂƐĞƐĐƌŝƚĂƐ
ϯƐĞƐŝŽŶĞƐ
ϭǐƐĞƐŝſŶĚĞ>͗ŵŝƚĂĚĚĞƐĞƉƚŝĞŵďƌĞ͊>͗ƉůĂŶƚĂϲ
ŶƚƌĞŐĂƌƉŽƌƉĂƌĞũĂƐĞůƉƌŽďůĞŵĂϭ͘ϱ͘ϭ ;ĞƐĐƌŝƚŽĂŵĂŶŽͿ
>ĞĞƌƉĄƌƌĂĨŽƐϭ͘ϯLJϭ͘ϱĚĞůĂƐŶŽƚĂƐĚĞĐůĂƐĞ
DKs/D/EdKĚĞƵŶĐƵĞƌƉŽсdZE^>/MEнZKd/MEн&KZD/ME
Αϭ͘ϳŝŶĞŵăƚŝĐĂĚĞůĂƉĂƌƚşĐƵůĂ
hŶĂƉĂƌƚşĐƵůĂ;Ž͞ƉƵŶƚŽŵĂƚĞƌŝĂů͟ͿĞƐƵŶŽďũĞƚŽĚĞĚŝŵĞŶƐŝŽŶĞƐĚĞƐƉƌĞĐŝĂďůĞƐƌĞƐƉĞĐƚŽĂůĂƐ
ĚŝŵĞŶƐŝŽŶĞƐĚĞƐƵƚƌĂLJĞĐƚŽƌŝĂ;ĚŝƐƚĂŶĐŝĂƐLJƌĂĚŝŽĚĞĐƵƌǀĂƚƵƌĂͿ͕LJƋƵĞŶŽŐŝƌĂƐŽďƌĞƐŝŵŝƐŵĂ͘
ůǀĞĐƚŽƌƉŽƐŝĐŝſŶ ĚĞƵŶĂƉĂƌƚşĐƵůĂ;Ž͞ƉƵŶƚŽŵĂƚĞƌŝĂů͟ͿĞƐĞůǀĞĐƚŽƌ͗
k̂
&
&
2
2
2
ˆ
ˆ
ˆ
r = ( x, y , z ) = xi + yj + zk , de módulo r = r = x + y + z
&
r = ( x, y, z)
ŶĞůĞƐƉĂĐŝŽ͕ůĂƉĂƌƚşĐƵůĂƚŝĞŶĞƚƌĞƐ ŐƌĂĚŽƐĚĞůŝďĞƌƚĂĚ͕LJĂƋƵĞƐĞ
ĵ
ŶĞĐĞƐŝƚĂŶϯŶƷŵĞƌŽƐƉĂƌĂĞƐƉĞĐŝĨŝĐĂƌƐƵƉŽƐŝĐŝſŶ;ĞŶĞůƉůĂŶŽ͕ƐſůŽϮͿ
iˆ
>ĂƚƌĂLJĞĐƚŽƌŝĂƚĞŵƉŽƌĂůĚĞůĂƉĂƌƚşĐƵůĂĞƐůĂĐƵƌǀĂ
&
r (t ) = ( x (t ), y (t ), z (t ) ) ĚĞƐĐƌŝƚĂƉŽƌůĂƉĂƌƚşĐƵůĂ͘ůĚĞƐƉůĂnjĂŵŝĞŶƚŽ ĞŶƚƌĞĚŽƐ
& & &
ƉƵŶƚŽƐĚĞůĂƚƌĂLJĞĐƚŽƌŝĂĞƐĞůǀĞĐƚŽƌ Δr = r2 − r1 = ( x2 − x1 , y2 − y1 , z2 − z1 )
&
^ŝůŽƐĚŽƐƉƵŶƚŽƐĞƐƚĄŶŝŶĨŝŶŝƚĂŵĞŶƚĞƉƌſdžŝŵŽƐ͗ dr = ( dx, dy , dz )
&
&
&
&
Δr
d r & § dx dy dz ·
ůǀĞĐƚŽƌǀĞůŽĐŝĚĂĚŝŶƐƚĂŶƚĄŶĞĂĚĞůĂƉĂƌƚşĐƵůĂ
v (t )
=
=r =¨
,
,
v = lim Δ t → 0
¸
Δt
dt
ĞƐůĂĚĞƌŝǀĂĚĂǀĞĐƚŽƌŝĂůĚĞůĂƉŽƐŝĐŝſŶ͗
© dt dt dt ¹
&
v (t ) ůǀĞĐƚŽƌǀĞůŽĐŝĚĂĚĞƐƚĂŶŐĞŶƚĞĂůĂƚƌĂLJĞĐƚŽƌŝĂĞŶĐĂĚĂƉƵŶƚŽ
&
;ƐĞǀĞŐƌĄĨŝĐĂŵĞŶƚĞƚŽŵĂŶĚŽĞůůşŵŝƚĞͿ
r (t )
&
&
& & dv d 2 r & § d 2 x d 2 y d 2 z ·
=
=r =¨ 2 , 2 , 2 ¸
>ĂĂĐĞůĞƌĂĐŝſŶĚĞůĂƉĂƌƚşĐƵůĂĞƐ͗ a = v =
dt dt 2
© dt dt dt ¹
ƐƚĂƐĚĞĨŝŶŝĐŝŽŶĞƐƐĞƉƵĞĚĞŶŝŶǀĞƌƚŝƌƉĂƌĂƐĂĐĂƌ;ƉŽƌĞũĞŵƉůŽͿůĂƉŽƐŝĐŝſŶĚĞůĂǀĞůŽĐŝĚĂĚ͗
&
&
&
r (t ) = ³ v (t )dt + C = ³ vx (t )dt + C x , ³ v y (t )dt + C y , ³ v z (t )dt + C z
&
&
WͲϭ͘ϯ͘ϭ;ƐŝŶƉƵŶƚŽ;ĚͿͿ͕WͲϭ͘ϳ͘Ϯ
a (t ) = (0,0,− g ) v (t = 0) = (2,0,1)
WͲϭ͘ϳ͘ϯнůŽŵŝƐŵŽĐŽŶLJ
(
)
&
v (t ) = v(t )vˆ(t ) = vvˆ
¯vˆ = vector tangente unitario
­
&
ůǀĞĐƚŽƌǀĞůŽĐŝĚĂĚĞƐƐŝĞŵƉƌĞƚĂŶŐĞŶƚĞĂůĂƚƌĂLJĞĐƚŽƌŝß
v (t )
®
&
&
dv d (vˆ v )
dv
dvˆ
ˆ
=
= vˆ + v
>ĂĂĐĞůĞƌĂĐŝſŶĞƐ a (t ) =
͘zĂƋƵĞ dv ⊥ vˆ
dt
dt
dt
dt
dt
aT vˆ
&
a
2
d (vˆ ⋅ vˆ)
dvˆ d ( vˆ )
n̂
;ĞƐƚŽƐŝŐƵĞĚĞͿ͕ůůĂŵĂŶĚŽĞůǀĞĐƚŽƌ
= 2vˆ ⋅ =
=0
dt
dt
dt
&
& &
dvˆ
dv
a (t ) = vˆ + v nˆ = aT + a N
ĚĞŵſĚƵůŽϭŽƌƚŽŐŽŶĂůß
v̂
dt
dt
&
&
r (t )
aN nˆ
Ÿ a ƚŝĞŶĞƵŶĂĐŽŵƉŽŶĞŶƚĞ
ǀĂƌŝĂĐŝſŶĚĞů ǀĂƌŝĂĐŝſŶĚĞ &
&
ƚĂŶŐĞŶƚĞĂůĂƚƌĂLJĞĐƚŽƌŝĂLJŽƚƌĂŶŽƌŵĂů
ŵſĚƵůŽĚĞ v ĚŝƌĞĐĐŝſŶĚĞ v
vˆ(t )
ĐĞůĞƌĂĐŝſŶŶŽƌŵĂůLJƌĂĚŝŽĚĞĐƵƌǀĂƚƵƌĂ͗
&
&
dθ
WĂƌĂƵŶĚĞƐƉůĂnjĂŵŝĞŶƚŽŝŶĨŝŶŝƚĠƐŝŵŽ͕ůĂĐƵƌǀĂƐĞĐŽŶĨƵŶĚĞ
dr
&
r (t )
d
r
ĐŽŶĞůƐĞŐŵĞŶƚŽƌĞĐƚŝůşŶĞŽLJĐŽŶĞůĂƌĐŽĚĞĐşƌĐƵůŽƚĂŶŐĞŶƚĞ͗
vˆ(t + dt )
&
d
θ
dr = d" = RCURV dθ ͕ĚŽŶĚĞĞƐĞů͞ƌĂĚŝŽĚĞĐƵƌǀĂƚƵƌĂ͟
RCURV
&
RCURV
dr
dvˆ
vˆ = 1
^ĞǀĞŐƌĄĨŝĐĂŵĞŶƚĞƋƵĞ͘ĂĚŽƋƵĞ͕
= dθ =
RCURV
& vˆ
dvˆ dθ
1 dr
v
dvˆ
&
v2
=
=
=
RCURV = v
Ÿ
͘WŽƌůŽƚĂŶƚŽLJ
a N = nˆ
dt
dt RCURV dt RCURV
dt
RC
& &
& & &
v v
a = aT + a N
ZĞƐƵŵĞŶĨſƌŵƵůĂƐ͗ vˆ(t ) = = &
v v
;ƚŽĚĂƐƐĞĐĂůĐƵůĂŶ
&
dvˆ
v2
&
dv
dvˆ dvˆ
&
ˆ
ˆ
=
=
a
n
v
n
ˆ
a
v
=
N
T
ˆ
n(t ) =
v (t )
ĂƉĂƌƚŝƌĚĞͿ
dt
RC
dt
dt dt
dvˆ
dvˆ = vˆ dθ = dθ
RC = v
dvˆ
dt
ĂƐŽƐŝŵƉŽƌƚĂŶƚĞƐ͗ŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůşŶĞŽƵŶŝĨŽƌŵĞ͕ƵŶŝĨŽƌŵĞŵĞŶƚĞĂĐĞůĞƌĂĚŽ ŽƉĞƌŝſĚŝĐŽ͕
ŵŽǀŝŵŝĞŶƚŽĐŝƌĐƵůĂƌƵŶŝĨŽƌŵĞ
WͲϭ͘ϳ͘ϰ;ƐſůŽĐŝƌĐƵůĂƌƵŶŝĨŽƌŵĞͿ͕WͲϭ͘ϳ͘ϱ
dĞŵĂϮ͗DĞĐĄŶŝĐĂĚĞƵŶĂƉĂƌƚşĐƵůĂ >ĂƐůĞLJĞƐĚĞĞƐƚĞƚĞŵĂƐŽŶǀĄůŝĚĂƐƉĂƌĂƉĂƌƚşĐƵůĂƐ͘DĄƐ
ĂĚĞůĂŶƚĞ;§ϯ͘ϮͿǀĞƌĞŵŽƐƋƵĞƚĂŵďŝĠŶǀĂůĞŶƉĂƌĂĚĞƐĐƌŝďŝƌĞůŵŽǀŝŵŝĞŶƚŽĚĞƚƌĂŶƐůĂĐŝſŶ ĚĞƵŶ
ĐƵĞƌƉŽƌşŐŝĚŽƐŽŵĞƚŝĚŽĂĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐ͘ŶĐŽŶĐƌĞƚŽ͕ůĂĂĐĞůĞƌĂĐŝſŶĚĞůĐĞŶƚƌŽĚĞŵĂƐĂƐĚĞů
ĐƵĞƌƉŽĞƐĚĂĚĂƉŽƌůĂϮǐ>ĞLJĚĞEĞǁƚŽŶ;ƚŽŵĂŶĚŽƐſůŽůĂƐĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐͿ͘
ΑϮ͘ϭLJϮ͘Ϯ>ĞLJĞƐĚĞEĞǁƚŽŶ͕ĨƵĞƌnjĂ͕ŵĂƐĂ͕ŵŽŵĞŶƚŽůŝŶĞĂů
ϭǐ>ĞLJĚĞEĞǁƚŽŶ͗ ƵŶĐƵĞƌƉŽƐŽďƌĞƋƵĞŶŽĂĐƚƷĂŶŝŶŐƵŶĂ͞ĐĂƵƐĂ͟;ĨƵĞƌnjĂͿ͕ƐĞŵƵĞǀĞĐŽŶ
&
ǀĞůŽĐŝĚĂĚĐŽŶƐƚĂŶƚĞ;ŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůşŶĞŽƵŶŝĨŽƌŵĞͿ;^dd/Æ F = 0 Ϳ͘
Ϯǐ>ĞLJĚĞEĞǁƚŽŶ͗ůĂĨƵĞƌnjĂƋƵĞĂĐƚƷĂƐŽďƌĞƵŶĂƉĂƌƚşĐƵůĂĞƐƉƌŽƉŽƌĐŝŽŶĂůĂůĂ
&
ĂĐĞůĞƌĂĐŝſŶĚĞůĂŵŝƐŵĂ͖ůĂĐŽŶƐƚĂŶƚĞĚĞƉƌŽƉŽƌĐŝŽŶĂůŝĚĂĚĞƐůĂŵĂƐĂĚĞ
& &.
F = ma = p
ůĂƉĂƌƚşĐƵůĂ͕ŽƐĞĂůĂĐĂŶƚŝĚĂĚĚĞŵĂƚĞƌŝĂ;ÅÆƌĞůĂĐŝŽŶĂĚĂĐŽŶƐƵƉĞƐŽͿ͘
&
&
ůǀĞĐƚŽƌƐĞůůĂŵĂŵŽŵĞŶƚŽůŝŶĞĂůŽĐĂŶƚŝĚĂĚĚĞŵŽǀŝŵŝĞŶƚŽ
p = mv
>ĂϮǐ>ĞLJŝŵƉůŝĐĂůĂϭǐ͘ŵďĂƐƐŽŶǀĄůŝĚĂƐƐſůŽĞŶƐŝƐƚĞŵĂƐĚĞƌĞĨĞƌĞŶĐŝĂŝŶĞƌĐŝĂůĞƐ
&
;ĞƐĚĞĐŝƌ͕ĨŝũŽƐŽĞŶŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůşŶĞŽƵŶŝĨŽƌŵĞͿ
&
F1
F
ŽŵŽůĂƐĂĐĞůĞƌĂĐŝŽŶĞƐ͕ƋƵĞƐŽŶǀĞĐƚŽƌĞƐ͕
ůĂƐĨƵĞƌnjĂƐƚĂŵďŝĠŶƐĞƐƵŵĂŶĐŽŵŽǀĞĐƚŽƌĞƐ͗
&
m
>ĂƵŶŝĚĂĚĚĞĨƵĞƌnjĂĞŶĞů^/ĞƐĞůŶĞǁƚŽŶ;EͿ͗ 1N = 1kg 2
F2
s
ĞĨŝŶĞůĂĨƵĞƌnjĂ͗ ůĂĨƵĞƌnjĂĞƐůĂĐĂƵƐĂĚĞůĂĂĐĞůĞƌĂĐŝſŶĚĞůĂƉĂƌƚşĐƵůĂ͖ƉƵĞĚĞƐĞƌ
&
& ĨƵŶĐŝſŶĚĞůĂƉŽƐŝĐŝſŶLJĚĞůĂǀĞůŽĐŝĚĂĚ;LJĚĞůƚŝĞŵƉŽͿ͕ƉĞƌŽŶŽ ĚĞůĂĂĐĞůĞƌĂĐŝſŶ͘
& &
F = ma ĞĨŝŶĞůĂŵĂƐĂ͗ ƐŝƐŽďƌĞůĂŵŝƐŵĂƉĂƌƚşĐƵůĂĂĐƚƷĂŶĨƵĞƌnjĂƐĚŝĨĞƌĞŶƚĞƐ͕ƋƵĞ
F
1 , F2 ...
& &
ĐĂƵƐĂŶĂĐĐĞůĞƌĂĐŝŽŶĞƐ͕ĞůĐŽĐŝĞŶƚĞĚĞůŽƐŵſĚƵůŽƐĞƐĐŽŶƐƚĂŶƚĞ͕ŝŐƵĂůĂůĂ
a1 , a2 ...
F1 F2
ŵĂƐĂ;ŝŶĞƌĐŝĂůͿ ĚĞůĂƉĂƌƚşĐƵůĂ͗
m=
a1
=
a2
= ...
&
&
& t2 &
F
^ĞĚĞĨŝŶĞĞůŝŵƉƵůƐŽƐƵŵŝŶŝƐƚƌĂĚŽƉŽƌƵŶĂĨƵĞƌnjĂĞŶĞůŝŶƚĞƌǀĂůŽĐŽŵŽ
I
t1 t2
I = Fdt
& & t1
ĞůĂϮǐ>ĞLJĚĞEĞǁƚŽŶƐŝŐƵĞŝŶŵĞĚŝĂƚĂŵĞŶƚĞĞů͗ dĞŽƌĞŵĂĚĞůŵŽŵĞŶƚŽůŝŶĞĂů͗ I =Δp
³
ΑϮ͘ϱƉůŝĐĂĐŝſŶĚŝƌĞĐƚĂĚĞůĂϮǐ>ĞLJĚĞEĞǁƚŽŶ
WͲϮ͘ϭ͘Ϯ͕WͲϮ͘ϭ͘ϯ͕WͲϮ͘Ϯ͘ϭ͕WͲϮ͘ϱ͘ϱ
WͲϮ͘Ϯ͘Ϯ͕
WͲϮ͘Ϯ͘ϯ͕WͲϰ͘ϭ͘ϭ
>ĂϮĂůĞLJĚĞEĞǁƚŽŶŶŽƐĚŝĐĞĐŽŵŽƐĞŵƵĞǀĞůĂƉĂƌƚşĐƵůĂ͗ ĞƐƵŶĂĞĐƵĂĐŝſŶĚŝĨĞƌĞŶĐŝĂůƋƵĞĞŶ
&
ĂůŐƵŶŽƐĐĂƐŽƐƉƵĞĚĞƐĞƌŝŶƚĞŐƌĂĚĂ;ĚŽƐǀĞĐĞƐͿƉĂƌĂĞŶĐŽŶƚƌĂƌůĂƚƌĂLJĞĐƚŽƌŝĂƚĞŵƉŽƌĂů͘
r (t )
& &
C1 C2
ŶĐĂĚĂŝŶƚĞŐƌĂĐŝſŶƐĞŝŶƚƌŽĚƵĐĞƵŶĂĐŽŶƐƚĂŶƚĞ;ǀĞĐƚŽƌŝĂůͿ͗LJ͘^ŝĐŽŶŽĐĞŵŽƐƉŽƐŝĐŝſŶLJ
& &
& &
& & &
&
r0 = r (t0 ) v0 = v (t0 )
ǀĞůŽĐŝĚĂĚĞŶƵŶŝŶƐƚĂŶƚĞŝŶŝĐŝĂůƚĞŶĚƌĞŵŽƐ͗
­ r0 = f ( C 1 , C 2 , t 0 )
t0
& &
& &
°
&
®&
∂f & &
(C1 , C 2 , t0 )
° v0 =
∂t
¯
&
& & &
r0 , v0
^ŝĐŽŶŽĐĞŵŽƐůĂĨƵŶĐŝſŶĨ͕ƐĞĐĂůĐƵůĂŶĂƉĂƌƚŝƌĚĞ͕ƉĂƌĂĞŶĐŽŶƚƌĂƌ͗
ů Ĩ
ſ Ĩ C1 , C2
ů ů&
r (t ) = r (r0 , v0 , t )
&
ũ͗͘ĨƵĞƌnjĂƋƵĞƐſůŽĚĞƉĞŶĚĞĚĞůƚŝĞŵƉŽ F = F (t )
&
§
·
d 2r &
1 ¨ &
&
¸ dt
m
F
(
t
)
r
(
t
)
F
(
t
)
dt
=
Ÿ
=
Æ /ŶƚĞŐƌĂĐŝſŶĚŝƌĞĐƚĂÆ
t
t
¸
dt 2
m ¨ &
©
¹
&
&
& F
m a = F Ÿ a = = ct
ĂƐŽƉĂƌƚŝĐƵůĂƌ͗ĨƵĞƌnjĂĐŽŶƐƚĂŶƚĞ͘ŶĞƐƚĞĐĂƐŽůĂϮĂ>ĞLJĚĞEĞǁƚŽŶĞƐ͗
&
m
/ŶƚĞŐƌĂŶĚŽϮǀĞĐĞƐ͗ r& (t ) = C& + C& t + 1 F t 2
1
2
2m
&
& &
&
r (t0 ) = r0 , r (t0 ) = v0
ŽŶůĂƐĐŽŶĚŝĐŝŽŶĞƐŝŶŝĐŝĂůĞƐ͕ƐĞŚĂƉƵĞƐ͗
³³
&
&
1 F 2½ ­ & & F
&
& & &
r (t0 ) = r0 = C1 + C2 t0 +
t0 ° °C2 = v0 − t0
m
2m ° °
&
&
¾Ÿ®
& F
& & &
1F 2
&
&
°
°
r (t0 ) = v0 = C2 + t0
C = r −v t +
t
°¿ °¯ 1 0 0 0 2 m 0
m
>ĂĞĐƵĂĐŝſŶĚĞůĂƚƌĂLJĞĐƚŽƌŝĂĞƐƉƵĞƐ͗
&
&
& &
1F
r (t ) = r0 + v0 (t − t0 ) +
(t − t0 ) 2
2m
ĂƐŽƐƉĂƌƚŝĐƵůĂƌĞƐĚĞĨƵĞƌnjĂƐĐŽŶƐƚĂŶƚĞƐ͗
ƉĞƐŽLJ ĨƌŝĐĐŝſŶĚŝŶĄŵŝĐĂ
&
&
&
&
&
&ƵĞƌnjĂƉĞƐŽ͗ m a = F = mg Ÿ a =
y
F &
= g = ct
m
m
y
^ŝƚŽŵĂŵŽƐƵŶƐŝƐƚĞŵĂĚĞƌĞĨĞƌĞŶĐŝĂĐŽŶĞůĞũĞƉĂƌĂůĞůŽĂůĂĨƵĞƌnjĂLJ
ƚĂůƋƵĞůĂǀĞůŽĐŝĚĂĚŝŶŝĐŝĂůĞƐƚĠĞŶĞůƉůĂŶŽ
( x, y ) ͕ůĂƚƌĂLJĞĐƚŽƌŝĂ
&
&
& &
1F
r (t ) = r0 + v0 (t − t0 ) +
(t − t0 ) 2 ĞƐƵŶĂƉĂƌĄďŽůĂĞŶƚĂůƉůĂŶŽ͘
2m
&
&
v
&
F
&
ZŽnjĂŵŝĞŶƚŽĚŝŶĄŵŝĐŽ͗ FRD = μ D N , FRD = − μ D Nvˆ
v0
&
&
F
v̂
FRD
^ŝсĐŽŶƐƚ͕сĐŽŶƐƚ͘^ŝĂĚĞŵĄƐсĐŽŶƐƚ͕сĐŽŶƐƚ
N
RD
x
v(t )
>ĂůĞLJŚŽƌĂƌŝĂ;ĚĞĐĞůĞƌĂĐŝſŶƵŶŝĨŽƌŵĞͿǀĂůĞƐſůŽŚĂƐƚĂƋƵĞůĂƉĂƌƚşĐƵůĂ
ƐĞƉĂƌĂ͕ĞƐĚĞĐŝƌŚĂƐƚĂƋƵĞǀ сϬ͕ƉŽƌƋƵĞĞŶƚŽŶĐĞƐůĂĨƌŝĐĐŝſŶĚĞƐĂƉĂƌĞĐĞ
t
ϭͿĂůĐƵůĂƌĂƋƵĞ ĄŶŐƵůŽƌĞƐƉĞĐƚŽĚĞůƐƵĞůŽŚĂLJƋƵĞůĂŶnjĂƌƵŶŽďũĞƚŽƉĂƌĂƋƵĞůůĞŐƵĞůŽŵĄƐůĞũŽƐƉŽƐŝďůĞ͕LJĐĂůĐƵůĂƌůĂǀĞůŽĐŝĚĂĚĚĞůŽďũĞƚŽ
v0 = 3 ms −1
ũƵƐƚŽĂŶƚĞƐƋƵĞƚŽƋƵĞĞůƐƵĞůŽ͖ϮͿ,ĂůůĂƌůĂĚŝŶĄŵŝĐĂĚĞƵŶĂŵĂƐĂƐƵũĞƚĂĂĨƌŝĐĐŝſŶĚŝŶĄŵŝĐĂ;ͿƋƵĞƐĂůĞĚĞůŽƌŝŐĞŶĐŽŶ
μ D = 0.1
KƚƌŽĐĂƐŽĚĞĨƵĞƌnjĂŝŶƚĞŐƌĂďůĞ͗ ĨƵĞƌnjĂĚĞƵŶŵƵĞůůĞ;ŵŽǀŝŵŝĞŶƚŽĂƌŵſŶŝĐŽͿ
x<0
&ƵĞƌnjĂĚĞƵŶŵƵĞůůĞ;ŽĚĞ,ŽŽŬĞͿ͗ FH = − k (l − l N )
x = l − lN
ĞĨŝŶŝĞŶĚŽƵŶŶƵĞǀŽƐŝƐƚĞŵĂĚĞƌĞĨĞƌĞŶĐŝĂĞŶƋƵĞ͕ƐĞŚĂ͗
k
x = − x
F = ml = mx = −kx ͕ŽƐĞĂ͕͗ĐƵLJĂƐŽůƵĐŝſŶŐĞŶĞƌĂůĞƐ͗
m
m
FH = 0
x(t ) = A sin(ω (t − t0 ) + ϕ0 ) ͕ĐŽŶ͗
ω=
k
Ÿ
m
x(t0 )
v(t0 ) 2
2E
2
sin
ϕ
=
=
+
=
A
x
(
t
)
0
͕
͕
0
2
A
ω
k
T = 2π ω
WƌĄĐƚŝĐĂϭĚĞůĂďŽƌĂƚŽƌŝŽ
FH > 0
x=0
m
x>0
FH < 0
WͲϮ͘ϱ͘ϯ
m
KƚƌĂƐĞƐƚƌĂƚĞŐŝĂƐƉĂƌĂĚĞƚĞƌŵŝŶĂƌĞůŵŽǀŝŵŝĞŶƚŽ͗ŵŽŵĞŶƚŽĂŶŐƵůĂƌLJĞŶĞƌŐşĂ
ΑϮ͘ϯDŽŵĞŶƚŽĚĞƵŶĂĨƵĞƌnjĂLJŵŽŵĞŶƚŽĂŶŐƵůĂƌ
&
&
^ĞĚĞĨŝŶĞĞůŵŽŵĞŶƚŽĚĞƵŶĂĨƵĞƌnjĂƌĞƐƉĞĐƚŽ
M ( P)
F
&
&
ĂƵŶƉƵŶƚŽ W ĐŽŵŽ͗
&
&
M (P)
M ( P ) = r( P ) × F
&
& &
P
r( P ) = r − rP ĞƐĞůǀĞĐƚŽƌƉŽƐŝĐŝſŶŵĞĚŝĚŽĚĞƐĚĞĞůƉƵŶƚŽĚĞ
ĂƉůŝĐĂĐŝſŶW ĚĞůĂĨƵĞƌnjĂ͘
&
P
^ĞĚĞĨŝŶĞĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌĚĞƵŶĂƉĂƌƚşĐƵůĂƌĞƐƉĞĐƚŽĂ
L( P )
&
&
&
ĐŽŵŽ͗ L( P ) = r( P ) × p
& dp&
ƉĂƌƚŝƌĚĞůĂ>ĞLJĚĞEĞǁƚŽŶ͕ŵƵůƚŝƉůŝĐĂŵŽƐĂŵďŽƐ
& F = dt
ƚĠƌŵŝŶŽƐ Ă ůĂ ŝnjƋƵŝĞƌĚĂ ƉŽƌ r( P ) ×
ƚĠƌŵŝŶŽƐĂůĂŝnjƋƵŝĞƌĚĂƉŽƌ
&
&
^ŝWĞƐƵŶƉƵŶƚŽĨŝũŽĚĞůƐŝƐƚĞŵĂĚĞƌĞĨĞƌĞŶĐŝĂ;ŝŶĞƌĐŝĂůͿ͕
r( P ) = v
&
F
&
F
&
r( P )
&
L( P)
P
&
p
&
&
p
r( P)
&
&
&
dp d &
& & &
&
= (r( P ) × p ) ͕LJĂƋƵĞ v × p = v × mv = 0 ͘ĞĞƐƚŽƐĞŽďƚŝĞŶĞƋƵĞ͗
ŶƚŽŶĐĞƐ r( P ) ×
dt dt
&
WͲϮ͘ϯ͘ϭ
^ŝĞŶƉĂƌƚŝĐƵůĂƌĞůŵŽŵĞŶƚŽĚĞůĂĨƵĞƌnjĂƋƵĞĂĐƚƷĂƐŽďƌĞůĂ
&
dL( P )
= M ( P ) ƉĂƌƚşĐƵůĂĞƐĐĞƌŽ͕ĞŶƚŽŶĐĞƐĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌƐĞŵĂŶƚŝĞŶĞ
WͲϮ͘ϯ͘ϰ
dt
ĐŽŶƐƚĂŶƚĞ;ĐƵŝĚĂĚŽ͗ƌĞƐƉĞĐƚŽĂůŵŝƐŵŽƉƵŶƚŽW͊͊Ϳ
WƌŽĚƵĐƚŽ
ǀĞĐƚŽƌŝĂů͗
iˆ
& &
A × B ≡ Ax
Bx
ˆj
Ay
By
kˆ
Az = ( Ay Bz − Az By , Az Bx − Ax Bz , Ax By − Ay Bx )
Bz
&
&
Y(P)
ĞĨŝŶŝĞŶĚŽĞůŝŵƉƵůƐŽĂŶŐƵůĂƌƐƵŵŝŶŝƐƚƌĂĚŽƉŽƌĞůŵŽŵĞŶƚŽĚĞƵŶĂĨƵĞƌnjĂĞŶĞů
M (P)
t1 t2
ŝŶƚĞƌǀĂůŽĐŽŵŽ͗
t2
&
&
&
&
Y( P ) = M ( P ) dt ͕ƚĞŶĞŵŽƐĞů dĞŽƌĞŵĂĚĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌ͗ Y(P) =ΔL(P)
³
t1
ůŵŽŵĞŶƚŽĂŶŐƵůĂƌƐĞŵĂŶƚŝĞŶĞĐŽŶƐƚĂŶƚĞĞŶƉĂƌƚŝĐƵůĂƌĞŶƚƌĞƐĐĂƐŽƐŝŵƉŽƌƚĂŶƚĞƐ͗
&
&
ϭͿ ^ŝ;ŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůŝŶĞŽƵŶŝĨŽƌŵĞͿ͕
L
F =0
( P ) = const ∀P
&
ϮͿ ŶƵŶŵŽǀŝŵŝĞŶƚŽĐŝƌĐƵůĂƌƵŶŝĨŽƌŵĞ͕͕ƐŝĞŶĚŽK
ĞůĐĞŶƚƌŽĚĞůĐşƌĐƵůŽ
L( O ) = const &
L
=
const
ϯͿ ŶĐĂƐŽĚĞĨƵĞƌnjĂĐĞŶƚƌĂůĚŝƌŝŐŝĚĂŚĂĐŝĂƵŶƉƵŶƚŽ͕ ( C )
&ƵĞƌnjĂƐĐĞŶƚƌĂůĞƐ
hŶĂĨƵĞƌnjĂĐĞŶƚƌĂůĞƐƵŶĂĨƵĞƌnjĂĐƵLJĂƌĞĐƚĂĚĞĂĐĐŝſŶƐŝĞŵƉƌĞƉĂƐĂ
ƉŽƌ ƵŶ ŵŝƐŵŽ ƉƵŶƚŽ ͘ ũĞŵƉůŽƐ͗ ůĂ ĨƵĞƌnjĂ ŐƌĂǀŝƚĂƚŽƌŝĂ ĚĞů ^Žů ƐŽďƌĞ
ƉŽƌƵŶŵŝƐŵŽƉƵŶƚŽ͘ũĞŵƉůŽƐ͗ůĂĨƵĞƌnjĂŐƌĂǀŝƚĂƚŽƌŝĂĚĞů^ŽůƐŽďƌĞ
ƵŶƉůĂŶĞƚĂ͖ůĂĨƵĞƌnjĂĞůĞĐƚƌŽƐƚĄƚŝĐĂĚĞƵŶĂĐĂƌŐĂĨŝũĂƐŽďƌĞŽƚƌĂ
ƉĂƌƚşĐƵůĂĐĂƌŐĂĚĂ͖ůĂƚĞŶƐŝſŶĚĞƵŶĂĐƵĞƌĚĂĨŝũĂĚĂĞŶƵŶĞdžƚƌĞŵŽ͘
&
mm
FGU = −G 1 2 2 rˆ
r
&
FC
C
WͲϮ͘ϯ͘ϰ
ŶĞůĐĂƐŽĚĞĨƵĞƌnjĂĐĞŶƚƌĂů͕ĞůŵŽŵĞŶƚŽĚĞůĂĨƵĞƌnjĂƌĞƐƉĞĐƚŽĂůƉƵŶƚŽ
&
ƐƐŝĞŵƉƌĞĐĞƌŽ͕LJƉŽƌƚĂŶƚŽ dL&
&
dL
(C )
= M (C ) Ÿ (C ) = 0
dt
dt
^ĞƉƵĞĚĞĚĞŵŽŶƐƚƌĂƌƋƵĞƚŽĚĂĨƵĞƌnjĂĐĞŶƚƌĂůĐƵŵƉůĞůĂ͗
&
Ϯǐ>ĞLJĚĞ<ĞƉůĞƌ;ĐŽŵŽĐŽŶƐĞĐƵĞŶĐŝĂĚĞůĂĐŽŶƐĞƌǀĂĐŝſŶĚĞͿ
L(C )
Æ sĞƌĞϮ͘ϯ͘Ϯ
&
TB
C
&
T
WͲϮ͘ϯ͘ϯ
WͲϮ͘ϯ͘ϱ
&
mm
>ĞLJĞƐĚĞ<ĞƉůĞƌ ;ƐĞĚĞŵƵĞƐƚƌĂŶĂƉĂƌƚŝƌĚĞůĂϮǐ>ĞLJĚĞEĞǁƚŽŶĐŽŶͿ
FGU = −G 1 2 2 rˆ
r
WƌŝŵĞƌĂ>ĞLJ
b
>ŽƐƉůĂŶĞƚĂƐĚĞƐĐƌŝďĞŶŽƌďŝƚĂƐĞůşƉƚŝĐĂƐĂůƌĞĚĞĚŽƌĚĞůƐŽů͕
ĐŽŶĞůƐŽůŽĐƵƉĂŶĚŽƵŶŽĚĞůŽƐĨŽĐŽƐĚĞůĂĞůŝƉƐĞ
a
^ĞŐƵŶĚĂ>ĞLJ
>ĂǀĞůŽĐŝĚĂĚĚĞƵŶƉůĂŶĞƚĂǀĂƌşĂĞŶĞůƚŝĞŵƉŽ͕ĚĞĨŽƌŵĂƋƵĞĞůǀĞĐƚŽƌƋƵĞƵŶĞĞůƐŽůĂůƉůĂŶĞƚĂ
&
ĐƵďƌĞĄƌĞĂƐŝŐƵĂůĞƐĞŶƚŝĞŵƉŽƐŝŐƵĂůĞƐ͘ƐƚĂůĞLJĞƐƵŶĂĐŽŶƐĞĐƵĞŶĐŝĂĚŝƌĞĐƚĂĚĞ L( C ) = const
ĞͲϮ͘ϯ͘Ϯ
dĞƌĐĞƌĂ>ĞLJ
ůĐƵĂĚƌĂĚŽĚĞůƉĞƌŝŽĚŽĚĞ
ƌĞǀŽůƵĐŝſŶĚĞƵŶƉůĂŶĞƚĂĞƐ
ĚŝƌĞĐƚĂŵĞŶƚĞƉƌŽƉŽƌĐŝŽŶĂůĂ
ůĂůŽŶŐŝƚƵĚĚĞůƐĞŵŝĞũĞŵĂLJŽƌ
ĂůĐƵďŽ͗ T 2 ∝ a 3
KƚƌĂ͞ĂƉůŝĐĂĐŝſŶ͟ĞŶůĂƋƵĞƚĂŶ
ƐſůŽƐĞĐŽŶƐĞƌǀĂůĂĚŝƌĞĐĐŝſŶ
ĚĞ
&
L ͗ƉĠŶĚƵůŽĚĞ&ŽƵĐĂƵůƚÆ
(C )
&
Fi
ΑϮ͘ϰdƌĂďĂũŽ͕ƉŽƚĞŶĐŝĂ͕ĞŶĞƌŐşĂƉŽƚĞŶĐŝĂů͕ĞŶĞƌŐşĂĐŝŶĞƚŝĐĂ
&
Wi ĚĞƵŶĂĨƵĞƌnjĂĂůŽůĂƌŐŽĚĞƵŶĂĐƵƌǀĂΓ
F
ůƚƌĂďĂũŽ
ĚĞƵŶ
i
&
&
&
r
r1
r2
ƉƵŶƚŽĂŽƚƌŽĞƐĞůŝŶƚĞŐƌĂů͗
& &
Wi = ³ Fi ⋅ dr
&
r1
2
&
Γ: r1
&
&
r2
&
dr
Γ
&
^ŝůĂĐƵƌǀĂΓ ĞƐůĂƚƌĂLJĞĐƚŽƌŝĂĚĞƵŶĂƉĂƌƚşĐƵůĂ͕͕LJĞůƚƌĂďĂũŽƉƵĞĚĞĞƐĐƌŝďŝƌƐĞĐŽŵŽ͗
dr = v dt
Wi
&
& &
F
℘i = Fi ⋅ v
;ŝŶƐƚĂŶƚĄŶĞĂͿĚĞƐĂƌƌŽůůĂĚĂƉŽƌůĂĨƵĞƌnjĂ i
Wi = ³℘i dt ͕ƐŝĞŶĚŽůĂƉŽƚĞŶĐŝĂ
t2
WͲϮ͘ϰ͘Ϯ
>ĂƵŶŝĚĂĚĚĞƚƌĂďĂũŽĞŶĞů^/ĠƐĞůũƵůŝŽ͗ J = N m ͖ůĂƵŶŝĚĂĚĚĞƉŽƚĞŶĐŝĂĞƐĞůǀĂƚŝŽ;tͿ͗ 1W = 1 J
͎ĐŽŵŽƐĞĐĂůĐƵůĂĞůƚƌĂďĂũŽ͍ůĂƚƌĂLJĞĐƚŽƌŝĂĞƐƵŶĂĐƵƌǀĂϭLJƐĞƉƵĞĚĞĚĞƐĐƌŝďŝƌĂƚƌĂǀĠƐĚĞƵŶ s
&
ƷŶŝĐŽƉĂƌĄŵĞƚƌŽƌĞĂů;Ɖ͘Ğũ͕͘ƵŶĄŶŐƵůŽ͕ŽĞůƚŝĞŵƉŽƚͿ͘>ĂƌĞůĂĐŝſŶƐĞůůĂŵĂĞĐƵĂĐŝſŶ
r (λ )
λ
ƉĂƌĂŵĠƚƌŝĐĂĚĞůĂĐƵƌǀĂΓ ĚĞƐĐƌŝƚĂƉŽƌůĂƚƌĂLJĞĐƚŽƌŝĂ͘^ŝƐĞĐŽŶŽĐĞƵŶĂĞdžƉƌĞƐŝſŶƉĂƌĂŵĠƚƌŝĐĂĚĞ
Γ͕ĞůŝŶƚĞŐƌĂůƋƵĞĚĞĨŝŶĞĞůƚƌĂďĂũŽƉƵĞĚĞĞƐĐƌŝďŝƌƐĞĞŶĨƵŶĐŝſŶĚĞůƉĂƌĄŵĞƚƌŽ λ
&
&
&
^ŝůĂĞdžƉƌĞƐŝſŶƉĂƌĂŵĠƚƌŝĐĂĚĞůĂĐƵƌǀĂĞƐƉ͘Ğũ͘
r
(
λ
)
=
(
x
(
λ
),
y
(
λ
),
z
(
λ
)),
P
=
r
(
λ
)
,
P
=
r
(λ2 ) ͕
1
1
2
&
&
r2
λ2
λ2
&
& &
& dr
ĞŶƚŽŶĐĞƐ͗
dy
dz ·
§ dx
Fi
Wi = ³ Fi ⋅ dr = ³ Fi ⋅ dȜ = ³ ¨ Fx
+ Fy
+ Fz
¸ dȜ
&
&
dȜ
dȜ
dȜ
dȜ
λ2
¹
Γ: r1
λ1
λ1 ©
dr
WͲϮ͘ϰ͘ϭƐŝŶ;ĐͿ
WͲϮ͘ϰ͘ϲ
&
r (λ )
Γ
λ1
͎WĂƌĂƋƵĠƐŝƌǀĞĞůƚƌĂďĂũŽ͍^ŝĐŽŶƐŝĚĞƌĂŵŽƐĞůƚƌĂďĂũŽĚĞůĂĨƵĞƌnjĂƚŽƚĂů
&
& &
& &
& & & & &
&
F = FTOT = F1 + F2 + ... = ma ƚĞŶĞŵŽƐ͗ dWTOT = FTOT ⋅ dr = F1 ⋅ dr + F2 ⋅ dr + ... = dW1 + dW2 + ...
t1
&
&
&
& &
dv &
d 1
1
WŽƌŽƚƌŽůĂĚŽ FTOT ⋅ dr = ma ⋅ dr = m ⋅ v dt = §¨ mv 2 ·¸dt = d §¨ mv 2 ·¸ ͘/ŶƚĞŐƌĂŶĚŽ͗
dt
dt © 2
¹
©2
¹
WͲϮ͘ϭ͘ϭ
1
§
1
2·
2
W = Δ¨ mv ¸ = ΔEc ͘>ĂĐĂŶƚŝĚĂĚƐĞůůĂŵĂĞŶĞƌŐşĂĐŝŶĠƚŝĐĂ͘ƐşƉƵĞƐ͗
W1 + W2 + ... = ΔEc
Ec = m v
2
©2
¹
&&
&
&
&
§ ∂U ∂U ∂U ·
∃ U (r )
F(r )
¸¸
hŶĂĨƵĞƌnjĂƐĞĚŝĐĞĐŽŶƐĞƌǀĂƚŝǀĂ
ƐŝƚĂůƋƵĞ͗
,
,
F = −∇U = −¨¨
x
y
z
∂
∂
∂
©
¹
>ĂĨƵŶĐŝſŶh ƐĞůůĂŵĂĞŶĞƌŐŝĂƉŽƚĞŶĐŝĂů U ĂƐƐŽĐŝĂĚĂĂůĂĨƵĞƌnjĂ͘ůƚƌĂďĂũŽĚĞůĂĨƵĞƌnjĂǀĂůĞ͗
' '
λ2
§ ∂U
∂U ·
∂U
&
dz ¸¸ = − ³ dU
dy +
dx +
W = ³ F ⋅ dr = ³ ( Fx dx + Fy dy + Fz dz ) = − ³ ¨¨
dr
∂z ¹
∂y
© ∂x 2
&
&
λ1
r (λ )
&
F
ŶƚŽŶĐĞƐ͕ƉĂƌĂĐŽŶƐĞƌǀĂƚŝǀĂ͗
W1→2 = − ³ dU = −[U (2) − U (1)] = −ΔU
F
i
1
dƌĂďĂũŽĚĞƵŶĂĨƵĞƌnjĂĐŽŶƐĞƌǀĂƚŝǀĂсǀĂƌŝĂĐŝſŶĚĞĞŶĞƌŐşĂƉŽƚĞŶĐŝĂů
&&
⇔
F(r ) ĞƐĐŽŶƐĞƌǀĂƚŝǀĂ⇔
& &
F
³ ⋅ dr = 0 ⇔
curva cerrada
³
B
A
W = −Δ U
& &
F ⋅ dr ĞƐŝŶĚĞƉĞŶĚŝĞŶƚĞĚĞůĂƚƌĂLJĞĐƚŽƌŝĂ
& &
>ĂĨƵŶĐŝſŶhƐĞĐĂůĐƵůĂĐŽŵŽ͗ U = − ³ F ⋅ dr + ct = − ³ ( Fx dx + Fy dy + Fz dz ) + ct
Ͳ Ŷϭ͗ĐĂĚĂĨƵĞƌnjĂĞƐĐŽŶƐĞƌǀĂƚŝǀĂLJĂƋƵĞƐŝĞŵƉƌĞƐĞƉƵĞĚĞĚĞĨŝŶŝƌU ( x) = − F ( x) dx + ct
³
WͲϮ͘ϰ͘ϱ͕YͲϮ͘ϰ͘ϰнĚŝƐĐƵƚŝƌƉƵŶƚŽƐĚĞĞƋƵŝůŝďƌŝŽĞƐƚĂďůĞĞŝŶĞƐƚĂďůĞ
& &
Ͳ ŶϮ͗dU = −F ·dr = −Fx dx − Fy dy ⇔ Fx = −
∂U
∂x
Fy = −
∂U
∂y
^ĞĚĞŵƵĞƐƚƌĂƋƵĞƵŶĂĨƵŶĐŝſŶĚŝĨĞƌĞŶĐŝĂďůĞĚĞĚŽƐǀĂƌŝĂďůĞƐƚŝĞŶĞůĂƉƌŽƉŝĞĚĂĚƋƵĞ͗
∂ 2U ∂ 2U ƐƚŽŝŵƉůŝĐĂƋƵĞƵŶĂĨƵĞƌnjĂĞŶϮĞƐĐŽŶƐĞƌǀĂƚŝǀĂ ⇔ ∂Fx ∂Fy
=
WͲϮ͘ϰ͘ϯ
=
∂y
∂x
∂x∂y ∂y∂x
WͲϮ͘ϰ͘ϰ͕WͲϮ͘ϰ͘ϳ
&
&
F
Ͳ Ŷϯ͗^ŝĞƐƌĂĚŝĂůLJƐſůŽĚĞƉĞŶĚĞĚĞŶŽĚĞŽƚƌĂƐĐŽŵďŝŶĂĐŝŽŶĞƐĚĞdž͕LJ͕nj͕ŽƐĞĂ
r =r
& &
&
Ɛŝ͕ĞŶƚŽŶĐĞƐĞƐĐŽŶƐĞƌǀĂƚŝǀĂLJ
F (r ) = F (r )rˆ
U (r ) = − ³ F ( r )rˆ ⋅ dr + ct = − ³ F (r )dr + ct
(
)
ĂƐŽƐŝŵƉŽƌƚĂŶƚĞƐ
;ΑϮ͘ϱLJΑϯ͘ϳͿ
ϭͿhŶĂĨƵĞƌnjĂĐŽŶƐƚĂŶƚĞ;ĞŶŵŽĚƵůŽLJĚŝƌĞĐĐŝſŶͿĞƐĐŽŶƐĞƌǀĂƚŝǀĂ͗
& &
&
dU = − F ·dr °½
&&
&
&
Fg = mg U = m g y + ct
¾ Ÿ U = − F ·r + ct ͘ũĞŵƉůŽ͗Æ
F = cte
°¿
&
&
k
k
mm
F = − 2 rˆ Ÿ U (r ) = −
FGU = −G 1 2 2 rˆ
ϮͿ͘ũĞŵƉůŽƐ͗>ĞLJĚĞŐƌĂǀŝƚĂĐŝſŶƵŶŝǀĞƌƐĂů
r
r
r
&
1 q1q2
ˆ
r
LJ>ĞLJĚĞŽƵůŽŵď FC = −
2
1
r
πε
4
2
0
ϯͿ FH = −kx Ÿ U(x) = −³ (−kx) dx = k x + ct
2
2
2
&
&
ˆ
ˆ
ϰͿ FRDS = − μ D Nv Ÿ W1→2 = ³ (− μ D N )v ⋅ dr = − μ D N ³ d" = − μ D N" 1→2
1
Å
1
^ſůŽǀĂůĞ
ƐŝE сĐŽŶƐƚ
ŶŵƵĐŚĂƐƐŝƚƵĂĐŝŽŶĞƐŚĂLJĨƵĞƌnjĂƐŽƌƚŽŐŽŶĂůĞƐĂůŵŽǀŝŵŝĞŶƚŽ;ŽƐĞĂ͕ĂůĂǀĞůŽĐŝĚĂĚͿ͖ŵƵLJĂ
ŵĞŶƵĚŽĠƐƚĞĞƐĞůĐĂƐŽĚĞůĂƐůůĂŵĂĚĂƐ͞ƌĞĂĐĐŝŽŶĞƐ͕͟ĐŽŵŽůĂĨƵĞƌnjĂŶŽƌŵĂůĚĞďŝĚĂĂƵŶĂ
͕
&
ƐƵƉĞƌĨŝĐŝĞŽůĂƚĞŶƐŝſŶĚĞƵŶĂĐƵĞƌĚĂĞŶƵŶŵŽǀŝŵŝĞŶƚŽĐŝƌĐƵůĂƌ͘/ŶĚŝĐĄŶĚŽůĂƐĐŽŶůĂůĞƚƌĂ͕ƐĞ
& & & & & & R&
& &
& & &
ŚĂ͗F = FTOT = ma = R1 + F2 + ... ͘DƵůƚŝƉůŝĐĂŵŽƐĞƐĐĂůĂƌŵĞŶƚĞƉŽƌ͗
& dr ma&⋅ dr &= R1 ⋅ dr + F2 ⋅ dr + ...
& &
&
ůƐĞƌ͕ůĂĨƵĞƌnjĂĚĞƌĞĂĐĐŝſŶĞƐŽƌƚŽŐŽŶĂůĂ͕ŽƐĞĂ͘ƐƚŽŶŽƐĚĞũĂ
dr
R1 ⋅ dr = 0
R ⊥ v = dr dt
&
&
&
&
ĐŽŶĞů͗
ƉƌŝŶĐŝƉŝŽĚĞĚ͛ůĞŵďĞƌƚ F2 + ... + Fn − ma ⋅ dr = 0
(
)
& &
& &
& &
& &
/ŶƚĞŐƌĂŶĚŽůĂĞĐ͘ĚĞĚ͛ůĞŵďĞƌƚ͕ ³ F ⋅ dr = ³ F1 ⋅ dr + ³ F2 ⋅ dr +... + ³ Fn ⋅ dr ͕ƐĞŽďƚŝĞŶĞůĂĞĐƵĂĐŝſŶ
ĨƵŶĚĂŵĞŶƚĂůĚĞůĂĞŶĞƌŐşĂ͗
ΔEc = −ΔU Fcons + WF NO cons Ÿ WF NO cons = Δ( Ec + U )
WͲϯ͘ϳ͘ϯ͕WͲϮ͘ϱ͘ϳ͕;WͲϯ͘ϭϬ͘ϰĐŽŶĂͿ
WͲϮ͘ϱ͘ϭ͕WͲϮ͘ϱ͘ϲ͕YͲϮ͘ϭ͘ϭ͕WͲϯ͘ϳ͘ϰ͕WͲϯ͘ϭϬ͘ϭ͕WͲϯ͘ϭϬ͘ϵ͕YͲϯ͘ϭϬ͘ϭ͕YͲϯ͘ϭϬ͘Ϯ
Ec + U = E = ct
^ŝŶŽŚĂLJĨƵĞƌnjĂƐĚŝƐŝƉĂƚŝǀĂƐ͕ĞŶƚŽŶĐĞƐ͕ŽƐĞĂƚĂŵďŝĠŶ͗
dE
=0
ŶĂůŐƵŶŽƐĐĂƐŽƐ;ƉŽƌĞũĞŵƉůŽƐŝĞůƐŝƐƚĞŵĂƚŝĞŶĞƵŶƐŽůŽŐƌĂĚŽĚĞůŝďĞƌƚĂĚͿ͕
dt
ĞƐƚĂĞĐƵĂĐŝſŶĞƐƐƵĨŝĐŝĞŶƚĞƉĂƌĂŚĂůůĂƌůĂƚƌĂLJĞĐƚŽƌŝĂƚĞŵƉŽƌĂůĚĞůƐŝƐƚĞŵĂ͘
WͲϯ͘ϭϬ͘ϯ
dĞŵĂϯ͗^ŝƐƚĞŵĂƐĚĞN ƉĂƌƚşĐƵůĂƐ;ŝ сϭ͕Ϯ͕͘͘͘NͿ
ϯǐ>ĞLJĚĞEĞǁƚŽŶŽ>ĞLJĚĞĂĐĐŝſŶLJƌĞĂĐĐŝſŶ͗ƐŝƵŶĐƵĞƌƉŽĂĐƚƷĂƐŽďƌĞƵŶƐĞŐƵŶĚŽĐŽŶƵŶĂ
&
&
ĨƵĞƌnjĂ͕ĞŶƚŽŶĐĞƐĠƐƚĞƷůƚŝŵŽŐĞŶĞƌĂƵŶĂĨƵĞƌnjĂŝŐƵĂůLJŽƉƵĞƐƚĂƐŽďƌĞĞůƉƌŝŵĞƌŽ
−
F
F
ũĞŵƉůŽ͗ĨƵĞƌnjĂƐĚĞĐŽŶƚĂĐƚŽ WͲϯ͘ϭϬ͘Ϯ
DƵĐŚĂƐǀĞĐĞƐĞƐƚĂƐĨƵĞƌnjĂƐĐƵŵƉůĞŶƚĂŵďŝĠŶŽƚƌĂĐŽŶĚŝĐŝſŶ͕ƋƵĞĞƐƋƵĞƐƵĚŝƌĞĐĐŝſŶĞƐ
&
&
& &
ƉĂƌĂůĞůĂĂůǀĞĐƚŽƌƋƵĞƵŶĞůĂƐƉŽƐŝĐŝŽŶĞƐĚĞůĂƐƉĂƌƚşĐƵůĂƐ͗ F
// r (= r − r )
1→ 2
1→ 2
2
1
>ĂƐĨƵĞƌnjĂƐĚĞŐƌĂǀŝƚĂĐŝſŶƵŶŝǀĞƌƐĂůLJŽƵůŽŵďĐƵŵƉůĞŶůĂϯǐ>ĞLJĚĞEĞǁƚŽŶĞŶĞƐƚĂĨŽƌŵĂ
͞ĨƵĞƌƚĞ͟;WͲϯ͘ϳ͘ϭͿ
& (ext )
mi &
F
&ƵĞƌnjĂƐĞdžƚĞƌŶĂƐ
͗ƐŽŶĐĂƵƐĂĚĂƐƉŽƌĂŐĞŶƚĞƐ
i
(ext )
Fi
ĞdžƚĞƌŝŽƌĞƐ͕ƋƵĞŶŽ ƉĞƌƚĞŶĞĐĞŶĂůƐŝƐƚĞŵĂĐŽŶƐŝĚĞƌĂĚŽ
mi
&
Fji
&
Fij
mj
&ƵĞƌnjĂƐŝŶƚĞƌŶĂƐ͗ƐŽŶůĂƐĨƵĞƌnjĂƐĚĞůĂƐƉĂƌƚşĐƵůĂƐĚĞůƐŝƐƚĞŵĂĞŶƚƌĞĞůůĂƐ
ƉŽƌůĂϯǐ>ĞLJĚĞEĞǁƚŽŶ͗
&
& ext
& int
& int
Fi → j = − F j →i
&
& &.
+ ¦ F jint→i = mi ai = p i
&
& ext
& ext dP
&. j &.
= ¦ p i = P ͕ŽƐĞĂ͗ FTOT = F =
dt
i
>ĂϮĂ>ĞLJĚĞEĞǁƚŽŶƉĂƌĂůĂƉĂƌƚşĐƵůĂŝ ĞƐ͗ Fi = Fi
&
&
& ext & ext
¦i Fi =¦i Fi = FTOT
&
&
Æ ^ŝ͕ĞŶƚŽŶĐĞƐ
F ext = 0
P = ct
^ƵŵĂŶĚŽƐŽďƌĞŝ͗ FTOT =
&
&
1
RCM =
mi ri
M = ¦ mi
ĞĨŝŶŝĞŶĚŽůĂĐŽŽƌĚĞŶĂĚĂĚĞůĞŶƚƌŽĚĞDĂƐĂƐ;DͿĐŽŵŽ͕͗ĐŽŶ͕
¦
M i
.
&
&
&
&
&
i
&
ĞŶƚŽŶĐĞƐ͗ P =
ext
LJ
p = MV
¦
i
i
CM
F
= P = MACM
ƉůŝĐĂĐŝſŶ͗ĨƵĞŐŽƐĂƌƚŝĨŝĐŝĂůĞƐ
WͲϯ͘ϭϭ͘ϭ͕WͲϯ͘Ϯ͘ϴ
YͲϯ͘Ϯ͘ϲ
WͲϯ͘ϲ͘ϳ
EŽƚĂŝŵƉŽƌƚĂŶƚĞƐŽďƌĞĞůĐĞŶƚƌŽĚĞŵĂƐĂƐ͘WĂƌĂĚŽƐƉĂƌƚşĐƵůĂƐ͗
&
1
RCM =
(m1r&1 + m2 r&2 ) = 1 (( M − m2 )r&1 + m2 r&2 ) = r&1 + m2 (r&2 − r&1 )
M
M
M
ƋƵşƐĞŚĂƵƚŝůŝnjĂĚŽ M = m1 + m2 Ÿ m1 = M − m2
& &
r2 − r1
×
1
&
r1
&
RCM
2
CM
WŽƌůŽƚĂŶƚŽ͕ĞůĐĞŶƚƌŽĚĞŵĂƐĂƐĚĞƵŶĐŽŶũƵŶƚŽĚĞĚŽƐ
ƉĂƌƚşĐƵůĂƐĞƐƚĄĂůŽůĂƌŐŽĚĞůĂƌĞĐƚĂƋƵĞůĂƐƵŶĞ͕ĞŶƵŶƉƵŶƚŽŝŶƚĞƌŵĞĚŝŽĞŶƚƌĞůĂƐĚŽƐ
&
&
&
&
Αϯ͘ϰ ůŵŽŵĞŶƚŽĂŶŐƵůĂƌƚŽƚĂůĚĞƵŶƐŝƐƚĞŵĂĚĞƉĂƌƚşĐƵůĂƐĞƐ͗ L =
Li (Q ) = ri (Q ) × mi vi
(Q )
&
&
& &
i
i
ĚŽŶĚĞ͘ĞƌŝǀĂŶĚŽƌĞƐƉĞĐƚŽĚĞůƚŝĞŵƉŽ͗
ri ( Q ) = ri − rQ WͲϯ͘ϰ͘ϭн RCM
&
сϬ
&
&
dL(Q )
&
&
&
&
&
&
&
&
&
&
&
= ¦ vi ( Q ) × mi vi + ¦ ri ( Q ) × mi ai = ¦ vi × mi vi − ¦ vQ × mi vi + ¦ ri ( Q ) × Fi = M (Q ) − vQ × ¦ mi vi
dt
i
i
i
i
i
& int &i
^ŝůĂƐĨƵĞƌnjĂƐŝŶƚĞƌŶĂƐĐƵŵƉůĞŶůĂĐŽŶĚŝĐŝſŶ͕ĞŶƚŽŶĐĞƐƐƵŵŽŵĞŶƚŽƚŽƚĂůĞƐĐĞƌŽ͕LJ
Fi → j // rij
&
& ext
&
&
&
M (Q ) = M ( Q )͘ŽŶĞƐƚŽ͕ůĂĞĐƵĂĐŝſŶĚĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌƋƵĞĚĂ͗ dL(Q ) dt = M (extQ) − v&Q × MVCM
¦
¦
ůϮǑƚĠƌŵŝŶŽĞƐĐĞƌŽƐŝ͗;ϭͿYĞƐƵŶƉƵŶƚŽĨŝũŽ͖Ž;ϮͿ͕YĐŽŝŶĐŝĚĞĐŽŶĞůĐĞŶƚƌŽĚĞŵĂƐĂƐ͘ƐşƉƵĞƐ͗
&
&
& ext
& ext dL(Q )
L
M
=
0
͕ĐŽŶYĨŝũŽŽYŁ
D͘ƐƚŽŝŵƉůŝĐĂƋƵĞƐŝ
͕ĞŶƚŽŶĐĞƐ
( Q ) = ct
(Q )
M (Q ) =
dt
ƉůŝĐĂĐŝŽŶĞƐ;YсDͿ͗ĨƷƚďŽůĂŵĞƌŝĐĂŶŽ͕ĂƌŵĂƐĚĞĨƵĞŐŽ͕
YƺĞƐƚŝŽŶƐ WĂƌĐŝĂůϮϬϭϬ
YƺĞƐƚŝŽŶƐWĂƌĐŝĂůϮϬϭϬ
WͲϯ͘Ϯ͘ϭ͕WͲϯ͘ϯ͘ϭ͕YͲϯ͘ϰ͘ϭ͕Y͘ϯ͘ϰ͘Ϯ͕YͲϯ͘ϰ͘ϯ
ŐŝƌŽƐĐŽƉŝŽ ĞƐƚĂĐŝŽŶĞƐ ƚĞƌƌĞŵŽƚŽƐ
ŐŝƌŽƐĐŽƉŝŽ͕ĞƐƚĂĐŝŽŶĞƐ͕ƚĞƌƌĞŵŽƚŽƐ
Αϯ͘ϱ ŽŶƐŝĚĞƌĞŵŽƐĞůƚƌĂďĂũŽĚĞƵŶĂƉĂƌĞũĂĚĞĨƵĞƌnjĂƐŝŶƚĞƌŶĂƐ͗
&
&
&
&
&
&
&
int
int
dW12 = F1int
→ 2 ⋅ dr2 + F2 →1 ⋅ dr1 = F1→ 2 ⋅ (dr2 − dr1 )
;ƐĞŚĂƵƚŝůŝnjĂĚŽůĂϯǐ>ĞLJĚĞEĞǁƚŽŶͿ^ŝůĂĨƵĞƌnjĂƐſůŽĚĞƉĞŶĚĞĚĞůĂ
&
& &
rrel = r2 − r1
ĐŽŽƌĚĞŶĂĚĂƌĞůĂƚŝǀĂLJĂĚĞŵĄƐĞƐĐŽŶƐĞƌǀĂƚŝǀĂ͕ĞŶƚŽŶĐĞƐ͗
&
& &
&
&
&
dW12 = F1→2 (rrel ) ⋅ drrel = −∇U (rrel ) ⋅ drrel = − dU
ƐƚŽƐŝŐŶŝĨŝĐĂƋƵĞĂĐĂĚĂƉĂƌĞũĂ;ŝ͕ũͿƐĞƉƵĞĚĞĂƐŽĐŝĂƌƵŶĂĞŶĞƌŐşĂƉŽƚĞŶĐŝĂůhƋƵĞƐſůŽ
ĚĞƉĞŶĚĞĚĞůĂĐŽŽƌĚĞŶĂĚĂƌĞůĂƚŝǀĂ͘^ŝƚŽĚĂƐůĂƐĨƵĞƌnjĂƐŝŶƚĞƌŶĂƐLJĞdžƚĞƌŶĂƐƐŽŶĐŽŶƐĞƌǀĂƚŝǀĂƐ͗
U = U iext +
U ijint
E = EC + U = ct ͕ĐŽŶ͘^ŝŚĂLJĨƵĞƌnjĂƐĞdžƚĞƌŶĂƐŶŽĐŽŶƐĞƌǀĂƚŝǀĂƐ͗
¦
i
WͲϯ͘ϱ͘Ϯ͕WͲϯ͘ϳ͘Ϯ
¦
parejas
WͲϯ͘ϳ͘ϰ͕WͲϯ͘ϱ͘ϯ͕YͲϯ͘ϯ͘ϭ͕YͲϯ͘ϱ͘ϭ͕YͲϯ͘ϱ͘Ϯ͕YͲϯ͘ϱ͘ϯ
WFext NO CONS = Δ( EC + U )
Αϯ͘ϲŚŽƋƵĞƐ
hŶĐŚŽƋƵĞĞƐƵŶĂĐŽůŝƐŝſŶĞŶƚƌĞĚŽƐĐƵĞƌƉŽƐ͕ŽƐĞĂƵŶĂŝŶƚĞƌĂĐĐŝſŶĚĞĐŽŶƚĂĐƚŽĚĞĚƵƌĂĚĂ
ůŝŵŝƚĂĚĂ;ƵƐƵĂůŵĞŶƚĞƐĞǀĞƌŝĨŝĐĂƋƵĞȴƚ сϬ͘Ϭϭр Ϭ͘ϬϬϭƐĞĐͿ͘EŽĐŽŶƐŝĚĞƌĂŵŽƐůĂƌŽƚĂĐŝſŶ͘
YͲϮ͘Ϯ͘ϭ dŽŵĞŵŽƐĐŽŵŽƉƌŝŵĞƌĞũĞŵƉůŽĞůĚĞƵŶĂƉĞůŽƚĂĚĞŵĂƐĂϴϬŐƋƵĞƌĞďŽƚĂĐŽŶƚƌĂ
ƵŶĂƉĂƌĞĚ͘ŽŶƐŝĚĞƌĞŵŽƐůŽƐŝŶƐƚĂŶƚĞƐũƵƐƚŽĂŶƚĞƐ;ͿLJũƵƐƚŽĚĞƐƉƵĠƐ;ͿĚĞůĂĐŽůŝƐŝſŶ͕LJ
ŽůǀŝĚĠŵŽŶŽƐĚĞŵŽŵĞŶƚŽĚĞůĂƐĞǀĞŶƚƵĂůĞƐĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐ͘^ŝĞůŵŽĚƵůŽĚĞůĂǀĞůŽĐŝĚĂĚ
ĂŶƚĞƐLJĚĞƐƉƵĠƐĚĞůĐŚŽƋƵĞĞƐĚĞϯϬŵͬƐ͕ĞůŝŵƉƵůƐŽŐĞŶĞƌĂĚŽƉŽƌůĂĨƵĞƌnjĂĚĞďŝĚĂĂůĂ
&
&
& &
ƉĂƌĞĚĚƵƌĂŶƚĞĞůĐŚŽƋƵĞĞƐ I = Δp = m(vD − v A ), en modulo 4.8 N s
>ĂĨƵĞƌnjĂŵĞĚŝĂƋƵĞƐĞŐĞŶĞƌĂĞŶĞůŝŵƉĂĐƚŽĞƐĞŶŽƌŵĞ
ƉĂƌĂƵŶĂƉĞůŽƚĂĚĞϴϬŐ͕LJǀĂůĞ͕ĐŽŶȴƚ сϬ͘ϬϬϯƐĞĐ
͞WZ/E/W/KĚĞůDZd/>>K͟
Fmedia =
&
I
Δt
= 1000 N
ŽŵƉĂƌĂĚŽĐŽŶĞƐƚŽ͕ůĂĨƵĞƌnjĂĚĞŐƌĂǀĞĚĂĚ;ƉŽƌĞũĞŵƉůŽͿƐŽďƌĞůĂƉĞůŽƚĂĞƐĚĞƐƉƌĞĐŝĂďůĞ͘
ŽŵƉĂƌĂĚŽ
ĐŽŶ ĞƐƚŽ ůĂ ĨƵĞƌnjĂ ĚĞ ŐƌĂǀĞĚĂĚ ;ƉŽƌ ĞũĞŵƉůŽͿ ƐŽďƌĞ ůĂ ƉĞůŽƚĂ ĞƐ ĚĞƐƉƌĞĐŝĂďůĞ
ŽŶƐŝĚĞƌĞŵŽƐĂŚŽƌĂĞůĐŚŽƋƵĞĞŶƚƌĞĚŽƐŽŵĄƐŵĂƐĂƐ͘^ĞƉƵĞĚĞƐƵƉŽŶĞƌƋƵĞŶŽŚĂLJĂ
ĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐ͕LJĂƋƵĞĞƐƚĂƐƐŽŶŵƵLJĚĠďŝůĞƐƉĂƌĂũƵŐĂƌƵŶƉĂƉĞůĚƵƌĂŶƚĞĞůĐŚŽƋƵĞ͘
WŽƌůŽƚĂŶƚŽ͕ůĂĐĂŶƚŝĚĂĚĚĞŵŽǀŝŵŝĞŶƚŽƚŽƚĂůĚĞůƐŝƐƚĞŵĂƐĞĐŽŶƐĞƌǀĂ͗
&
& ext
F ≈ 0 &Ÿ P
& = ct
Ÿ PD = PA
WͲϯ͘ϲ͘ϴ;ĐŚŽƋƵĞĐŽŵƉů͘ŝŶĞůĄƐƚŝĐŽͿ
&
v1(ini)
m1
m1
DŝĞŶƚƌĂƐƋƵĞĞůŵŽŵĞŶƚŽůŝŶĞĂůƐĞĐŽŶƐĞƌǀĂƐŝĞŵƉƌĞĞŶƵŶĐŚŽƋƵĞĞŶƚƌĞ &
v2(ini )
ƉĂƌƚşĐƵůĂƐ͕ůĂĞŶĞƌŐşĂĞŶŐĞŶĞƌĂůŶŽ͘>ůĂŵĂŵŽƐĞůĄƐƚŝĐŽƵŶĐŚŽƋƵĞĞŶ
m2
ƋƵĞůĂĞŶĞƌŐşĂĐŝŶĠƚŝĐĂƚŽƚĂů͕LJƉŽƌůŽƚĂŶƚŽůĂĞŶĞƌŐşĂƚŽƚĂů͕ƐĞĐŽŶƐĞƌǀĂ
;ƐŝŶĚĞƐƉůĂnjĂŵŝĞŶƚŽƐ͕ůĂĞŶĞƌŐşĂƉŽƚĞŶĐŝĂůĚĞ&ĞdžƚĞƌŶĂƐŶŽĐĂŵďŝĂͿ͘
WĂƌĂϮƉĂƌƚşĐƵůĂƐ͕ĐŚŽƋƵĞĞůĄƐƚŝĐŽ ⇔
m2
1
1
1
1
m1v12, D + m2 v22, D = m1v12, A + m2 v22, A
2
2
2
2
N
m
&
vA
, i = 1,2
&
P
= ct
ĚĞŵĄƐĂƋƵşƚĞŶĚƌĞŵŽƐ͗
//
i,D
v
m
&
vD
n̂
/ŵƉŽƌƚĂŶƚĞ͗WĂƌĂƵŶĂĐŽůŝƐŝſŶĞŶƚƌĞƵŶĐƵĞƌƉŽLJƵŶĂƉĂƌĞĚĨŝũĂ͗ vD// = v A//
EſƚĞƐĞƋƵĞĞŶĞƐƚĞĐĂƐŽŶŽ ƐĞĐŽŶƐĞƌǀĂĞůŵŽŵĞŶƚŽůŝŶĞĂůƚŽƚĂů
/ŐƵĂůŵĞŶƚĞ͕ƉĂƌĂƵŶĐŚŽƋƵĞĞŶƚƌĞĚŽƐĐƵĞƌƉŽƐ͗
&
v2( fi)
WͲϯ͘ϲ͘ϯ
^ŝƐĞĐŽŶŽĐĞůĂŐĞŽŵĞƚƌşĂĚĞůĂĐŽůŝƐŝſŶLJŶŽŚĂLJƌŽnjĂŵŝĞŶƚŽ͗
ĞŶĞƐƚĞĐĂƐŽƉŽĚĞŵŽƐĐŽŶŽĐĞƌůĂĚŝƌĞĐĐŝſŶĚĞůĂƐĨƵĞƌnjĂƐŝŵƉƵůƐŝǀĂƐ
ŐĞŶĞƌĂĚĂƐĞŶĞůŝŵƉĂĐƚŽ͕ƋƵĞĞƐŽƌƚŽŐŽŶĂůĂůƉůĂŶŽĚĞĐŚŽƋƵĞ͘ƐƚŽ
ŝŵƉůŝĐĂƋƵĞŶŽŚĂLJŶŝŶŐƵŶĂĨƵĞƌnjĂƉĂƌĂůĞůĂĂĞƐƚĞƉůĂŶŽ͕LJƋƵĞ
ĞŶƚŽŶĐĞƐůĂƐĐŽŵƉŽŶĞŶƚĞƐƉĂƌĂůĞůĂƐĚĞůĂƐǀĞůŽĐŝĚĂĚĞƐƐĞĐŽŶƐĞƌǀĂŶ͘
&
v1( fi)
&
v1, D
&
v1, A
1
=v
//
i, A
&
v2 , A
&
v2 , D
2
n̂
WŽƌůŽƚĂŶƚŽůĂƐƷŶŝĐĂƐŝŶĐſŐŶŝƚĂƐƋƵĞƋƵĞĚĂŶƐŽŶůĂƐĐŽŵƉŽŶĞŶƚĞƐĚĞůĂǀĞůŽĐŝĚĂĚŽƌƚŽŐŽŶĂůĞƐ
ĂůƉůĂŶŽĚĞĐŚŽƋƵĞ͘^ĞĚĞĨŝŶĞ ĞůĐŽĞĨŝĐŝĞŶƚĞĚĞƌĞƐƚŝƚƵĐŝſŶĞ ĐŽŵŽ͗
e=
v1⊥( D ) − v2⊥( D )
v1⊥( A) − v2⊥( A)
^ŝĐŽŶŽĐĞŵŽƐĞůǀĂůŽƌĚĞĞ͕ĞůƉƌŽďůĞŵĂƐĞƌĞƐƵĞůǀĞƉŽŶŝĞŶĚŽĂƐŝƐƚĞŵĂůĂƐĞĐƵĂĐŝŽŶĞƐƋƵĞ
ƌĞƐƵůƚĂŶĚĞůĂĐŽŶƐĞƌǀĂĐŝſŶĚĞůŵŽŵĞŶƚŽLJĚĞůĂĚĞĨŝŶŝĐŝſŶĚĞĞ͘
^ĞƉƵĞĚĞĚĞŵŽŶƐƚƌĂƌƋƵĞƉĂƌĂƵŶĐŚŽƋƵĞĞůĄƐƚŝĐŽ͕Ğсϭ
WͲϯ͘ϲ͘ϭ͕WͲϯ͘ϲ͘ϰ
WͲϯ͘ϲ͘ϲ͕WͲϯ͘ϲ͘ϵ͕WͲϯ͘ϲ͘ϭϬ
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