trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008 PHÇN I C¸CH CHøNG MINH C¤NG THóC TÝNH VËN TèC Vµ SøC C¡NG D¢Y CñA CON L¾C §¥N PH¦¥NG PH¸P:

Oα

B

α OαA

mgh

=

+

A

mgh B

1. C«ng thøc tÝnh vËn tèc t¹i vÞ trÝ bÊt kú: I Do con l¾c chuyÓn ®éng trong tr−êng träng lùc nªn c¬ n¨ng b¶o toµn Chän mèc thÕ n¨ng h=o t¹i vÞ trÝ c©n b»ng O. ¸p dông ®Þnh luËt B¶o toµn c¬ n¨ng cho 2 vÞ trÝ A vµ B ta cã

H

A

2. m v 2

(1) . WA=WB hay :

B

B

)

IO IH l

= −

=

−

. . ( l co s α o

Chó ý : con l¾c ®¬n ®−îc th¶ kh«ng vËn tèc ban ®Çu tõ vÞ trÝ A Nªn vA=O h A

O

= −

=

B

)]

)]

[ mg l

. ( lco s

[ mg l

. ( lco s

−

=

−

+

α

α o

2. m v 2

2.

.

)

v

. ( co s

) α

=

−

α o

B

[ . ( g l co s

]

Trong ®ã . ) . ( l co s α IO IB l Bh − Nªn thay vµo biÓu thøc (1) ta cã:

(2)

0oα=

T−¬ng ®−¬ng : Tõ ®ã ta cã c¸c tr−êng hîp sau x¶y ra :

2.

)

g l

=

−

cos(oo)=1 suy ra

. ( co s α o

[ . 1

]

1

a. T¹i vÞ trÝ c©n b»ng gãc v . ma x (3) ( T¹i VTCB vËn tèc ®¹t gi¸ trÞ cùc ®¹i )

0

trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008 10

α

≤

2

) 1 2.sin

. ( co s

α = −

1 ≈ −

0 10 , α≤ o α 2

NNÕÕuu ggããcc

2

2

) 1 2.sin

. ( co s

α = −

1 ≈ −

0

ta sö dông c«ng thøc gÇn ®óng : 2 α 2

α 0 2

α 0 2

.

=

−

2 2 g l α α 0

Bv

 

  ((44))

VVµµ cc««nngg tthhøøcc vvËËnn ttèècc ccùùcc ®®¹¹ii llóócc nnµµyy llµµ :: tthhaayy vvµµoo ((33)) ::

2

2

2.

2.

. 2.sin

)

g l

g l

=

. 1 (1 2.sin − −

=

v . ma x

α 0 2

α 0 2

  

  

  

  

0

2

0

2.

. .2.

.

g l

. g l

≈

=

sin

α 0

v . ma x

Vµ Thay tÊt c¶ vµo (2) ta cã :

2 α 4

2 α α 0 ≈ 2 4

0α α=

.min

o=

(5) Do

Bv

nªn

(cid:1) (cid:1) (cid:1) . P T m a + =

T¹i vÞ trÝ biªn 22..CC««nngg tthhøøcc ttÝÝnnhh ssøøcc cc¨¨nngg dd©©yy TT tt¹¹ii vvÞÞ ttrrÝÝ bbÊÊtt kkúú :: xxÐÐtt tt¹¹ii vvÞÞ ttrrÝÝ bbiiªªnn AA ttaa ccãã cc¸¸cc llùùcc tt¸¸cc ddôônngg llªªnn vvËËtt mm llµµ ssøøcc cc¨¨nngg ssîîii dd©©yy TT vvµµ ttrräänngg llùùcc PP.. TThheeoo ®®ÞÞnnhh lluuËËtt IIII NNIIUUTT¥¥NN ttaa ccãã ::

2

((66))

trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008 CChhiiÕÕuu ((66)) llªªnn pphh−−¬¬nngg ssîîii dd©©yy hh−−íínngg vvµµoo ®®iiÓÓmm ttrreeoo II cchhiiÒÒuu

2

. ( Pco s

. T m

−

+ =

) α

Bv l

3.

. ( co s

. ( co s

α

α

T mg =

) 2. −

dd−−¬¬nngg nnhh−− hh××nnhh vvÏÏ :: ((77))

[

]0 )

((88))

I

TT¹¹ii VVTTCCBB

TThhaayy ((22)) vvµµoo ((77)) :: 0oα= cos(oo)=1 nªn :

T

)

mg

=

3 2. −

.

. ( co s α 0

ma xT

[

]

α OαA

((99))

H

A

T

0α α=

nªn :

B

B

)

. . ( mg co s

) α

=

=

α 0

]

[ . ( mg co s

((1100))

TT¹¹ii vvÞÞ ttrrÝÝ hhaaii bbiiªªnn T min

O

α

0

P

PX

10

α

≤

2

) 1 2.sin

. ( co s

α = −

1 ≈ −

0 10 , α≤ o α 2

NNÕÕuu ggããcc

2

2

) 1 2.sin

. ( co s

α = −

1 ≈ −

0

ta sö dông c«ng thøc gÇn ®óng : 2 α 2

α 0 2

α 0 2

Vµ

0

3(1

)

mg

T mg ≈

−

) 2.(1 −

−

=

+

2 α

0

2 α 2

2 α 2

2 α 2

 . 1 3. −  

  

  

  

0oα=

Suy ra :

3

cos(oo)=1 nªn : (11) TT¹¹ii VVTTCCBB

trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008

0

)

)

mg

. ( co s

mg

=

3 2. −

=

−

.

α 0

ma xT

[

]

2 α 2

  

2

=

.

0

ma xT

mg α 1 +

 . 3 2.(1 −     ((1122))

0α α=

HHaayy ::

nªn :

0

)

.(1

)

.(1

)

T

. ( . mg co s

mg

mg

=

=

≈

−

=

−

) α

min

α 0

[ . ( mg co s

]

2 α 2

2 α 2

TT¹¹ii vvÞÞ ttrrÝÝ hhaaii bbiiªªnn

. ) t

0 (1 l

2.

T

2 . π

π

=

=

PPHHÇÇNN 22 ((1133)) BBiiÕÕnn tthhiiªªnn cchhuu kkúú ccññaa ccoonn ll¾¾cc ®®¬¬nn tthheeoo nnhhiiÖÖtt ®®éé,, ®®éé ccaaoo vvµµ vvÞÞ ttrrÝÝ ttrrªªnn ttrr¸¸ii ®®ÊÊtt PPHH¦¦¥¥NNGG PPHH¸¸PP:: DDùùaa vvµµoo cc««nngg tthhøøcc ::

l g

+ α g

0

0

c

0

t = 0t c

ol :: llµµ cchhiiÒÒuu ddµµII dd©©yy ttrreeoo ccoonn ll¾¾cc ëë l :: llµµ cchhiiÒÒuu ddµµII dd©©yy ttrreeoo ccoonn ll¾¾cc ëë α :: llµµ hhÖÖ ssèè nnëë ddµµII ëë 0000CC BBµµii ttoo¸¸nn 11:: XX¸¸cc ®®ÞÞnnhh tthhêêii ggiiaann ccoonn ll¾¾cc cchh¹¹yy ssaaii ttrroonngg mmççii cchhuu kkúú.. TTHH11:: KKhhii ëë ®®éé ccaaoo nnhhÊÊtt ®®ÞÞnnhh ((ccïïnngg ®®«« ccaaoo )) ccãã gg==ccoonnsstt vvµµ nnhhiiÖÖtt

t≠ )) 2

t ®®éé kkhh¸¸cc nnhhaauu (( 1 CCoonn ll¾¾cc cchh¹¹yy ®®óónngg ëë nnhhiiÖÖtt ®®éé tt11 ,, ttaa ccãã cchhuu kkúú TT11

4

TTrroonngg ®®ãã ::

trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008

(1

l 0

. ) t 1

2.

=

π

T 1

+ α g

(1

l 0

. ) t 2

=

2.

=

π

T 2

VVµµ cchhuu kkýý TT22

1 1

+ +

. t α 1 . t α

+ α g

T 1 T 2

2

SSuuyy rraa :: ((1144))

1ε< ≤ tthh×× ::

1 ≈ +

HHaayy :: ¸¸pp ddôônngg cc««nngg tthhóócc ggÇÇnn ®®óónngg :: vvííii 0

1 1

ε ε 2 1 − 2 2

+ +

ε 1 ε 2

)

t

2

1 ≈ +

((1155))

T 1 T 2

)

)

t

t

2

2

≈

1 − ≈

tthhaayy ((1155)) vvµµoo ((1144)) ttaa ccãã

( t α − 1 2 ( t − α 1 2

( t α − 1 2

T T − 1 2 T 2

)

t

2

=

=

((1166)) ⇔ HHaayy

T 1 T 2 NNHHËËNN XXÐÐTT:: ++))NNÕÕuu tt11>>tt22 ssuuyy rraa TT11>>TT22 cchhuu kkúú ggii¶¶mm ®®åånngg hhåå cchh¹¹yy nnhhaannhh.. ++))NNÕÕuu tt11<

T T − 1 2 T 2

T ∆ T 2 KKÕÕtt lluuËËnn :: MMççii nnggµµyy ®®ªªmm ®®åånngg hhåå cchh¹¹yy ssaaii mméétt kkhhoo¶¶nngg

)

t

2

86400.

86400.

=

=

θ

TTõõ ((1166)) ssuuyy rraa ((1177))

( t α − 1 2

T ∆ T 2

5

((1188))

trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008 TTHH22:: kkhh¸¸cc ®®éé ccaaoo (( kkhhii gg## ccoonnsstt ,, nnhhiiÖÖtt ®®éé vvµµ cchhiiÒÒuu ddµµii == ccoonnsstt))

2

=

=

2 . π

2 . π

T 0

l g

. l R . G M

0

++))ëë mmÆÆtt ®®ÊÊtt ®®åånngg hhåå cchh¹¹yy ®®óónngg :: ((1199))

2

)

2 . π

2 . π

=

=

T h

TTrroonngg ®®ãã gg00 llµµ ggiiaa ttèècc rr¬¬ii ttùù ddoo ëë ggÇÇnn mmÆÆtt ®®ÊÊtt .. MM,, RR llµµ kkhhèèii ll−−îînngg vvµµ bb¸¸nn kkÝÝnnhh ttrr¸¸ii ®®ÊÊtt .. ll llµµ cchhiiÒÒuu ddµµii dd©©yy ttrreeoo ccoonn ll¾¾cc.. aa)) ëë ®®éé ccaaoo hh cchhuu kkúú ccññaa ccoonn ll¾¾cc

l g

.( l R h + . G M

h

g

=

0

((2200))

g

=

h

2

GGiiaa ttèècc ttrräänngg ttrr−−êênngg ëë mmÆÆtt ®®ÊÊtt ((2211))

(

)

.G M 2 R . G M R h +

GGiiaa ttèècc ttrräänngg ttrr−−êênngg ëë ®®éé ccaaoo hh :: ((2222))

h

h

1

=

=

<

R

g g

R R h +

0

SSuuyy rraa :: ssuuyy rraa TT00<

O

T 0

T h

1

=

=

<

− T h

CChhuu kkúú tt¨¨nngg ®®åånngg hhåå cchh¹¹yy cchhËËmm §§éé bbiiÕÕnn tthhiiªªnn tt−−¬¬nngg ®®èèii ttrroonngg mmççii cchhuu kkúú :: T h ∆ R h T + h

6

MMççii nnggµµyy ®®ªªmm ®®åånngg hhåå cchh¹¹yy cchhËËmm mméétt kkhhoo¶¶nngg ::

trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008

86400.

86400.

=

=

θ

h R h + ((2233))

T ∆ T h

2

=

=

2 . π

2 . π

'

T h

') '

l g

'

h

'

g

=

h

2

((2244)) TTrroonngg ®®ãã gghh’’ llµµ ggiiaa ttèè bb)) ëë ®®éé ssaauu hh’’((ssoo vvííii mmÆÆtt ®®ÊÊtt)) cchhuu kkúú ccññaa qquu¶¶ ll¾¾cc :: .( l R h − . G M

' ')

. G M R h −

( ll−−îînngg ccññaa pphhÇÇnn ttrr¸¸ii ®®ÊÊtt ggiiííii hh¹¹nn bbëëii mmÆÆtt ccÇÇuu ccãã bb¸¸nn kkÝÝnnhh ((RR--hh’’))

g

=

0

ttrräänngg ttrr−−êênngg ëë ®®éé ss©©uu hh’’ ((2255)) MM’’ llµµ kkhhèèii

.G M 2 R

GGiiaa ttèècc ttrräänngg ttrr−−êênngg ëë mmÆÆtt ®®ÊÊtt :: ((2266))

2

3 ') .

. ( π

ρ

R h −

'

2

'

4 3

.

.(

)

=

=

'

'

  

  

g h g

M M

R R h −

R R h −

0

.

3 . . R π ρ

4 3 ((ëë ®®©©yy vv×× ttrr¸¸ii ®®ÊÊtt hh××nnhh ccÇÇuu nnªªnn kkhhèèii ll−−îînngg ®®−−îîcc ttÝÝnnhh nnhh−− ttrrªªnn))

'

'

=

TTõõ((2255)) vvµµ ((2266)) ssuuyy rraa

hg g

R h − R

0

'

2

'

)

1

1

=

=

=

=

−

1 ≈ −

<

'

. ' G M .( . G M R h

g h g

R h − R

' h R

' h 2. R

0

'

HHaayy ((2277)) llÊÊyy ((1199)) cchhiiaa cchh00 ((2244)) vvÕÕ tthheeoo

7

vvÕÕ vvµµ ®®ÓÓ ýý ®®ÕÕnn ((2277)) ttaa ccãã :: T R 0 T − h

trÇn quang thanh-k15-ch-lý-®h-vinh-8-2008

1 2

1

1 ≈ ±

ε

( 1 ± = ±

) ε

ε 2

T

T

0

'

h

=

=

(( DDoo ¸¸pp ddôônngg cc««nngg tthhøøcc ggÇÇnn ®®óónngg ::

' h 2 R

− T

'

'

h

h

T

86400.

86400.

=

=

θ

VVËËyy mmççii nnggµµyy ®®ªªmm SSuuyy rraa TT11<

' h 2 R

'

∆ T h

®®åånngg hhåå cchh¹¹yy ssaaii mméétt kkhhoo¶¶nngg ::

1 ≈ +

CChhóó ýý :: cc¸¸cc cc««nngg tthhøøcc ggÇÇnn ®®óónngg ssöö ddôônngg ttrroonngg bbµµii ::

1ε< ≤ tthh××

1 1

ε ε 2 1 − 2 2

+ +

ε 1 ε 2

1 2

1

1 ≈ ±

ε

( 1 ± = ±

) ε

vvííii 0

ε 2

n

n ε

1 ≈ ±

hhooÆÆcc ::

∓

1

≈

hhooÆÆcc:: ( 1

ε 2

1

≈ + −

hhooÆÆcc ::

' ε ε 2 2

hhooÆÆcc ::

) ± ε 1 1 1 1

± ε ε + ' ε −

1

. 1

ε

ε

+

+

' 1 ≈ + +

' ε ε 2 2

8

hhooÆÆcc ::