Solución de Problemas de Conservación de La Energía

 

4.

]k ]kau aumi mikg kgdr drik ik b bcc _r _rkh khac ac`d `dss bc mmkg kgsc scrv rvdm dmiý iýg g bc aad d cg cgcr crfà fàd5 d5

_rkhac`d5

Sg d`krtifudbkr bc dutk`ývia mkgsistc cg ug miaigbrk aacgk bc dmcitc bc bcgsibdb  τ 0

( ) lf

9 C 8

`=

  ,

bkgbc c`hkgd ug pistýg quc ticgc ug krijimik, mk`k sc iaustrd cg ad jifurd. Ads ärcds trdgsvcrsdacs 8 8 8 bca miaigbrk, bca cec y bca krijimik skg  D p 043 m` , D ] 08 m`  y D ?04.? m` .Mdamuac ad jucrzd J rcqucribd pdrd c`puedr ca pistýg bcgtrk bca miaigbrk d ugd vcakmibdb Z  p04.?

Bdtks5

 τ 09?? lf / `  D p 043 m`  D s08 m`

=

8

8

8

 D? 04 m` 4

Z  p0 ` / s ]kaumiýg5

J

Bijcrcgmid bc prcsikgcs5

 J 0( _  _∘ _ 8) ( D p∘ D? )   ...(4) Hcrgkuaai5 8

 _ Z 8  _ 8   ... (8) 0 +  τf 8 f  τf Mkgscrvdmiýg bc ad `dsd5

` ˙ 0Z 8 τ D ?0Z  p ( D p ∘ D ? ) τ ↘ Z 80 Z  p Bc (4), (8) y (=) sc khticgc5

( D » » p ∘ D ) ?

 D?

» …(=)

 ` . s

 

 τ 8 ( D  p∘ D? )  J 0  Z  p 8 8  D ?

=

 

_rkhac`d.  Cg ug cxpcri`cgtk `ébimk sc bcscd igycmtdr räpibd`cgtc ugd pcqucôd mdgtibdb bc jauibk rdbikdmtivk cg ad vcgd bc ug dgi`da. Ca fdstk vkau`étrimk T bc ad ecrigfd quc sc iaustrd cg ad jifurd scrä5

 Ϗt  T0T ? scg   < ? ≢t ≢ς  ς  »??%, csti`c ad sdaibd bc  pktcgmid caémtrimd. ]kaumiýg5

` ˙ 0 τ ∏ 04???∖4?? 04????? 8

lf s

8

Z 4  _4 Z   _ + + z 4 + n pu`p 0 8 + 8 + z 8 + nturhigd+ na 8 f  τf 8 f  τf nturhigc 0 z 4∘na

˙ 0`˙ f ( 48?∘=3 )04????? x 9.>4 x >3 0» >=.2@V  Cturhigc Fcgcrdmiýg caémtrimd 0 ?.>*>=.2 @V 0 ::.6 @V

 

_rkh. 2.4> (Ed`cs D. Jdy) ]kaumiýg5 _adgtcd`ks pdrd ca rcqucri`icgtk (d) ad cmudmiýg bc ad mkgscrvdmiýg bc ad `dsd yd quc gks skaimitd ad vcakmibdb bca dirc d trdvés bc ad pucrtd. Mkgscrvdmiýg bc ad `dsd pdrd ca dirc5

b

❔

∠ ❔ ∖b 0 ˙` ∘` ˙ d

i

k

❔

bt 

❔

b bt 

❔ ∖b 0∘ τ ZD ∘ j  `˙ ∠ ❔ d

d

b ∏d

∘ τ d ZD ∘j  `˙

bt 

 τd

 0

( ?)

Mkgscrvdmiýg bc ad `dsd pdrd aks prkbumtks bc ad mk`hustiýg5 ❔

b bt 

❔ ∖b 0`˙ ∘`  ˙ ∠ ❔

 τ p

 p

i

k

b ∏ p bt 

  0`  ˙ ( 4 + j )

b ∏ p

 `˙ (4 + j )

bt 

 τ p

 0

( ?)

∏ p + ∏d 0M 

 

b ∏ p

∘b  ∏ d

bt 

bt 

 0

Ad cmudmiýg (=) sc rcc`padzd cg ad cmudmiýg (8)5

( ?)

 

b ∏d

∘ ˙` (4 + j )

bt 

 τ p

 0

( ?)

Ad cmudmiýg (2) sc rcc`padzd cg ad cmudmiýg (4)5

 ˙ (4 + j ) ∘ τd ZD ∘ j  `˙ ∘` 0  τ p

 τ d

Z 0

\

 `˙  ( 4 + j  ) ) ∘  j 

 D

( ?)

 τ p

 P

( ?)

 τ d

 h) Dpaimd`ks Hcrgkuaai Hcrgkuaai pdrd ca jauek bc dirc5

 8  _ Z   _ d 0 +  τd 8  τ d

( ?)

Igscrtd`ks ad cm. (:) cg (6).

 _∘ _ d0

 \

 τd ` ˙ 8  D

 ( 4 + j  ) )   j  ∘  τ p

 P

( ?)

 τd

_rkh. 2.82 (Ed`cs D. Jdy) ]kaumiýg _adgtcd`ks ad cmudmiýg bc Hcrgkuaai pdrd ad aàgcd bc mkrricgtc bcsbc ad pdrtc igjcrikr ndstd ad pdrtc supcrikr bca mnkrrk. 8

8

Z   _ Z 8  _8 +   + f U 80 4 + 4 + f U 4  τ  τ 8 8 8

Z h ( _ d+ τf A= )  _d +   0   + f ( A4 + A8)  τ  τ 8

A4+ A8+ A= ) Z h 0∕ 8 f (  A

 

_rkh. 2.8: (Ed`cs D. Jdy) ]kaumiýg5 (d) _adgtcd`ks ad cmudmiýg bc ad mkgscrvdmiýg bc ad `dsd `dsd pdrd ca tdgquc 45

b

❔

∠ ❔ ∖b 0`˙ ∘` ˙  p

bt 

 τ D t 

i

k

❔

b n4

  0 τT ∘ τZ D p

bt 

(4)

_adgtcd`ks Hcrgkuaai da igfrcsk bc ad tuhcràd trdgsvcrsda5

8

8

Z 8  _8 Z   _ +   + f U 80 4 + 4 + f U 4  τ  τ 8 8

8

nfτ Z   _  0 +  τ  τ 8

(8)

_adgtcd`ks Hcrgkuaai d ad sdaibd bc ad tuhcràd trdgsvcrsda5

8

8

Z   _ Z 8  _8 +   + f U 80 4 + 4 + f U 4  τ  τ 8 8

 _  _k 0  τ  τ

(=)

Bkgbc5 _ cs ad prcsiýg bcgtrk bc ad tuhcràd. _? cs ad prcsiýg d ad sdaibd bc ad tuhcràd cg ca tdgquc 8

 _?0 νfτ  0 _

( 2)

Bkgbc  cs ad daturd bc ad mkau`gd bc dfud cg ad sdaibd bc ad tuhcràd trdgsvcrsda cg ca tdgquc 85

 

( 3)

n4 + n 808 n? Xcc`padzd`ks ad cm. (2) cg (8)5 8

Z  8

0

f ( n∘ ν )

( :)

4

n 0n4∘ h ν 0 n8∘h

n ∘ν 0n 4∘n8

n ∘ν 0n 4∘(8 n? ∘n4)< n ∘ν 08 ( n » » 4∘n? ) » ( 6)

n ∘ν 08 ( n » » 4∘n? ) » Xcc`padzd`ks ad cm. (6) cg ad cm. (:)5

Z 0 8 ∕ f ( n4∘ n? )

( >)

Xcc`padzd`ks ad cm.(>) cg ad cm. (4)5

 Dt 

b n4

  0T ∘8 D  p ∕ f ( n ∘n ) 4

bt 

(9)

?

Mudgbk sc damdgzd ca cstdbk cstdmikgdrik sc ticgc5

( )

 T n ∗0 8  D p

8

4

f

( 4?)

+n

?

 h)

b n4

8 D p

bt 

 D t 

  0T ∘

∕ f ( n ∘n ) 4

?

(44)

 

Hakf pcrskgda skhrc Igfcgicràd Gdvda y djigcs5 nttps5//bcggysbcadtkrrc.hakfspkt.mk`/