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FISIKA MATEMATIKA I

CONTOH SOAL DAN JAWABAN PERSAMAAN

DIFERENSIAL DAN PENERAPAN FUNGSI GAMMA, BETA

DAN FAKTORIAL

DISUSUN OLEH

YOLANDA BIDARI DWI SELANI

4143121067

FISIKA DIK D 2014

Fakultas Matemat ka !a" Ilmu #e"$eta%ua" Alam

&" 'e(s tas Ne$e( Me!a"

201)



Contoh Soal dan Jawa an dala! P"#$a!aan D%&"#"n$%al

1* Suatu ta"$k s l "!e( +e(,a( -,a( 2.) m mem/u" a lu+a"$ /a!a alas" a 2) mm* D keta%u +a% a

a ( aka" me"$al ( melalu lu+a"$ !e"$a" ke e/ata" me"!ekat v= 2,5 √ h m / s . % a!ala%

ke!alama" a ( !alam ta"$k * a( la% aktu a"$ ! /e(luka" u"tuk me"$ s "$ka" ta"$k melalu

lu+a"$ te(se+ut*

#e" elesa a" 5

lume a ( a"$ me"$al ( kelua( /e( !et k ! m salka" se+a$a ' lume s l "!e( !e"$a" 2) mm

!a" t "$$ " a v* De"$a" +e$ tu. maka ' lume a ( a"$ kelua( /a!a saat !t !et k a!ala% 5

2,5 ¿2

dhataudt =− ( 2,50,025 )

2 dh2,5 √ h =− 4000

dh√ h

0,025 ¿2 (2,5 √ h)dt =− π ¿π ¿

a"ta(a t 0. % 3 !a" t t. % 0*

∫0

t

dt =− 4000 dh√ h

=− 4000 ∫3

0 dh√ h

= [− 8000 √ h]30= 8000 √ 3 detik

¿3 jam 34 detik

2* Bal k es !e"$a" +e(at 10 k$ melele% !alam l "$ku"$a" a"$ tem/e(atu(" a "a k* La,u

/e"$u(a"$a" +e(at es /e( !et k a!ala% se+a"! "$ !e"$a" 20 ! ku(a"$ +e(at es a"$ te(s sa*

Setela% 60 !et k. +e(ate s a!ala% 8.) k$* Be(a/ +e(ate s setela% 120 !!et k 9La"$ka% 1 : Susu" /e(samaa" ! ;e(e"s al" a 5

( ) ( ) ( ) ).860.10020 ==×−= M M M k dt

dM

La"$ka% 2 :I"te$(alka" 5

La"$ka% 3 : <a! ka" M se+a$a su+,ek 5ckt ckt e M e M ckt M ++ −==−=+=− 20.20.20l"

La"$ka% 4 : Susu" kem+al /e(samaa" M !e"$a" k "sta"ta a"$ +e(sa"$kuta" 5.20.20 kt ckt Ae M ee M −=−=!e"$a"

ce A =

La"$ka% ) : a( " la k "sta"ta

( )∫ ∫ −=−−−= dt k M dM M k dt dM 2020

ckt M +−=−− 20l"



=u"aka" " la k "! s a al 5 M(0)=10, M(60)=9,5102010 0 =−= A Ae

0).1020).8 60 =−= k Ae

0).1l"60 =k . k = 0,000813

maka

t e M 000>13.01020 −=

La"$ka% 6 : ?e"tuka" s lus " a 5( ) 120000>13.0000>13.0 1020120.1020 ×−=−= e M e M t

( ) kg M 87).>120 =

3* Be(a/a ke(,a m " mum a"$ ! /e(luka" u"tuk mem+e"tuk /e(mukaa" +e(ke"aa" !e"$a"

t t k ka+ut a ( !e"$a" ,a( -,a( )@m 9* A"$$a/ +a% a 0.001m 3 a ( /a!a )0 * te$a"$a"

/e(mukaa" a ( /a!a )0 a!ala% 0.063 Nm -1*

#e" elesa a" 5

Ras /e(mukaa" te(%a!a/ ' lume se+ua% tetes a tu 5

( ) r r

r

V

A 3

3A4

43

2

==π

π

De"$a" r (a! us + la* &"tuk ' lume 0.001 m 3 !a" ,a( -,a( ) 10 -6 m

( ) 26 600

10)001.033

m xr

V A === −

( )( )∫ −=−=−=−= − A J m Nm AdAW

0

21 >.37600063.0γ γ

<a! . ke(,a 3.7> < %a(us ! lakuka" /a!a a (*

4* De"$a" me"$a/l kas ka" %ukum $e(ak Ne t " a"$ ke!ua /a!a t(a"las se+ua% +e"!a

te$a(. $am+a(ka" asal mula suku-suku e"e($ k "et k !a" e"e($ / te"s al !alam /e(samaa"

e"e($ u"tuk se+ua% s stem meka" s mu(" *C

#e" elesa a" 5



ukum Ne t " ke!ua u"tuk s stem " 5

dt du

mma F ==

De"$a" u ke e/ata" +e"!a !a" F $a a lua( t tal a"$ +eke(,a /a!a +e"!a /a(a(el te(%a!a/

/e(/ "!a%a"" a. !ll* Maka ke(,a t tal a"$ ! lakuka" le% +e"!a tu u"tuk $a a te(se+ut

a tu 5

∫ −= Fd W

∫ ∫ ∫

−=×−=−= 2

2mud dumud dt

dumW

! " mu

W ∆−≡

∆−=

2

2

#e(samaa" ! atas me(u/aka" /e(" ataa" umum a"$ sem/u("a +a$ ke(, meka" k t tal

a"$ ! lakuka" le% +e"!a te$a( !alam t(a"slas !a" /e(samaa" te(se+ut t !ak +e(!asa(

/a!a a/a/u" me"$e"a s ;at !asa( $a a F * Namu". F ! a"$$a/ !e"$a" +a k se+a$a

/e",umla%a" !ua ,e" s $a a. $a a +e"!a F # !a" $a a /e(mukaa" F $*

F = F # % F $

=a a +e"!a ! se+ut +e$ tu ka(e"a +eke(,a ! selu(u% ' lume s stem. $a a /e(mukaa"

+eke(,a /a!a !ae(a% /e(mukaa" +atas s stem* Da( /e(samaa" ! atas. maka ke(,a t tal !a/at

! a"$$a/ se+a$a ,umla% ke!ua +e"tuk ke(,a 5

W = W # % W $

De"$a"

∫ −= d F W # #

∫ += d F W$ $



=a a +e"!a me(u/aka" $a a k "se('at ;* I" +e(a(t $a a te(se+ut !a/at ! tu(u"ka" !a(

se+ua% ;u"$s / te"s al ( ) . a"$ +e($a"tu"$ %a" /a!!a /a!a l kas s stem. me"u(ut

/e"u(u"a" te(%a!a/ k (! "at / s s * <a! . u"tuk kasus " 5

( )d

d F #

Φ−=

( ) ∆Φ=Φ= Φ−−= ∫ ∫ d d d

d W #

Ka(e"a. +e!a %a" a +e($a"tu"$ /a!a / s s a al !a" ak% ( s stem. !a" t !ak /a!a ,alu(

a"$ ! kut ! a"ta(a ke!ua / s s " . maka ke(,a a"$ ! lakuka" u"tuk $a a +e"!a t !ak +e($a"tu"$ /a!a ,alu(* De"$a" me"e"tuka" e"e($ / te"s al " & se+a$a " & G . maka !a/at

! tul s se+a$a 5

W # = '" &

=a a /e(mukaa" se a(aa umum t !ak k "se('at ; !a" + asa" a me"ul s /ee(" ataa" u"tuk

ke(,a a"$ ! lakuka" te(%a!a/ $a a-$a a !em k a"* K m+ "as !a( /ee(samaa" ! atas

!e"$a" /e(samaa" se+elum" a mem+e( ka" 5

$ & W " W +∆≡

#e(samaa" ! atas me(u/aka" /e(" ataa" !alam +e"uk la " !a" !a( ke!ua /e(samaa"

te(se+ut !a/at ! te(a/ka" /a!a /( ses a"$ sama* De"$a" melakuka"" a !!a" me" usu"

ula"$. maka ! !a/atka" /e(samaa" e"e($ 5

( ! $ " " W ∆+∆=−

A/a+ la t !ak a!a $a a. maka /e(samaa" ! atas te(e!uks 5

0=∆+∆ ( ! " " atau

ta"k)*$ " " &k =+

Ya"$ le+ % ! ke"al se+a$a H/( "s / kekekala" e"e($ meka" ka klas k* Ist la% ke(,a /a!a

%ukum te(m ! "am ka + asa" a. te"tu sa,a +uka" se+ua% +e"!a te$a(*



)* #e(samaa" ! ;e(e"s al a"$ me"$%u+u"$kaa" teka"a" % !( stat k !e"$a" ke!alama" ;lu !a

. gd+ d( ρ −= !e"$a"

ρ ke(a/ata" massa l kal. $ /e( e/ata" $(a' tas l kal !a" d+

ket "$$ a" ! atas /e(mulaa" !asa( ;lu !a* ?e(a/ka" /e(samaa" " u"tuk me" usu"

/e(" ataa" +a$ teka"a" atm s; ( se+a$a ;u"$s ket "$$ a"! atas /e(mukaa" +um *

#e" elesa a" 5

Ka(e"a su%u !a" teka"a" ! se(taka. /a!a !asa(" a u!a(a +e(laku se+a$a $as !eal a"$

u"tuk tu V = - * u+u"$a" a"ta(a ρ

!a" V . ' lume m la( u!a(a a tu 5V M A= ρ

.

!e"$a" M massa m la( u!a(a* Su+st tus ke !alam (umus % !( stat k mem+e( ka" 5

d+ V

Mg d( −=

d+ ,- Mg

( d( −=

∫ −= &

& ,- Mg

( d(

00

atau

−=

00 eB/

,- Mg+

( (

De"$a 0 teka"a" /a!a /e(mulaa" !asa(. a tu + " l* #e(%at ka" +a% a $ ! a"$$a/ +e+as

te(%a!a/ J* #e(samaa" ! atas ! ke"al se+a$a /e(samaa" +a( met( !a" !a/at ! /aka

la"$su"$ a/a+ la - t !ak +e($a"tu"$ /a!a J* k "! s " ! !a/at ! !ae(a% atm s; ( a"$

! ke"al !e"$a" st(at s;e(. /a!a ket "$$ a" a"ta(a 11 km !a" 2) km*

( d(

- d-

−=1γ

γ

! ,

Mg d+ d- ≡

−−=

γ γ 1



!+ - d+

! ( d(

+

−=01γ

γ

∫ ∫ + −= + ( & !+ -

d+ ! ( d(

00

0 1γ γ

+=

00 1

- !+

( (

K ta tela% me"!a/atkaa" !ua /e(" ataa" ! atas !e"$a" /e(samaa" se+elum" a a"$

+e(!asa( /a!a !ua ekst( m 'a( as su%u a"$ mu"$k " ! atm s;e( * #e(samaa" em/ ( s a"$le+ % umum a"$ me"$%u+u"$ka" ? !a" # a!alaa% 5

( ) δ δ 1

00

−

=

- -

( )1A

0

0 1−

+=δ δ

- !+

( (

Maka 5

−−==

, Mg

d+ d-

k δ

δ 1

6* Massa a ( a ( se+esa( m =5 kg a"$ mula-mula +e(a!a !alam keset m+a"$a" te(mal

!e"$a" atm s;e( /a!a 20 a"$ mula-mula +e(a!a !alam keset m+a"$a" te(mal !e"$a"

atm s;e( /a!a 20 sam/a 4 . ! ! "$ "ka" /a!a teka"a" k "ssta" sam/a 4 !e"$a"

memaka / m/a kal ( a"$ +eke(,a a"ta(a a ( !a" atm s; (* Be(a/a ke(,a m " mum a"$

! /e(luka" 9 &"tuk a ( am+ llaa% . &= /,18/ kJ kg 1 C -1

#e" eles a" 5



Ke(,a m " mum ! /e(luka" , ka /( es " (e'e(s +el !a" k ta !a/aat mem+a a"$ka"

se(a"$ka a" / m/a-/ m/a kal ( (e'e(s +el a"$ +eke(,a u"tuk me"$elua(ka" kal ( !( a (

/a!a +e(+a$a le'el su%u sela$ a ( me"! "$ " !a( 20 sam/a 4 !a" mem+ua"$ kal (

ke atm s;e( /a!a ? 0 20 273.1) 283.1) K* Set a/ / m/a kal ( me"$elua(ka" se+a$ a"

kal ( !a" me"$u(a"$ su%u a ( se a(a +a$ a" /e( +a$ a"* &"tuk se+ua% / m/a kal (

(e'e(s +el mem+e( ka" 5

1−=−=.

2

.

. 2

. - -

- - -

3

W

Seka(a"$ " tas - 2 me",a! - 0 !a" - me",a! -, su%u a ( se+a$a tam+a%a" .

/e"$elua(a" se+a$ a" ,umla% kal ( a"$ !e"$a" ! lakuka"" a se+a$ a" ,umla% ke(,a

W u"tuk su%u ? te(te"tu. meme(luka" /e(samaa" a"$ ! tul s !alam +e"tuk ! ;e(e"s al*

<a! .

10 −=- -

3

W

δ δ

3-

3- W δ δ δ −= 0

NN0 2 4 - W ∆−∆=

De"$a" S !a" me(u/aka" /e(u+a%a"-/e(u+a%a" s ;at a (* Ke!ua" a ! % tu"$

me"u(ut '2 =m. & '- a"$ mu" ul a/+ la . & k "sta" !a"

1

2

l"N - -

m. 4 &=∆

( )−−= 121

20 l" - -

- -

- m. W &

! !- - 1).2771).27341).2831).27320 21 =+==+=



( )( ) kJ W >1.8161).2831).277

l"1).2831>4.4) −=+

=

7* ?e"tuka" aktu a"$ ! +utu%ka" u"tuk me"! "$ "ka" +e"!a /a!at a al" a ! >0

u"tuk > " ! tem/atka" !alam lema( es !e"$a" u!a(a "te( ( ! /e(ta%a"ka" /a!a

) * < ka k e; s e" P 0.002 m 2 s !a" + !a"$ k tak a"ta(a /a!at !a" u!a(a ! "$ " !

!alam lema( es a!ala% A 0.2 m 2

#e" elesa a" 5

D keta%u 5 ? >0 . > . !a" )

P 0.002 m 2 s

A 0.2 m 2

D ta" a 5 ?e"tuka" aktu a"$ ! +utu%ka" u"tuk me"! "$ "ka" +e"!a /a!at9

<a a+ 5

De"$a" me"$$u"aka" /e(samaa" ! ;e(e"s al (!e /e(tama maka ! !a/atka"

dT (t )dt

=− (0,002 ) (0,2 ) [T (t )− 5]=− 0,0004 [T (t )− 5 ]

De"$a" k "! s

?Q0 >0

Maka aka" me",a!

dT (t )[T (t )− 5 ]=− 0,0004 dt

De"$a" me"$ "te$(alka" ke!ua s s 5



dT (t )[T (t )− 5 ]= ¿− 0,0004 ∫ dt +c

∫ ¿

Maka akan didapatkan

T (t )− 5 = e−0,0004 t +c 1= c e−0,0004 t

T (t )= 5 +75 e− 0,0004 t

< ka /a!a saat su%u > aktu a"$ ! +utu%ka" a!ala%

T (t )= 8 = 5 +75 e− 0,0004 t

Maka ! !a/atka" aktu" a a!ala% t >047 s

>* Se (a"$ /e"e(+a"$ la a"$ ! le"$ka/ !e"$a" /a(asut !a" /e(alata" /e"t "$ la "" a

u"tuk me"!a(at ! /e(mukaa" ta"a%* ? tal +e(at /e"e(,u" !a" /e(alata"" a a!ala% 160

l+ * Se+elum /a(asut ! +uka. ke e/ata" a"$ " 12

v . ! ma"a ' a!ala%

ke e/ata"Q;eet !et k * #a(asut ! +uka setela% ) !et k /e"e(,u" te(se+ut te(,u" !a( atas*

Setela% /a(asut ! +uka. ke e/ata" u!a(a me",a!58

v2 . tu"$ ke e/ata" /e"e(,u"

QA se+elum /a(asut ! +uka !a" B setela% /a(asut ! +uka C

#e" elesa a" 5F (mula 5 K ta mem l % a s / s t ; a!ala% 'e(t al me"u(u" !e"$a" t t k asal a!ala%

t t k ! ma"a /e"e(,u" /a!a / s s a al

* A! ke e/ata" /e"e(,u" QA se+elum /a(asut ! +uka

1* F1. +e(at 160 l+. ka(e"a +e($e(ak tu(u" se% "$$a +e(" la / s t ;*



2* F2. ke e/ata" a"$ " 12

v . ka(e"a a"$ " +e($e(ak "a k se% "$$a +e(" la

"e$at 'e

Q− 1

2 v¿

ukum II Ne t "

F m a

F1 F 2 m a ! ma"a m wg

; g = 32

)dvdt

= 160 − 12

v

dvv− 320

= − 110

dt

l" Q'-320 − 110 t +c

' 320 e110

ka(e"a saat se+elum /a(asut ! +uka. t 0 . 'Q0 0 se% "$$a -320

<a! ke e/ata" /e"e(,u" QA se+elum /a(asut ! +uka a!ala% ' 320Q1- e1

10 ¿

Waktu sam/a saat /a(asut ! +uka a!ala% 0 t )

Ke e/ata" saat /a(asut ! +uka a tu saat t )

' 320Q1- e1

10 ¿≈ 126

* A! ke e/ata" /e"e(,u" QB setela% /a(asut ! +uka



F1 160

F2

− 58

v2

Se% "$$a ! /e( le% #e(samaa" D ;e(e"s al" a 5

)dvdt

= 160 − 58

v2

dv

v2− 256= − 1

8 dt

v− 16v+16

=− ¿ 18

t +c

132

ln ¿

v− 16v+16

=− ¿ 4 t +c

ln ¿

v− 16v+16

= ce− 4 t

v= 16 (c e−4 t +1)1− ce− 4 t

Ka(e"a /a(asut ! +uka ) !et k setela% /e"e(,u" tu(u". ! ma"a ' ' 1 saat t )

se% "$$a ' ' 1 ≈ 126 maka c= 110142

e 20

<a! ke e/ata" /e"e(,u" setela% /a(asut ! +uka Qt T ) a!ala%

v=16 (110

142 e20 − 4 t +1)

1 − 110

142 e20 − 4 t



8* Se (a"$ te"ta(a /e"e(,u" /a u"$ !e"$a" mem+a a amu" s mem l k +e(at se+esa(

322 N mel m/at !a( /esa at te(+a"$ /a!a ket "$$ a" 10000 kak !e"$a"

me"$a+a ka" a(a% $e(ak a"$ " asums ka" +a% a %am+ata" u!a(a RQt a"$ /e"e(,u"

/a u"$ alam a!ala%v (t )¿2

R(t )= c ¿ ! ma"a k e; s e" 1) te"tuka"la% 5

a* #e(samaa" a"$ te/at u"tuk ke e/ata" tu(u" seket ka /e"e(,u" /a u"$ +* Fu"$s !a( ke e/ata" /a!a saat me"u(u" 'Qt

* Waktu a"$ ! +utu%ka" u"tuk me"!a(at!* Dam/ak ke e/ata" saat me"!a(at

#e" elesa a" 5

a* #e(samaa" a"$ te/at u"tuk ke e/ata" tu(u" seket ka /e"e(,u" /a u"$

#a!a /e" elesa a" s al " ! $u"aka" /e(samaa" ! ;e(e"s al u"tuk ke e/ata" 'Qt

D keta%u 5

322 Nv( t )¿2

R(t )= 15 ¿

m 322 10 32.2 k$

D ta" a 5 #e(samaa" u"tuk ke e/ata" tu(u" sesaat9

<a a+ 5

Ke e/ata" tu(u" sesaat. 'Qt !a/at ! /e( le% !e"$a" me"$$u"aka" /e(samaa" se+a$a

+e( kut 5

v(t )¿2

¿15 ¿

dv (t )dt

+¿

Atau !a/at ,u$a ! tul ska" !e"$a"

v(t )¿2

10dv (t )

dt = 322 − 15 ¿



De"$a" k "! s 'Q0 0

+* Fu"$s !a( ke e/ata" /a!a saat me"u(u" 'Qt

#e" elesa a" /e(s ala" + me"$$u"aka" k "! s 'Q0 0

vt = 4,634 (e 13,9 t − 1 )e 13,9 t +1

* Waktu a"$ ! +utu%ka" u"tuk me"!a(at

<a(ak a"$ ! +utu%ka" le% /e"e(,u" u"tuk tu(u" a!ala% se/e(t /e(samaa" +e( kut5

xt =∫0

1

v (t )dt =∫01

4,634 (e13,9 t

− 1 )e 13,9 t +1

dt

Setela% me"!a/atka" /e(samaa" ,a(ak" a maka k ta !a/at me"$eta%u aktu a"$

! +utu%ka" le% /e"e(,u" /a u"$ u"tuk sam/a ! ta"a% !a( ket "$$ a" 1000 kak

!* Dam/ak ke e/ata" saat me"!a(at

Setela% me"$eta%u aktu a"$ ! +utu%ka" /e"e(,u" u"tuk me" a/a ta"a% maka k ta

!a/at me"$eta%u ke e/ata" /e"e(,u" saat me"!a(at ! ta"a% !e"$a" /e(samaa"

se+a$a +e( kut 5

vt g=4,634 (e13,9 t g− 1 )

e13,9 t g+1

10* u+u"$a" s ;at !asa( u"tuk sel elekt( k m a. a tu 5d6 (dV -d$d7 t t t ε +−=

De"$a" & t. S t !a" t s ;at-s ;aat t tal sel.

ε em; sel Qte$a"$a" sel (e'e(s +el . !" U

muata" sel* ?ul ska" !am/aak !a" ( te( a kesakka" !a( %u+u"$a" !asa( u"tuk sel

elekt( k m a te(se+ut C

#e" elesa a" 5

!e"$a" /eme( ksaa" k ta /e( le% 5



6V t

t

t 4 7

- .

∂∂=

4 t

t

t V 7

.

∂∂−=

t t V 4

t

67

.

∂∂=ε

kemu! a" 5

6V t

64 t

t t 4 (

V -

..

∂∂=

∂∂

6V t

V 4 t t t 4 6

-

..

∂∂=

∂∂ ε

64 t

V 4 t t t V 6

(

..

∂∂−=

∂∂ ε

11* a( la% se+ua% /e(samaa" a"$ %a(us ! /e"u% le% sem+a(a"$ ;a t ( "te$(as l Q .

u"tuk /e(" ataa" ! ;e(e"s al t !ak eksak( ) ( )d8 8 x N dx 8 x M + .. +=δ

C

#e" elesa a" 5

Ka(e"a I 9Nd8 9Mdx + +=δ

me",a! eksak.

( ) ( ) x

9N 8

9M

∂

∂=∂

∂

Eks/a"s me"u(ut atu(a" /e(kal a" !a" /e" usu"a" ula"$ mem+e( ka" /e(samaa"

! ;e(e"s al a"$ ! a( u"tuk 5

9 x N

8 M

8 9

M x 9

N

∂∂−∂

∂=∂∂−∂

∂

12* De"$a" memaka !e; " s . maka !a/at ! tul s se+a$aVd( d2 3 :*gk −=*δ

*/e(" ataa" " t !ak eksak* De"$a" me"$a"$$a/ se+aa$a ;u"$s !a( ? !a" #.



a( la% se+ua% /e(" ataa" u"tuk

( *- (

2 ∂∂ =u"aka" ke" ataa" +a% a 1 ? me(u/aka"

se+ua% ;akt ( "te$(as u"tuk:*gk 3δ

*

#e" elesa a" 5

#e(kal a" /e(" ataa" ! atas !e"$a" 1 ? mem+e( ka" 5

d&- V

- d2

-

3d4 :*gk −==

δ

?eta/ me(u/aka" ;u"$s !a( ? !a" #. se% "$$a

d- -

2 d&

( 2

d2 ( -

∂∂+

∂∂=

Maka ! /e( le% 5

Ndt Md&d- -

2

- d&

-

V

(

2

- d4

( -

+=

∂

∂+−

∂

∂= 11

Kemu! a". ka(e"a 5

22

2 111

- V

- V

- - 2

- ( - 2

- - M

( - (

+

∂∂−

∂∂−∂∂

∂=

∂∂

!a"- (

2 - (

N

- ∂∂∂=

∂∂ 21

Me" "ta a$a( 5

011

22 =+

∂∂−

∂∂−

- V

- V

- ( 2

- ( -

atau

V - V

- ( 2

( -

+

∂∂−=

∂∂

13* Kea!aa" te(m ! "am k +ata"$ a"$ te(teka" se a(a aks al !e"$aa" ' lume k "sta"

!a/at ! $am+a(ka" le% t $a k (! "at ?.ε σ .

!e"$a" σ st(ess !a" ε st(a " alam *



Dua !a( k (! "at " a"$ ma"a/u" !a/at ! am+ l se+a$a 'a( a+el +e+as*

"k$&a*$:v:ta$ :*earα

!a" M du u$ ; u*g " mate( al ! !e; " s ka" se+a$a 5

σ

ε α

∂∂≡

- -

"

∂∂≡

ε σ

a( la% se+ua% %u+u"$a" a"ta(aα

!a" E C

#e" elesa a" 5

De"$a" me"$am+ lσ = x

. =-,ε = +

<

σ ε

ε ε σ σ

∂∂

∂∂−=

∂∂

- - -

Atau 5

α σ

ε

" -

−=

∂∂

14* E"tal/ !a/at ! %u+u"$ka" !e"$a" e"e($ = ++s !a" tu(u"a"" a te(%a!a/ su%u

melalu /e(samaa" :>>$?2e m@ t+ * Bukt ka" +a% a 5

( dt

- =

- - =

- = 2 ∂

−=

∂∂−−=

Q2

#e" elesa a" 5

-4 = 2 −=

( - =

4

∂∂−=

1)* ?e"tuka" s lus , ka se+ua% +ata"$ mema",a"$ !a( 0 sam/a V . su%u a al" a uQ .t

;Q Q0 ( V !a" s a(at +atas" a ! u,u"$ k ( a!ala% uQ0.t 0 QtT0

#e" elesa a" 5



Ka(e"a +e(k sa( a"ta(a 0 sam/a V. se% "$$a ;Q0 uQ0.0 0 !a" ,u$a k ta aka"

mem/e( le% 5

Fs Qut ∂

^

u s

∂ t 2

F Qu - 2 2

Fs Qu - 2 2 ^

us Q .t

S lus /e(samaa" ! ;;e(e"s al (!e /e(tama " a!ala%

^

us Q .t Q e− c2 w2 t

De"$a" s a(at a al

uQ .0 ;Q k ta mem/e( le%^

us Q .0 ^

f s Q Q !e"$a" !em k a" 5

^

us Q .t ^

f s Q e− c2 w2 t

De"$a" me"$am+ l t(a"s; (m s klus ke+al ka" !a" me"su+st tus ka"

Fs Q √ 2π ∫0

∞

f ( p )sin dp

K ta mem/e( le% s lus a"$ k ta a( 5

uQ .t 2π 0

∞

f ( p)sin e c2 w2 t sin wxdp dx

16* Meka" ka stat st k me" e! aka" se+ua% %u+u"$a" a"ta(a /e"$$am+a(a"

te(m ! "am k Qmak( sk / k suatu s stem le at ;u"$s /a(t st X* Satu /e(umusa"

meka" ka satt sst k u"tuk se+uaa% s stem # ? te(tutu/ mem+e( ka" /e(" ataa"

+e( kut u"tuk e"e($ elm% ltJ 5 A= - *

De"$a" R k "sta"ta $as u" 'e(sal !a" X se+ua% ;u"$s !a( ? !a" sa,a* a( la%

/e(" ataa" u"tuk #. S. &. !a" = !alam ?. !a" X*

#e" elesa a" 5



- -

,- V A

(

∂∂=

∂∂−= Xl"

V V ,

- A4 ∂∂+= ∂∂−= ?

l"XR?Xl"

V

,- -4 A7

∂∂=+=

?

Xl"2

- V V ,-V ,- (V 7 2

∂∂+

∂∂=+= Xl"

?

Xl"2

∂∂+−=+= Xl"

R?Xl" ,- (V A=

17* &"tuk se+ua% /( ses (e'e(s +el !alam s stem # ? te(tutu/. tu",,ukka" +a% a 5( )*V (d d*W :: +−=

∑ µ δ

#e" elesa a" 5

&"tuk se+ua% s stem te(tutu/ a"$ t !ak +e($e(ak. %ukum /e(tama mem "ta a$a(

( )*7 d 3W −=δ δ * &"tuk /( ses-/( ses (e'e(s +el. %ukum ke!ua mem "ta

( )*4 -d 3 =δ *

<a! 5

( ) ( ) ( )*V (d d**7 d *4 -d W :: +−=−= ∑ µ δ

Da( /e(samaa" " !a/at k ta l %at +a% a ke(,a sela " ke(,a /emua a" !a/at

! %as lka" !a( /e(u+a%a"-/e(u+a%a" ,umla% m l u"su(-u"su( k m a a"$ a!a* Dalam

se+ua% s stem te(tutu/. /e(u+a%a"-/e(u+a%a" !em k a" %a" a !a/at ! %as lka" !a( (eaks

k m a* Ke(,a te(se+ut /al "$ se( "$ ! u,u!ka" se+a$a ke(,a l st( k !a" s stem te(se+ut

!alam kasus !em k a" a tu sel elekt( k m a*



1>* #a!a !ae(a% 28>->4>K. ka/as tas kal ( kua(sa (4:B C ) /a!a teka"a" atm s; ( ! !ekat

se+a$a 5

( ) ( )- - . ( )3

1032.>1060.44)0.40 ×−×+= −

De"$a" ? !alam Kel' " !a" # !alam <*m l -1*K -1* < ka 1000 k$ kua(sa !! /a"aska"

!a( 300 sam/a 700 K /a!a teka"a" atm s; (. +e(a/a +a" ak kal ( a"$ ! +utu%ka" 9

#e" elesa a" 5

3 2 =∆u"tuk /( ses /a!a teka"a" k "sta"* Sela",ut" a. mem+e( ka" d2=. d-

u"tuk /( ses /a!a teka"a" k "sta"* as l" a 5

∫ = 2

2

-

- ( d- . 3

Su+st tus u"tuk # !a" "te$(as mem+e( ka" /e(samaa" 5

( ) ( ) ( )

−×−−×+−=

−

21

)21

22

3

12

111032.>

21060.44

)0.40- -

- - - - 3

De"$a" me"su+st tus ka" %a($a-%a($a "ume( k su%u. k ta !a/atka" 5

= C3,535 kJ km 1

Kua"t tas S O 2 a tu Q1000 k$ Q60 k$*km l -1 16.67 km l* Ka(e"a kal ( t tal a"$

! /e(luka" a tu Q16.67 Q23)3) 328300*

18* < ka teka"a" ua/ # sat a (a" te(te"tu ! ak l le% se+ua% /e(samaa" a"$ +e(+e"tuk 5

- (

A ( $at −=l"

De"$a" A !a" B k "sta"ta. !a" ? su%u mutlak. tu",ukka" +a% a u"tuk mate( al " 5

2- #(

V 4 $at

vv ∆=∆

#e" elesa a" 5

v

v $at

V 4

d- d(

∆∆=

atau

v $at

v $at

V ( 4

d- ( d

∆∆=l"



( )( )( ) 113

*2.401340

1).661.3>76*)32.0 −−

==∆ kg kJ !

k(a ! kg m 2 v

a($a a"$ ! la/ (ka" a tu /0C,8 kJ kg 1 *

21* Data +e( kut te(se! a u"tuk a ( a ( /a!a 2) !a" teka"a" atm s; ( 5

16102)6 −− °×= . β .

( ) .106.8A 26 −− °×=∂∂ . - ( β

13 .003.1 −= g cmV . ?e"tuka" e;ek

teka"a" /a!a # a tu. % tu"$la% %aa($a-

(

( .

∂∂

u"tuk a ( /a!a k "! s " C

#e" elesa a" 5

#e(samaa" a"$ ! +utu%ka" 5

∂∂+−=

∂∂

( -

(

- V-

( . β β 2

Su+st tus %a($a-%a($a "ume( k mem+e( ka" 5

( )( ) ( ) ( )[ ]261613 106.8102)61).28>*003.1 −−−−− ×+×−=

∂∂

! ! ! g cm ( .

-

(

113 **002>8.0 −−−= ! g cmatau

1114 ***10>8.2 −−−−×− >ar ! kg kJ

<a! . ke"a ka" teka"a" se+esa( 1 +a( /a!a k "! s a"$ ! " ataka"

me"$ak +atka" /e"u(u"a" # a"$ sa"$at ke l* Ka(e"a ka/as tas kal ( a ( /a!a 2)

sek ta( /,18 kJ kg 1 ! 1 . /e"u(u"a" /e(se"tase" a u"tuk ke"a ka" teka"a" se+esa( 1 +a(

a tu %a" a sek ta( 0.007 *



22* a($a-%a($a s ;at u"tuk a (a" ! "$ " atau te(teka" ! % tu"$ !a( %a($a-%a($a s ;at

u"tuk a (a" ,e"u% /a!a su%u a"$ sama le% "te$(as /e(samaa"-/e(samaa" 5

d( - V

- V d2 (

∂∂−=

!a"

d( - V

d4 (

∂∂−=

Se( "$kal !ata u"tuk !a"( ) ( - 7 ∂∂ A

se+a$a ;u"$s !a( !alam !ae(a% a ( t !ak

te(se! a !a" asums ! +uat +a% a %a($a-%a($a " k "sta" /a!a %a($a-%a($a ,e"u%*

Kemu! a" "te$(as mem+e( ka" 5

( ) $at

( (

d-

dV - V 2 2 −

−=−

(- k *$ta*)

( ) $at

( ( d-

dV 4 4 −−=−

(- k *$ta*)

Ka(e"a. ' lume ! a"$$a/ t !ak +e($a"tu"$ /a!a #. tu(u"a" t tal" a ! tul s*

?e(a/ka" /e(samaa"-/e(samaa" ! atas /a!a a ( a (. !e"$a" me"$$u"aka" %a($a-%a($a

s ;at-s ;at ,e"u% !a" % tu"$la% !a" S u"tuk a ( /a!a ? 100 !a" Qa # 2) +a(.Q+ # 200 +a(*

#e" elesa a" 5

#a!a 100 !a( !ata a"$ ! !a/atka" 5

# sat 1.0132 +a( l 418.1 k<*k$ -1

S l 1.3068 k<*k$ -1K -1 l 1.0437 m 3*$-1

11388101 **000>.02

0428.1044).188101

−−=−=−−= ! g cm

V V d-

dV

a Su+st tus la"$su"$ %a($a-%a($a ke !alam Ql mem+e( ka" 5

( )({ } ( )[ ] 113134 ***018.00132.12)*10>1).3730437.1 −−−− −=−×−=− ! g cm>ar >ar g cm 2 2

0,0159 kJ kg 1 ! 1

Da" !a( s tu ! /e( le% S 1.3068 Z 0.0018 1.30)0 k<*k$ -1K -1



+ #( se!u( a"$ sama u"tuk teka"a" 200 +a( me"$%as lka" 51*>.14 −=− kg kJ 2 2

atau 2 = /33,9 kJ kg 1

11

*01)8.0 −−

−=− ! kg kJ 4 4

atau 2 = 1,C910 kJ kg 1 ! 1

as l-%as l Qa !a" Q+ /a!a !asa(" a sesua !e"$a" eks/e( me"* Namu". /( se!u(

te(se+ut aka" t !ak te/at selu(u%" a !alam !ae(a% k( t s. tem/at s ;at-s ;at a (a" +e(u+a%

se a(a e/at te(%a!a/ teka"a"*

23* 0.001 k$ ua/ a ( /a!a 260 . memua se a(a (e'e(s +el !a" s t%e(mal !alam se+ua% /e(a"$kat t (ak !a" s l "!e( !a( ' lume mula-mula 2 L ke ' lume ak% ( 20 L* Be(a/a

+esa( usa%a a"$ ! ke(,aka" le% ua/ a ( tu 9

#e" elesa a" 5

&sa%a (e'e(s +el ! % tu"$ !a( 5

∫ = 2

1

V

V rev (dV *W

De"$a" ' lume m la( ua/ a ( !a" " ,umla% t tal m l ua/ a ( 5

++= 32

1

V .

V #

V ,- (

Su+st tus /e(samaa" ! atas ke !alm Ql . meme+e( ka" 5

++−+=

21

21

21

12

1

2

2l"

V V V V .

#V V

V V V V

*,- W rev

Da( /e(" ataa" s al 5

m) km) * ))48.0000))48.002.1>01.0 ===



11 *604.3

))48.02 −== m) DV 1

12 *04.3610 −== m) DV V

1*1422.0 −−= m) D # 22 *00714.0 −−= m) D.

Su+st tus " la -" la ke !alam /e(samaa" ! atas !e"$a" R >.314 <*m l -1K -1 !a"

? )33.1) K. me"$%as lka" W (e' ))7) <* k ta !a/at meme( ksa kea+sa%a" u"tuk

/e"e(a/a" " !e"$a" me"$% tu"$ teka"a" a al !a" ak% (. !e"$a"

R 0.0>314 L*+a(*m l -1K -1 5

1 = 11,80 >ar C = 1,CC5 >ar

#em+a"! "$a" teka"a"-teka"a" " me"u",ukka" +a% a /e(samaa" ' ( al t $a

suku se a(a keselu(u%a" uku/ mema!a !alam !ae(a% " *

24* &!a(a /a!aa teka"a" 680 k#a !a" su%u 20 memasuk se+ua% / /a % ( J "tal

te(selu+u"$ ua/ a ( !e"$a" ke e/ata" 30ms -1* #e"$uku(a" ! u,u"$ kelua( / /a

me"u",ukka" +a% a u!a(a tu ! /a"aska" sam/a 200 !a" +a% a teka"a"" a 668

k#a* Be(a/a /e(se"tase /e"u(u"a" teka"a" ! !alaam / /a a"$ !a/at ! ka tka"

!e"$a" $eseka" ;lu !a 9 A"$$a/ u!a(a +e(/e( laku se+aa$a suatu $as !eal*

#e" elesa a" 5

Kese m+a"$a" e"e($ meka" k + as ! tul s !alam +e"tuk ! ;e(e"s al se+a$a 5

δψ δ +++=− d+ g duuVd( W $ **

Ka(e"a t !ak a!a ke(,a ( "$ !a" t !ak a!a /e(u+a%a" ket "$$ a". /e(samaa" "

me",a! 5

V du

V u

d( δψ −−=

Me"u(ut /e(samaa" k "t "u tas.



2

22

1

11

V Au

V Au =

Atau.ka(e"a A 1 A 2 5

ta"2

2

1

1 k)*$V u

V u

V u ===

#e(" ataa" u"tuk d le% ka(e"a tu + as ! tul s 5

V du

V u

d( δψ −

−=1

1

Da" "te$(as mem+e( ka" 5

∫ −∆

−=∆

2

11

1

V u

V u

( δψ

#e"u(u"a" teka"a" t tal ! +e( ka" se+a$a ,umla% !a( !ua suku !a" k ta + sa

me"$%u+u"$ka" a"$ /e(tama !e"$a" /e"u(u"a" teka"a" a"$ ! se+a+ka" suatu

ke"a ka" ke e/ataa" !a" a"$ ke!ua !e"$a" /e"u(u"a" teka"a" a"$ +e(ka ta" !e"$a"

$eseka" ;lu !a* F(aks a"$ ! /e(luka" le% ka(e"a tu sama !e"$a" 5

( u

V u

∆∆

+

1

11

K " . ' lume m la( a al u!a(a ! +e( ka" me"u(ut /e(samaa" $as !eal se+a$a 5

( )( ) 1333

1

1 *10)32.310680

1).283314.> −−×=×

= m) m (

,-

Massa m la( u!a(a se a(a /e"!ekata" a tu

.*1028 1−× m) kg

maka 5

yolanda bidari dwi selani_4143121067_fisika dik d 2014_fisika matematika ii_contoh soal persamaan...

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