Bab 9 Teori Relativitas Khusus

memformulasikan teori relativitas khusus untuk waktu, panjang, dan massa, serta kesetaraan massa dengan energi yang diterapkan dalam teknologi

Setelah mempelajari bab ini, Anda harus mampu:

menganalisis berbagai besaran fisis pada gejala kuantum dan batas-batas berlakunya relativitas Einstein dalam paradigma Fisika Modern.

Hasil yang harus Anda capai:

Teori Relativitas Khusus

A. Semua Gerak

Bersifat Relatif

B. Teori Relativitas

Einstein

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)'' E:G6K6HM6C<7:FC6B6)'(%#)'%'(B6BEI7:F?6A6C9:C<6C @:8:E6H6CDF9:86=6M6/6=I@6=C966E6=I7IC<6C6CH6F6?6F6@9:C<6C @:8:E6H6C86=6M6%>@6GI6HI7:C967:F<:F6@9:C<6C@:8:E6H6C86=6M6 H:DF>F:A6H>J>H6G@=IGIG6@6C7:FA6@I.6A6=G6HI>BEA>@6G>96F>E:C:F6E6C H:DF>F:A6H>J>H6G@=IGIG696A6=H:F?69>CM6@DCHF6@G>E:CMIGIH6CE6C?6C< G:=>C<<6?6F6@H:BEI=CM6G:B6@>CE:C9:@

.:A6>C @DCHF6@G> E6C?6C< B6G>= 76CM6@ A6<> E:C<6FI= M6C< 9>6@>76H@6C DA:= 7:C96 M6C< 7:F<:F6@ 9:C<6C @:8:E6H6C B:C9:@6H> @:8:E6H6C86=6M66CM6@=6AM6C<H:F@:G6CH>96@B6GI@6@6A@6C H:H6E>?>@69>H>C?6I9:C<6CB:C<<IC6@6CH:DF>F:A6H>J>H6G@=IGIGH:FCM6H6 >AB>6=96C96E6H9>7I@H>@6C!:CDB:C6 ;:CDB:6B:C6F>@H:FG:7IH6@6C

Pesawat ruang angkasa Starship Enterprise dalam serial televisi Star Trek seringkali berjalan dengan kecepatan cahaya.

Bab

9

Gambar 9.1

Menurut pengamat di luar kereta api, penumpang sedang bergerak dengan kecepatan yang sama dengan kecepatan kereta api.

A. Semua Gerak Bersifat Relatif

.:BI67:C969>@6H6@6C7:F<:F6@?>@6@:9I9I@6C7:C96>HI7:FI76= H:F=696EH>H>@68I6C6H6I@:F6C<@668I6CM6C<9>H:H6E@6C.:DF6C<G>GK6 M6C<G:96C<9I9I@9>96A6B@:F:H66E>M6C<7:F<:F6@9>@6H6@6C9>6B H>96@7:F<:F6@?>@6@:F6C<@668I6CCM6@:F:H66E>@6CH:H6E>G:DF6C< E:C<6B6H9>AI6F@:F:H66E>B:C<6H6@6CG>GK6H:FG:7IH7:F<:F6@9:C<6C @:8:E6H6CM6C<G6B69:C<6C@:8:E6H6C@:F:H66E>.6B6=6ACM6<:F6@ G:7I6=E:F6=IH:FG:7IH6A>F6C6>FGIC<6>+:F6=I96C6FIGGIC<6>7:F<:F6@ H:F=696EIB>DCHD=9>6H6GB:CIC?I@@6C76=K6H>96@6967:C96 M6C<7:F<:F6@BIHA6@M6C<696=6CM6A6=<:F6@F:A6H>;

/:DF>):KHDCH:A6=7:F=6G>AB:C?:A6G@6CE:F>GH>K6G:7I6=7:C96M6C< 7:F<:F6@9:C<6C@:8:E6H6CF:C96=G:E:FH>8DCHD= 8DCHD=<:F6@H:FG:7IH @6CH:H6E>H:DF>CM6H>96@7:F=6G>AB:C?:A6G@6CE:F>GH>K67:C96M6C< 7:F<:F6@B:C9:@6H>@:8:E6H6C86=6M6G:E:FH>G:?IBA6=:A:@HFDCM6C< 9>E:F8:E6HB:A6AI>7:96EDH:CG>6A(:16@6CB:C86E6>@:8:E6H6C

(:CIFIHB:@6C>@6):KHDC76=K6@:8:E6H6C:A:@HFDC6@6CB:C?69> ?>@6 9>E:F8:E6H 9:C<6C EDH:CG>6A G:7:G6F (:1 &:CM6H66C B:CIC?I@@6C76=K6@:8:E6H6C:A:@HFDCH:H6EG6?6G:7:G6F%:A6GA6= B:@6C>@6):KHDCH>96@96E6H9>E6@6>ICHI@7:C96 7:C96M6C<7:F<:F6@ B:C9:@6H>@:8:E6H6C86=6M6

1. Transformasi Galileo

+F>CG>EF:A6H>J>H6G"6A>A:D9>@:C6A?I<6G:76<6>EF>CG>EF:A6H>J>H6G@A6G>@ @6F:C6 =6CM6 7:F@6>H6C 9:C<6C =I@IB =I@IB <:F6@ ):KHDC DCHD=CM6E:C<6B6H6CH:F=696EG:DF6C<G>GK6M6C<G:96C<9I9I@9> 96A6B@:F:H66E>M6C<7:F<:F6@DA:=9I6DF6C<E:C<6B6HG:76<6>9I6 @:F6C<@668I6C7:F7:96

DCHD=H:FG:7IH96E6H9><6B76F@6C9:C<6CB:C<<IC6@6CG>GH:B @DDF9>C6H@:F6C<@668I6C=6AH:FG:7IH9>AI@>G@6CG:E:FH> +! / 6A6BHF6CG;DFB6G>M6C<6@6C9>HIFIC@6C>C>G:A6C<K6@HIM6C<9>86H6H DA:=E:C<6B6H9>96C9>6C<<6EG6B6

+! / B:BE:FA>=6H@6C9I6@:F6C<@668I6C96C&:F6C<@6

68I6CH:F=696E68I6C7:F<:F6@@:@6C6CG:E6C?6C<GIB7I -9:C<6C @:8:E6H6C H:H6E + +696 @:F6C<@6 68I6C >C:FG>6A 7:FA6@I* *" !""#,)$#&:F6C<@668I6C>C:FG>6AH>96@B:C<6A6B>E:F8:E6H6C (>G6A@6C9I6DF6C<E:C<6B6H96C B6G>C< B6G>C<7:F69696A6B @:F6C<@668I6C.96C+:C<6B6HM6C<7:F6969>GH6G>IC9>6C<<6E 7:F696E696@:F6C<@668I6CM6C<9>6B96EICE:C<6B6H 7:F696

Y

S'

O' X' = X V t

Z

Y' S x

P

O Z'

x 8

Y'

S'

Z

O' t = 0 X

v

7 6

Z

Y

S

X O t = 0

$!$*2++$+.$* ( /')-,0$.$-/'$* 1'3'1 0&2020)$/( ) ,* &0- *0- *!$/')21# * +!2)2* 1'& ,

Tes Kompetensi Awal

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M6C<B:C<<I<IF@6CH:DF>696CM6:H:F

Gambar 9.2

(a) Kerangka acuan S dalam keadaan diam; (b) Kerangka acuan S' dalam kecepatan konstan v; (c) Setelah selang waktu t, berada

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.:7I6=7:C967:F<:F6@@:@6C6C9:C<6C@:8:E6H6C*+696G66H)

G:@DC7:C96H:F=696E@:F6C<@668I6CB:B>A>@>@DDF9>C6H-./96C H:F=696E@:F6C<@668I6CB:B>A>@>@DDF9>C6H-./G:=>C<<6

--0+)..// O

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)) O

$/0 + , 7 96C7 9>@:C6A 9:C<6C HF6CG;DFB6G> "6A>A:D /F6CG;DFB6G>@:76A>@6CCM6696A6=

--+)..//)) O *A:=@6F:C6))HIFIC6C-6H6I-H:F=696E)G6B69:C<6CHIFIC6C

-6H6I-H:F=696E)%>@6$/0 + ,79>HIFIC@6CH:F=696EK6@HI)

6@6C9>E:FDA:=@:8:E6H6C

- +) +)

-

-) ) ) ) )

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@:8:E6H6C7:C96H:F=696E96C₎ +) + 6@6C9>E:FDA:=G:7I6= 7:CHI@E:FG6B66C

*_-*_-0+ O

+696GIB7I .7:FA6@I..E696GIB7I /7:FA6@I//

. . / / ) ) ) ) *_.*_.*_/*_/

H6I *_-*_-O+*_.*_.*_/*_/ O

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*_-*_-+*_.*_.*_/*_/ O &:H:F6C<6C

*_-@:8:E6H6C7:C96G:?6?6F -*_.@:8:E6H6C7:C96G:?6?6F. *_/@:8:E6H6C7:C96G:?6?6F/

&:8:E6H6C9>HIFIC@6CH:F=696EK6@HI6@6CB:C<=6G>A@6CE:F8:E6H6C

- *₎- *₎-

_. *.

) *).. O

_/*/

) *) / /

Tantangan

untuk Anda

Berilah contoh ilustrasi transformasi Galileo yang ada di sekitar Anda.

:F96G6F@6C$/0 + ,796E6H9>@6H6@6C76=K6E:F8:E6H6C 7:C9696A6B@:F6C<@668I6CG6B69:C<6CE:F8:E6H6C7:C9696A6B @:F6C<@668I6C(6GG67:C969>E6C96C<96F>@:F6C<@668I6C

96C9>6C<<6EH:H6EG6B6B6@6<6M6E696@:F6C<@668I6C96C B:C?69>G6B6EIA6M6>HI"96C"G:=>C<<6

6F>IF6>6CH:FG:7IH96E6H9>6B7>A@:G>BEIA6C76=K6* *",)$# )#)#' #%'("#' (*)*#))%("!"("*

'# *##'(!

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4 !

:C<6CB:C<<IC6@6C$/0 + ,7*_-*_-0+

*F6C<M6C<9>6B9>H:E>F:AG:76<6>@:F6C<@668I6C&:F:H66E>M6C<7:F<:F6@ H:F=696EDF6C<M6C<9>6BG:76<6>@:F6C<@668I6C&:8:E6H6CE:CIBE6C<H:F=696E @:F6C<@668I6C696A6=*_-@B?6B:C<6C9:B>@>6C@:8:E6H6CE:CIBE6C<

*_-H:F=696EM6C<9>6B696A6=

*-*-+@B?6B

Contoh

9.1

2. Percobaan Michelson–Morley

":?6A6M6C<H6BE6@E696<:ADB76C<H6A><:ADB76C<E:FBI@66C6>F 6H6I<:ADB76C<7ICM>B:CIC?I@@6C76=K6<:ADB76C<96E6HB:F6B76H B:A6AI>GI6HIB:9>IBM6C<B:C?6A6F@6C<:H6F6C96F>GI6HIH:BE6H@: H:BE6H A6>CCM6 *A:= @6F:C6 86=6M6 ?I<6 96E6H 9>6C<<6E G:76<6> <:ADB76C<6E6@6=M6C<B:BIC<@>C@6CB:F6B76HCM6<:ADB76C<86=6M6 96F>(6H6=6F>@:IB>G:96C<@6CG:76<>6C7:G6F96:F6=6CH6F6(6H6=6F> 96CIB>696A6=FI6C<=6BE6

0CHI@ B:C?6K67 E:FH6CM66C >C> E696 6769 @: E6F6 6=A> B:C<6?I@6CGI6HI=>EDH:G>GM6C<9>@:C6A9:C<6C%$)(()'(:CIFIH =>EDH:G>G>C><:ADB76C<86=6M696F>(6H6=6F>@:IB>96E6HB:F6B76H B:A6AI>GI6HIB:9>IBM6C<9>C6B6@6C:H:F(:9>IB>C>H:F96E6H9>B6C6 B6C69>6A6B>C>H:FB6GI@FI6C<=6BE6:C<6C696CM6=>EDH:G>G>C> ICHI@ G:B:CH6F6 96E6H B:C<=>C96F@6C @:GI@6F6C H:CH6C< B:9>IB E:F6B76H6C<:ADB76C<86=6M6

+696'"&$*0-,96C-/*$5B:A6@I@6C:@GE:F>B:C

ICHI@B:B7I@H>@6C@:7:F6966C:H:F @GE:F>B:C>C>9>A6@I@6C9:C<6C B:C<I@IFG:A>G>=K6@HI6CH6F6@:A6?I6C86=6M6@:H>@67:F<:F6@G:?6?6F 96CH:<6@AIFIG:H:F<6FC9696E6HB:B6=6B>:@GE:F>B:C(>8=:AGDC (DFA:M>C>E:F=6H>@6C>AIGHF6G>H:CH6C<<:F6@E:F6=IM6C<7:F<:F6@G:6F6= 9:C<6C 6A>F6C GIC<6> 96C <:F6@ E:F6=I M6C< 7:F<:F6@ B:CM:7:F6C<> GIC<6>H:<6@AIFIG9:C<6C6A>F6CGIC<6>+696 +! /E:F6=I 7:F<:F6@EIA6C<E:F<>B:CM:7:F6C<>GIC<6>96F>H>H>@(@:)96C@:B76A> @:(A6<>9:C<6C6F6=H:<6@AIFIG6A>F6CGIC<6>M6C<7:F<:F6@9:C<6C @:8:E6H6C+0CHI@B:A6@I@6CE:CM:7:F6C<6C>C>6F6=<:F6@6CCM6=6FIG 9>B>F>C<@6C@:@>F>ICHI@B:C<>B76C<>6FIGGIC<6>G:E:FH>9>E:FA>=6H@6C

E696 +! /:G6FF:GIAH6C96F>@:8:E6H6CE:F6=I96C@:8:E6H6C

6A>F6CGIC<6>+96E6H9>HIA>G@6CG:76<6>7:F>@IH

Gambar 9.3

Gerak perahu bolak-balik yang tegak lurus aliran sungai.

Gambar 9.4

Diagram arah kecepatan perahu dan aliran sungai.

N

v

c

M N

v

aliran sungai

c

M

2 2 ’ v c v

v

c + v

c – v

M O

aliran sungai

Gambar 9.5

Gerak perahu yang searah dan berlawanan dengan aliran sungai.

v’ c2v2 O

%>@6E6C?6C<A>CH6G6C(@:)696A6=!K6@HIM6C<9>E:FAI@6CICHI@ B:CM:7:F6C<>GIC<6>696A6= + %69>ICHI@EIA6C<E:F<>B:CM:7:F6C<>

GIC<6>9>7IHI=@6CK6@HI

)

+

+

O .:@6F6C<E:F=6H>@6C +! /+:F6=I 7:F<:F6@7DA6@ 76A>@ 96F>H>H>@(@:*A6AI@:B76A>@:(A6<>+6C?6C<A>CH6G6C96F>(@:* G6B6 9:C<6C E6C?6C< A>CH6G6C 96F> ( @: ) M6>HI! +696 G:E6FI= E:F?6A6C6CCM6M6C<E:FH6B6@:8:E6H6CCM69>76CHIDA:=6A>F6CGIC<6> G:=>C<<6 ICHI@ B:C:BEI= G:?6I= 9>E:FAI@6C K6@HI

+ @6C

H:H6E>E696K6@HIG:E6FI=E:F?6A6C6CCM6ICHI@@:B76A><:F6@CM6?IGHFI B:A6K6C6FIGG:=>C<<69>E:FAI@6CK6@HI ₊ ICHI@B:C:BEI=?6F6@ M6C<G6B6M6>HIG:E6C?6C<:C<6C9:B>@>6CG:AIFI=E:F?6A6C6C

E:F6=IEIA6C<E:F<>B:B:FAI@6CK6@HI

) _{+ +}

:CHI@H:FG:7IH96E6H9>HIA>G?I<69:C<6C

+ +

) _{+ +}

+ + )

+

+

+:FG6B66CH:FG:7IH96E6H?I<69>HIA>G@6C

)

+

)

+

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)O)

+

O

+

)O)

+

O

+

O

a2 _{– b}2 _{= (a + b)(a – b)}

0CHI@B:CM:A:G6>@6CE:FG6B66CH:FG:7IH9><IC6@6C9:F:H7>CDB>6A M6>HI

-#

# - # - # - ## -#

96C

# # # # #

_O

.I@I +

E696$/0 + ,796E6H9>HIA>G@6C96A6B7:CHI@

9:F:H7>CDB>C6A9:C<6CB:C<6C6AD<>@6C+G:76<6>-96CE6C<@6H

OG:76<6>#B6@66@6C9>96E6H@6C

+

#-##O

-

+

+

*A:=@6F:C6_{BG96C+7:FDF9:@:8>AC>A6>}

+

G6C<6H@:8>AG:@6A>

G:=>C<<6+ %69>E:FG6B66C9:F:HCDB>C6AH:FG:7IH96E6H9>HIA>G@6C

+

+

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+

+

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)O)

+

O

+

)O)

+ O

+ )O)

+

O

+696 E:F8D766C (>8=:AGDCO(DFA:M M6C< G@:B6CM6 H:FA>=6H E696

+! /E6C?6C<A>CH6G6CM6C<9><IC6@6C696A6=B96EIC

@:A6?I6C:H:F+696A6=@:A6?I6CFDH6G>IB>M6>HIP_BG*A:=

@6F:C6>HIG:A>G>=K6@HI86=6M6M6C<G:?6?6F:H:F96CH:<6@AIFIG:H:F 96E6H9>=>HIC<

)O)

B

P BG

P BG P BG

PO_G

Tantangan

untuk Anda

Agar Anda lebih memahami tentang deret binomial. Kerjakan contoh-contoh berikut.

1. (1 – x2₎3 2. (1 + 2x)3

Gambar 9.6

Skema percobaan Michelson-Morley. cermin 2

cermin 1 perjalanan A

aliran eter layar

pengamat kaca

setengah cermin sinar datang

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Tes Kompetensi

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Kata Kunci

• transformasi Galileo • kerangka acuan • eter

Gambar 9.7

Dua kerangka acuan S dan S', dengan S' memiliki kecepatan (v) relatif terhadap S.

1. Transformasi Lorentz

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Tokoh

H.A. Lorentz (1853–1928)

Hendrik Antoon Lorentz (1853– 1928) adalah seorang fisikawan Belanda yang kali pertama me-ngemukakan tentang pemantulan dan pembiasan cahaya, ditinjau dari teori bahwa cahaya termasuk gelombang elektromagnetik. Teori ini dikembangkan oleh James Clerk Maxwell yang menyatakan bahwa cahaya terdiri atas gelombang listrik dan gelombang magnetik dengan sudut yang berbeda. Lorentz juga memprediksi keberadaan elektron. Beliau menerima hadiah Nobel tahun 1902 dalam bidang fisika atas penelitiannya mengenai medan magnet. Dalam bab teori relativitas,

Lorentz berjasa dalam teorinya mengenai "Transformasi Lorentz".

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2. Transformasi Kecepatan dari Lorentz

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Tantangan

untuk Anda

Dengan cara yang sama seperti pada

Persamaan (9–21), turunkan persamaan transformasi kecepatan relativistik sehingga didapatkan hasil seperti Persamaan (9–22).

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Tantangan

untuk Anda

Dengan menggunakan transformasi Lorentz, buktikan bahwa kecepatan cahaya untuk pengamat yang diam dan pengamat yang bergerak dengan kecepatan v adalah sama.

Gambar 9.8

Sebuah meteran yang bergerak sebesar 87% dari kecepatan cahaya, panjangnya hanya terlihat setengahnya oleh pengamat yang diam.

3. Akibat Prinsip Relativitas Khusus

a. Kontraksi Lorentz

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Gambar 9.9

Semakin besar kecepatan bola, ukuran bola akan semakin mengecil.

v = 0 v = 0,995 c

v = 0,999 c v = 0,87 c

v = c

Tantangan

untuk Anda

Perhatikan kembali Persamaan (9–26) dan Persamaan (9–29).

Diskusikan dengan teman sekelas Anda tentang pengertian fisis persamaan-persamaan tersebut. Kemudian, carilah analogi tentang kejadian pada persamaan tersebut dalam kehidupan sehari-hari.

Gambar 9.10

(a) Sebuah bangunan yang dilihat oleh pengamat yang diam; (b) Bangunan yang dilihat oleh pengamat yang bergerak sangat cepat; (c) Diagram yang memperlihatkan bagaimana bagian sisi dinding dapat terlihat.

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%69> E6C?6C< 7:C96 G:DA6= DA6= B:CMIGIH ":?6A6 >C> 9>G:7IH @DCHF6@G> E6C?6C< #6A >C> 9>@:BI@6@6C DA:=-/$,16 G:=>C<<6 @DCHF6@G>CM69>G:7IH $#)' ($'#)/

b. Dilatasi Waktu

',01$',B:CM6H6@6C76=K6K6@HI696A6=G:GI6HIM6C<F:A6H>;#6A

>C>96E6H9>>AIGHF6G>@6C9:C<6C<:F6@86=6M69>96A6BGI6HI@:F6C<@6 68I6C M6C< 7:F<:F6@ &:BI9>6C <:F6@ 86=6M6 H:FG:7IH 9>6B6H> 96F> @:F6C<@668I6CM6C<9>6BG:E:FH>E696 +! /

&:F6C<@668I6C7:F<:F6@9:C<6C@:8:E6H6CF:A6H>;H:F=696E@:F6C<@6

.:DF6C< E:C<6B6H E696 @:F6C<@6 B:C<6F6=@6C G:7I6= GIB7:F 86=6M6@:G:7I6=8:FB>CM6C<?6F6@CM6H:FA>=6HG:E:FH> +! /

Contoh

9.3

Contoh

9.4

6 7

8

A B C

C

C’ D

v

B

B’ A

+696<6B76FH:FA>=6H76=K686=6M6M6C<9>E6C86F@6C9>9>E6CHIA@6C @:B76A>G:E6C?6C<<6F>GM6C<G6B6G:E:FH>M6C<9>6B6H>96A6B@:F6C<@6

26@HIEIA6C<E:F<>@:A>E86=6M69>96F>@:8:FB>C96C@:B76A> @:696A6=

6H6I )

)

O

&:?69>6C 9> ?>@6 9>E6C96C< DA:= E:C<6B6H M6C< 7:F696 E696 @:F6C<@696E6H9>A>=6HE696 +! /!M6C<B:BE:FA>=6H@6C76=K6 A>CH6G6C86=6M6K6@HIE:F<>H>96@A6=G6B69:C<6CK6@HIEIA6C<%>@6G:A6C< K6@HIEIA6C< E:F<>B:CIFIHE:C<6B6H9>696A6=)?6F6@6CH6F6

96C696A6=+ ) 96EICA>CH6G6C86=6M6696A6= 6F><6B76FH:FA>=6H

76=K6

9:C<6C + )

)

+ )

)

O

%>@6$/0 + , 7 9><67IC<@6C @: 96A6B$/0 + , 7 9>E:FDA:=

) _{+ )}

)

O

%>@6$/0 + ,79>@I69F6H@6C9>E:FDA:=

)

) _{+ )}

)

)+ )

)

O+ )

)

)

_O

+

)

.:A6C?IHCM6 9>E:FDA:= =I7IC<6C 6CH6F6t 96C t

M6C< 96E6H

9>FIBIG@6CG:76<6>7:F>@IH

) )

+

O

&:H:F6C<6C

+ @:8:E6H6CF:A6H>;E:C<6B6HM6C<7:F<:F6@H:F=696EE:C<6B6HM6C< 9>6B

)

G:A6C<K6@HI9>I@IFDA:=E:C<6B6HM6C<F:A6H>;7:F<:F6@

)

G:A6C<K6@HI9>I@IFDA:=E:C<6B6HM6C<F:A6H>;9>6B

Gambar 9.11

(a) Kelip cahaya pulang balik dipancarkan di O dipandang oleh pengamat di S'; (b) Kelip cahaya di S' dipandang dari S.

cermin cermin

Dt₀

Dt v Dt

c 6

7

d

d waktu

waktu

waktu 1 waktu 2

pengamat

O O'

.I6HIE6FH>@:AB:B>A>@>K6@HI=>9IEPO_{G:@DCM6>HIK6@HIE6FH>@:A>HIBIA6>}

696=>C<<6BIGC6=G6B6G:@6A>:F6E6@6=K6@HI=>9IEE6FH>@:A>HI?>@6>67:F<:F6@ 9:C<6C@:A6?I6C

4 !

>@:H6=I>)PO_G:@DC+O

9:H>@

) )

+

5. Momentum dan Massa Relativistik

#I@IB):KHDCH:CH6C<<:F6@B:CM6H6@6C76=K6?>@6E696G:7I6= 7:C967:FB6GG6"7:@:F?6G:7I6=<6M6@DCGH6C7:C96H:FG:7IH6@6C 7:F<:F6@9>E:F8:E6H7:F6HIF6C:G6FCM6@:8:E6H6C7:C966@6C7:FH6B76= H:FIGG:GI6>K6@HI7:F96G6F@6C=I7IC<6C_{+ "})%>@6<6M67:@:F?6 8I@IEA6B6G:=>C<<6=6F<6)G6C<6H7:G6FH>96@A6=BIGH6=>AA6?I7:C96 B:C?69>G6C<6H7:G6F6F>=6G>AE:F8D766C76<6>B6C6EIC7:G6FCM6:C:F<> M6C<9><IC6@6C7:G6FCM6@:8:E6H6CG:7I6=7:C96H>96@E:FC6=B:C86E6> @:8:E6H6C 86=6M6 .:H>6E 7:C96 M6C< 7:FB6GG6" 7:F<:F6@ 9:C<6C @:8:E6H6C+B:B>A>@>BDB:CHIBA>C:6FM6C<B:CIFIHB:@6C>@6@A6G>@ 9>FIBIG@6C

%"+ O

%>@6@:8:E6H6C7:C96+GI6HIG66HB:C9:@6H>@:8:E6H6C86=6M6 B6GG67:C966@6C7:FI76= I76=+696G66H>HIBDB:CHIB7:C969>G:7IH BDB:CHIBF:A6H>J>GH>@

0CHI@<:F6@6C <:F6@6C7:C969:C<6C@:8:E6H6CM6C<F:A6H>;@:8>A H>96@ H:F?69> E:FI76=6C B6GG6 +:FI76=6C >HI 76FI H6BE6@ ?>@6 @:8:E6H6CCM6B:C9:@6H>@:8:E6H6C86=6M6*A:=#()#=I7IC<6C B6GG6 9>6B 96C B6GG6 7:F<:F6@ M6C< 9>H>C?6I 96F> GI6HI E:C<6B6H 9>FIBIG@6CG:76<6>7:F>@IH

" "

+

O

&:H:F6C<6C

" B6GG67:C9696A6B@:6966C7:F<:F6@@<

" B6GG67:C9696A6B@:6966C9>6B@<

+ @:8:E6H6C7:C96BG

@:8:E6H6C86=6M69>I96F6BG

Contoh

9.5

.6A6=G6HI7I@H>@:7:C6F6CH:DF>F:A6H>J>H6G696A6=;6@H6@:7:F6966C"*$#9>IB> 6F>A6=>C;DFB6G>H:CH6C<"*$#>C>96F>7:F76<6>GIB7:F96C7I6HA6=F:GIB:96F> >C;DFB6G>M6C<C9696E6H@6C&IBEIA@6CF:GIB:C96@:E696<IFI-:GIB: H:F76>@9>EF:G:CH6G>@6C96C9>9>G@IG>@6C9>9:E6C@:A6G

%>@6$/0 + ,79>GI7GH>HIG>@6C@:96A6B$/0 + ,7B6@6

" + %

+

O

#I@IB&:@:@6A6C(DB:CHIB7:FA6@I96A6BF:A6H>J>H6G@=IGIGG:E:FH> 96A6B;>G>@6@A6G>@)6BIC#I@IB$$):KHDCB:C<:C6><:F6@96A6B H:DF>F:A6H>J>H6G96E6H9>HIA>G@6CG:76<6>7:F>@IH

" + % "+

) ) ) ₊

O +696E:FG6B66C ₎ " + " +₎ +"₎ =6F<6 "₎ H>96@CDA?>@6B6GG6 7:C967:FI76=H:F=696EK6@HI"6M6F:GIAH6CE6967:C96G:A6AIG6B6 9:C<6CA6?IE:FI76=6CBDB:CHIBH:F=696EK6@HI

%>@6B6GG69>6B7:C96@<=>HIC<A6=B6GG67:C967:F<:F6@9:C<6C@:8:E6H6C

4 !

:C<6CB:C<<IC6@6C$/0 + ,79>E:FDA:=

_@<

" "

+

96=6AB:C6F>@M6C<7:F@6>H6C9:C<6CB6GG6F:A6H>J>GH>@>C>E6@6= C96E:F86M6696E6FH>@:AM6C<H>96@B:B>A>@>B6GG6H:H6E>B:B>A>@> :C:F<>(:CIFIH;>G>@6@A6G>@=6A>C>H>96@BIC<@>CH:F?69>?>@6"

"+ ₊

@6CH:H6E>?>@6=6A>C>9>H>C?6IB:CIFIHH:DF>F:A6H>J>H6G96E6H9>7I@H> @6CG:86F6B6H:B6H>G76=K6696E6FH>@:AM6C<H>96@7:FB6GG6H:H6E> B:B>A>@>:C:F<>(>G6A@6C696G:7I6=E6FH>@:AM6C<7:FB6GG6"96C 7:F<:F6@9:C<6C@:8:E6H6C+

" +

" +

" +

" +

6A6B B6H:B6H>@6 696A6=7:CHI@H>96@H:CHIG:=>C<<6C>A6>CM6 G:B76F6C<7:F6E6G6?6&:G>BEIA6CCM6696A6=696E6FH>@:AM6C<H>96@ 7:FB6GG6B6GG69>6BH:H6E>B:B>A>@>:C:F<>6A6B@:=>9IE6CG:=6F> =6F>E6FH>@:AM6C<B:B>A>@>@6F6@H:F>GH>@G:E:FH>H:FG:7IH696A6=;DHDC 96CC:IHF>CD

6. Kesetaraan Massa dan Energi

#I7IC<6C E6A>C< H:F@:C6A M6C< 9>E:FDA:= >CGH:>C 96F> EDGHIA6H F:A6H>J>H6G@=IGIG696A6=B:C<:C6>B6GG696C:C:F<>#I7IC<6CCM696E6H 9>HIFIC@6CG:86F6A6C<GIC<96F>9:;>C>G>:C:F<>@>C:H>@96F>GI6HI 7:C96M6C<7:F<:F6@

(

(

9:C<6C B:CM6H6@6C @DBEDC:C <6M6 M6C< 7:@:F?6 96A6B 6F6= E:FE>C96=6CG:FH6(96C(B:CM6H6@6C?6F6@M6C<9>H:BEI=:C<6C B:B6@6>7:CHI@F:A6H>J>GH>@=I@IB<:F6@@:9I69>E:FDA:=

"+

₎

G:=>C<<6FIBIG6C:C:F<>@>C:H>@B:C?69>

( _"+

₎ (

"+ + " +

+ "+ + ₊

:C<6CB:C<<IC6@6CFIBIG6C>CH:<F6AE6FG>6A6@6C9>E:FDA:=

"_0"

O

#6G>A>C>B:CM6H6@6C76=K6:C:F<>@>C:H>@GI6HI7:C96G6B69:C<6C E:FH6B76=6CB6GG6CM66@>76H<:F6@F:A6H>;CM69>@6A>@6C9:C<6C@I69F6H @:A6?I6C86=6M6+:FG6B66C96E6H?I<69>HIA>G

"_0"

O

%>@69>B>G6A@6C"_{G:76<6>:C:F<>HDH6A7:C967:C9696A6B@:6966C}

9>6BH:H6E>7:C96H:H6EB:B>A>@>:C:F<>"_{:C<6C9:B>@>6C}

" _{9>G:7IH :C:F<> 9>6B}

96F> 7:C96 M6C< B6GG6 9>6BCM6"

$/0 + ,7B:C?69>

O

&:H:F6C<6C

:C:F<>9>6B?DIA:

%>@67:C967:F<:F6@:C:F<>HDH6ACM6696A6=

" "

+

O

Tantangan

untuk Anda

Dapatkah Anda menduga, apakah ada partikel yang bergerak dengan kecepatan 0,95 c dan memiliki energi sebesar E tetapi tidak memiliki massa?

*A:=@6F:C6B6GG696C:C:F<>7I@6CB:FIE6@6C@I6CH>H6GM6C< 7:76G#I@IB&:@:@6A6C(6GG696C C:F<>G:7:C6FCM6B:C?69>G6HI M6>HI #I@IB &:@:@6A6C (6GG6 C:F<> (6GG6 96E6H H:F8>EH6@6C 96C H:FBIGC6=@6CH:H6E>?>@6=6A>C>H:F?69>G:?IBA6=:C:F<>M6C<G:H6F6=>A6C< 6H6IBIC8IA96CG:76A>@CM6%69>B6GG696C:C:F<>B:FIE6@6C6GE:@ M6C<7:F7:9696F>GI6HI@I6CH>H6G :F6E6@6=:C:F<>HDH6A:C:F<>@>C:H>@96CBDB:CHIBG:7I6=EFDHDC" (:1M6C<7:F<:F6@9:C<6C@:8:E6H6C+ 4 !

C:F<>HDH6A96E6H9>=>HIC<9:C<6CB:C<<IC6@6C$/0 + ,7

#6AM6C<B:C6F>@ICHI@9>76=6GB:C<:C6>:C:F<>F:A6H>J>GH>@696A6= 76<6>B6C6B:C<=>HIC<:C:F<>F:A6H>J>GH>@GI6HIE6FH>@:A?>@6B6GG6CM6 H>96@9>@:H6=I>6H6I?>@6E6FH>@:AH:FG:7IHH>96@7:FB6GG6+:F=6H>@6C @:B76A>E:B76=6G6CE696GI7767G:7:AIBCM6B:C<:C6>E6FH>@:AM6C< H>96@7:FB6GG6H:H6E>B:B>A>@>:C:F<>+:FG6B66C:C:F<>96CBDB:C HIBF:A6H>J>GH>@96E6H9>HIA>G@6CG:76<6>7:F>@IH " % " + + %>@6@:9I6E:FG6B66CH:FG:7IH9>@I69F6H@6C6@6C9>96E6H@6C

_{" O}

% " + + O

Contoh

9.7

6

(:1 _(:1 " +

_HDHP_:1PO_?DIA::1

PO_?DIA:

7 C:F<>@>C:H>@CM6O"

O(:1(:1 8 (DB:CHIBCM6

(:1 ?DIA:(:1 BG "

PO_@<

G:=>C<<6 " + % +

@< BG _@<BG %

%69>BDB:CHIBCM6696A6=PO_@<BG

Perumusan integral parsial adalah:

x dy xy y dx

Ingatlah

Pembahasan Soal

Agar energi kinetik benda bernilai 25% dari energi diamnya dan c adalah kelajuan cahaya ruang hampa, benda harus bergerak dengan kelajuan .... a.

4

c _d. 3

4 c

b.

2

c _e. 4

5 c

c. 3₅c

UMPTN 1998 Pembahasan:

Energi kinetik relativitas adalah

E_k = m₀c2

2 2 1 ₁ 1 v c

Dengan menggunakan E_k =

1 4m0c

2 _{= m} 0c 2 2 2 1 ₁ 1 v c Jadi, 2 2 1 ₁ 1 v c

= 1₄; atau

22 1 1 v

c

= 5

4

22 1 v

c =

16 25 v2 =

9 25 c2

v = 3_{5 c}

m₀₁ m₀₂ v₂ v₁

6 7

m₀

Gambar 9.12

(a) Sebuah benda yang memiliki massa diam m₀; (b) Benda tersebut pecah menjadi

&6A>@6C$/0 + ,79:C<6C_9>E:FDA:=

%

" + +

&IF6C<@6C$/0 + ,79:C<6C$/0 + ,7G:=>C<<6

_O%"O

" + +

*A:=@6F:C6"

" +

B6@6

" " + %

+ +

+ "

%

+

_O%"

_"

% O

:C<6C B:C<<IC6@6C$/0 + , 7 C96 96E6H B:C<=>HIC< 7:F6E6:C:F<>F:A6H>J>GH>@GI6HIE6FH>@:AM6C<H>96@7:FB6GG6M6>HI

_%8

% O

7. Hukum Kekekalan Energi dan Momentum Relativistik

>&:A6G3$C96H:A6=B:BE:A6?6F>H:CH6C<#I@IB&:@:@6A6C C:F<>96C#I@IB&:@:@6A6C(DB:CHIB#I@IB&:@:@6A6C C:F<> B:CM6H6@6C76=K6:C:F<>H>96@96E6H9>8>EH6@6C6H6I9>BIGC6=@6C H:H6E>:C:F<>96E6H7:FI76=7:CHI@B:C?69>:C:F<>A6>C:C<6C@6H6 A6>CE696GI6HIG>GH:B:C:F<>6K6ACM6G6B69:C<6C:C:F<>6@=>FCM6 :<>HI?I<6#I@IB&:@:@6A6C(DB:CHIB?>@6E696GI6HIG>GH:BH>96@ 7:@:F?6<6M6AI6FBDB:CHIB6K6AG6B69:C<6CBDB:CHIB6@=>F

&:9I6=I@IB>C>7:FA6@I?I<6E696GI6HIG>GH:BF:A6H>J>GH>@ C:F<> 6K6AF:A6H>J>GH>@G6B69:C<6C:C:F<>6@=>FF:A6H>J>GH>@96CBDB:CHIB 6K6AF:A6H>J>GH>@G6B69:C<6CBDB:CHIB6@=>FF:A6H>J>GH>@.IE6M6C96 96E6H A:7>= B:B6=6B> #I@IB &:@:@6A6C C:F<> 96C (DB:CHIB -:A6H>J>GH>@E:F=6H>@6C +! / 7:F>@IH

+! / B:CIC?I@@6CG:7I6=7:C96M6C<B:B>A>@>B6GG69>6B

"G:86F6GEDCH6CE:86=B:C?69>9I676<>6C"96C"&:9I6E686=6C >C>7:F<:F6@9:C<6C@:8:E6H6CM6C<7:F7:96M6>HI+96C+0CHI@ B:C<<6B76F@6C@DCG:E>C>9:C<6CA:7>=?:A6G7:G6F6C 7:G6F6C""

+96C+6@6C9>7:F>C>A6>M6>HIB6GG6"@<7:F<:F6@9:C<6C

@:8:E6H6C+96CB6GG6"@<7:F<:F6@9:C<6C@:8:E6H6C

+:F96G6F@6C#I@IB&:@:@6A6C C:F<>7:FA6@I C:F<>6K6A C:F<>6@=>F

"

" +

O

" +

O %>@6$/0 + ,79>76<>9:C<6C_9>E:FDA:=

"

" +

O

" +

O %69>96F>$/0 + ,7 C9696E6HB:C<:H6=I>B6GG69>6B96F> 7:C96G:7:AIBE:86=

"

c

" _@<

6F>=6G>AE:F=>HIC<6CH:FG:7IH6E6@6=C96B:C:BI@6CG:GI6HIM6C< B:C6F>@#6G>AE:F=>HIC<6CB:CIC?I@@6CB6GG69>6BE696@:6966C6K6A "@<7:F7:969:C<6CB6GG69>6BE696@:6966C6@=>F@< #6A>C>G:?6A6C9:C<6C>BEA>@6G>H:DF>F:A6H>J>H6G@=IGIGM6C<B:CM:767@6C E:BI6>6CB6GG6E6967:C96M6C<7:F<:F6@B:C9:@6H>@:8:E6H6C86=6M6

&:BI9>6C 76<6>B6C6 9:C<6C BDB:CHIBCM6 .:A6B6 H>96@ 696 E:C<6FI=<6M6AI6F6@6C7:FA6@I#I@IB&:@:@6A6C(DB:CHIB

(DB:CHIB6K6ABDB:CHIB6@=>F

" + +

" + +

" + +

O

8. Aplikasi Kesetaraan Massa dan Energi pada Reaksi Fisi dan

Fusi Nuklir

E6M6C<96E6HC96G>BEIA@6C96F>E:B76=6G6CH:CH6C<#I@IB &:@:@6A6C C:F<>96C(DB:CHIB-:A6H>J>GH>@&:G>BEIA6C96F>#I@IB &:@:@6A6C C:F<>-:A6H>J>GH>@696A6=B6GG69>6BH>96@@:@6A

"@<

"@<

"

"O"O@<@<

+:C<IF6C<6CB6GG6>C>:@>J6A:C9:C<6C:C:F<>

"

&:BI9>6C76<6>B6C69:C<6C:C:F<>@>C:H>@CM6 C:F<>@>C:H>@E696 @:6966C6K6A96C@:6966C6@=>F96E6H9>HIA>G@6CG:76<6>7:F>@IH

6K6A

6@=>F

" +

O" _"

+

Gambar 9.13

Tumbukan foton dan elektron (a) sebelum tumbukan; (b) sesudah tumbukan. foton dengan, p

6

7

', p'

foton terhambur

elektron p_e

v

Menurut Planck, energi pada foton adalah E = hf. Jadi, hf = pc

P = hf_c = h

Ingatlah

8

"

+

₈

%69>6@=>F_G:=>C<<6

6@=>FO6K6A

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Kata Kunci

• relativitas khusus • transformasi Lorentz • kontraksi Lorentz • dilatasi Lorentz • dilatasi waktu • momentum relativitas • massa diam

• kesetaraan massa dan energi • reaksi fusi

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Refleksi

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Kata Kunci

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Setelah mempelajari bab ini, tentu Anda telah memahami bahwa gerak bersifat relatif. Anda juga tentu telah memahami teori relativitas Einstein dan pengaruh-nya terhadap besaran panjang, waktu, kecepatan dan massa benda. Dari keseluruhan materi yang ada pada bab ini, bagian manakah yang menurut Anda sulit

Refleksi

dipahami? Coba Anda diskusikan bersama teman atau guru Fisika Anda.

Selain Anda mengetahui teori relativitas khusus, coba Anda cari manfaat lain mempelajari materi bab ini. Kemudian, diskusikan hasilnya bersama teman Anda. B:C<6@>76H@6C

Tes Kompetensi

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