Chapter 6 Gravitation

CHAPTER 6 GRAVITATION Contents: 1. NEWTON’S UNIVERSAL GRAVITA GRAVITATIONAL TIONAL LAWS LAWS 2. CALCULATION OF MASS AND DENSITY OF EARTH VARIATION OF “g” DUE TO HEIGHT AND DEPTH 3. VARIATION 4. WEIGHTLESS NESTS IN SATELLITES GRAVITY 5. ARTIFICIAL GRAVITY . E!UATIONS ". DIMENSIONS #. SHORT !UEASTIONS AND ANSWERS

Gravitation: G$%&'(%(')* '+ % *%(,$%- /0*)0*)* *%(,$%- /0*)0*)*   /'/ )60(+ '(/ '(/ %++ %++ %(($%( %(($%( )*0 %*)(/0$. I( '+ $0+)*+'-0 7)$ 800'*g (/0 E%$(/ %*9 (/0 )(/0$ -%*0(+ '* (/0'$ )$'(+ %$),*9 )$'(+ %$),*9 (/0 S,*: 7)$ 800'*g (/0 M))* (/0 M))* '* '* '(+ )$'( %$),*9 (/0 E%$(/; 7)$ (/0 7)$%(')* )7 ('90+: ('90+: 7)$ )*&0(')* )*&0(')* < /'/ /)( 7-,'9+ $'+0=: 7)$ /0%('*g (/0 '*(0$')$+ )7 7)$'*g +(%$+ %*9 -%*0(+ () &0$ /'g/ /'g/ (00$%(,$0+: (00$%(,$0+: %*9 7)$ &%$'),+ )(/0$ /0*)0*% /0*)0*% (/%( (/%( 0 )+0$&0. T/0 +'-0$ +'-0$ N0()*>+ N0()*>+ -% )7 ,*'&0$+%- g$%&'(%(')* $)&'90+ %* %$)?'%(')* %$)?'%(')* 7)$ )+( %-,-%(')*+. %-,-%(')*+. T/0 (0$+ g$%&'(%(')* %*9 g$%&'( %$0 )+(- '*(0$/%*g0%-0 '*(0$/%*g0%-0 '* 0&0$9% ,+0; ,( % 9'+('*(')* % 0 %90 '* +'0*('7' ,+%g0. @G$%&'(%(')*” '+ % g0*0$%- (0$ 90+$''*g (/0 /0*)0*)*  /'/ )9'0+ '(/ %++ %$0 %(($%(09 () )*0 %*)(/0$; /'-0 “g$%&'(@ “g$%&'(@ $070$+ +0'7'%-- () (/0 g$%&'(%(')*%- 7)$0 0?0$(09  (/0 E%$(/ )* )60(+ '* '(+ &''*'(. &''*'(. T/'+ %+ +(,9'09  % +'0*('+( *%09 S'$ I+%% N0()* '* 1.

DESCRIPTIVE PART PART Universal Gravitational Law: 1. Newton’s’ Universal I* (/0 ,*'&0$+0 0&0$ )'*( 0&0$ )'*( %++ %(($%(+ %++ %(($%(+ 0&0$ )(/0$ )'*( %++  % 7)$0 )'*('*g 7)$0 )'*('*g %-)*g (/0 -'*0 6)'*'*g -'*0 6)'*'*g (/0'$ 0*(0$+. T/)+0 () )9'0+ %(($%( 0%/ )(/0$ '(/ % 7)$0 $))$(')*%- () (/0'$ $)9,( %++0+ %*9 '*&0$+0- $))$(')*%- () (/0 (/0 +,%$0 )7 9'+(%*0 9'+(%*0 0(00* (/0. (/0. E!lanation: C)*+'90$ () %++0+ 1 %*9  2; '7 %* )*0 %++ '*$0%+0+ (/0* $)9,( '*$0%+0+ %*9 %* )*0 %++ 90$0%+0+ (/0* $)9,( 90$0%+0+. 90$0%+0+. A)$9'*g A)$9'*g () (/0 7'$+( %$( %$( )7 (/0 +(%(00*(; +(%(00*(;

T/,+;

F µ 1  2

T/0 +0%$%(')* )7 %++0+ '+ “$” 9'+(%*0 9'+(%*0 %%$(. I7 (/0 9'+(%*0 0(00* 0(00* (/0 %++0+ '+ %90 /%-7; (/0 7)$0 '*$0%+0+ 7),$ ('0+ )$ 9'+(%*0 %90 9),-0 (/0 7)$0 0)0 )*0B 7),$; %)$9'*g () (/0 +0)*9 %$( )7 (/0 +(%(00*(. 1 Fµ 2 $ "at#e$ati%al Derivation:

Fµ

1  2 $ 2

T/'+ '+ %(/0%('%- 7)$ )7 N0()*’+ g$%&'(%(')*%- -%.   FCG 12 2 $ Ve%tor Ve%tor &or$: & or$: T/0 9'$0(')* )7 7)$0 '+ $0$0+0*(09  ,*'( &0()$. A*9 A*9 (/0 *0g%('&0 +'g* '*9'%(0+ (/0 7)$0 )7 %(($%(')*. r

H0*0;

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F C B G

1  2 $ , $ 2 

A!!li%ations T/0$0 %$0 *,0$),+ %-'%(')*+ )7 N0()*>+ U*'&0$+%- G$%&'( E,%(')*.

Prof: Najeeb Mughal , Composed by Tarvesh Kumar Page 1

Eart# an' "oon W/%( '+ (/0 g$%&'(%(')*%- 7)$0 70-( )* (/0 M))* 7$) (/0 E%$(/ T/0 E%$(/ %*9 M))* %$0 3.#4?1 5 8'-)0(0$+ %%$(. S'*0 (/0 ,*'(+ )7 G %$0 '* 0(0$+; ), *009 () /%*g0 (/0 () 3.#4?1# . N)(0 (/%( (/0 9'+(%*0 '+ 7$) 0*(0$ () 0*(0$. T/0 7)$0 )* (/0 +,$7%0 )7 (/0 E%$(/ ),-9 0 +-'g/(- -0++; ,( +'*0 (/0 9'+(%*0 () (/0 M))* '+ +) ,/ g$0%(0$ (/%* (/0 $%9',+ )7 (/0 E%$(/; (/0 9'770$0*0 9'770$0*0 ),-9 0 *0g-'g'-0.

T/0 %++ )7 (/0 E%$(/ '+ 5.#124 8g %*9 (/%( )7 (/0 M))* '+ ".35?122 8g. T/0 7)$0 )7 %(($%(')* )* (/0 E%$(/ 7$) (/0 M))* '+ F  GMRJ <."?1B11 NBJ8gJ=<5.#?124 8g=<".35?122 8g=<3.#4?1# =J  1.?12 N T/'+ )*+'90$%-0 7)$0 '+ /%( /)-9+ (/0 M))* '* )$'( %$),*9 (/0 E%$(/. E&&e%t o& "oon on !erson T/0 g$%&'(%(')*%- ,-- 7$) (/0 M))* )* (/0 5 8g <11 ),*9= g'$- '+ F  GMRJ  <."?1B11 NBJ8gJ=<5 8g=<".35?122 8g=<3.#4?1# =J  1."?1B3 N  .1" N S/0 ),-9 *)( *)('0 (/0 ,-- 7$) (/0 M))*; +'*0 (/0 g$%&'(%(')* ,-- )* /0$ ()%$9 (/0 E%$(/ '+ 4 N. K,( +('--; +/0 '+ %(($%(09 )$0 ()%$9 (/0 ))* (/%* (/0 ) /) %+ +'(('*g *0?( () /0$.

() Cal%*lation o& $ass o& t#e Eart#: "at#e$ati%al Derivation: N0()* +%'9 0&0$ )9 %(($%(+ 0&0$ 0&0$ )(/0$ )9 '(/ % 7)$0; 7)$0; (/%( )9'0+ '+; F C G

 M0 R 02

.

N0()* %-+) +%'9 (/%( (/%( (/0 %0-0$%(')* %0-0$%(')* )7 %* )60( '+ '+ 9'$0(- $))$(')*%$))$(')*%- () (/0 *0( 7)$0 7)$0 %('*g )* (/0 )60(; )60(; '+ '* (/0 9'$0(')* )7 (/0 *0( 7)$0; %*9 '+ '*&0$+0- $))$(')*%$))$(')*%- () (/0 %++ )7 (/0 )60(. F   g  M0 K )%$'*g %)&0 () 0,%(')*+;  g C G R 20 gCG

M 0 R 20

W0 8*) (/%(; g.#+0 2; R 0  .3# ?1; G."3?1B11 N2g2 %*9 %-,-%(09 &%-,0 )7 %++ )7 (/0 0%$(/ M 0 5.#?124 g

+) Cal%*lation o& 'ensit, o& t#e Eart#: "at#e$ati%al Derivation: W0 8*) (/%( 90*+'( '+ (/0 %++ 0$ ,*'( &)-,0; 90*)(09 ; “ ρ@. T/0 90*+'( )7 (/0 0%$(/ '+ g'&0* ;  C 4 T/0 &)-,0 )7 (/0 0%$(/ '+; V C N R 3

T/0$07)$0;

C

M0

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M0 V

3 0

3  M0

   4  ( N R 30 )   

R 0  .3# ?1 M 0  5.#? 124 g %*9 (/0 %-,-%(09 %-,-%(09 &%-,0 )7 90*+'( )7 (/0 0%$(/; ρ  5.52?13 g.B3

-) Variation Variation o& ./0 '*e to altit*'e: "at#e$ati%al Derivation: T/0 %0-0$%(')* T/0 %0-0$%(')* 9,0 9,0 () g$%&'( g '+ % &%$'%-0 ,%*('( %*9 '( &%$'0+ 7$) -%0 () -%0. W0 /0$0; '-- 90$'&0 (/0 $0-%(')* (/%( 90+$'0+ &%$'%(')* &%$'%(')* )7 g '(/ %-('(,90. L0( ,+ )*+'90$ % )9 )7 %++ “” -'*g )* (/0 +,$7%0 )7 0%$(/ )7 %++ %*9 $%9',+ R 0 $%9',+ )7 0%$(/. L0( g 0 &%-,0 )7 %0-0$%(')* )7 %0-0$%(')* 9,0 9,0 () g$%&'( )* (/0 +,$7%0 )7 0%$(/. W0 8*) (/%( M 0 gCG B B B B B B B B B <1= R 20 S,)+0 (/0 )9 '+ (%80* () /0'g/( /> %)&0 (/0 +,$7%0 )7 0%$(/ /0$0 (/0 &%-,0 )7 %0-0$%(')* 9,0 () g$%&'( '+ g/. T/0*

g/ C G

M0

( R0 P / )

2

B B B B B B B B B <2=

D'&'90 0,%(')* <2=  0. <1= 0 g0(;


 M0   2  ( R 0 P / )   M  G  20  R 0 

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g/ g

C

C

R 02

( R0 P /)

2

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g/ g

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(/,+; %0-0$%(')* 9,0 () g$%&'( 90$0%+0+ '(/ '*$0%+0 '* /0'g/(.

1) Variat riatio ion n o& o& ./0 ./0 '*e '*e to to 'e! 'e!t# t#:: "at#e$ati%al Derivation: S,)+0 “” '+ (/0 %++ )7 %* )60( -%09 )* () (/0 0%$(/’+ +,$7%0 /%&'*g “R 0” $%9',+. W0 8*) (/%(

g0 C G

W/0$0;

C

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M0 C

T/0 0,%(')* (1) %* 0 $'((0* %+ O$

M 0

B B B B B B B B B (1)

R 20

3  M0

   4  ( N R 30 )    4 3

4 3

G

g 0

C

g 0

CG

3

 N R 0

 

 N R 30 R 02

4 3

 

B B B B B B B B B (2)

 N R 0

W/0* )60( '*() 0 (%80* %( 90(/ “9”; '* +'90 (/0 0%$(/; M 0

g9 C G

W/0$0 M 0 C

4 3

 N

( R0 B 9)

B B B B B B B B B 3)

2

3

T/0 0,%(')* (3) %* 0 $'((0* %+; O$

( R0 B 9)

g9 C

4 3

G

 N

 

3

( R0 B 9 ) 

( R0 B 9)

4 3

g9 C G 

 N

2

 

( R 0 B 9 )


B B B B B B B B B 4)

g9

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g9 g0

 4  N R B 9 ( 0 ) 3  4  G   N R0  3  

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T/'+ +/)+ (/%(; '7 R 0  9 T/0*; g9 g (1 Q 1) g9   I( 0%*+ (/%( &%-,0 )7 “g” '+ 0$) %( (/0 0*(0$ )7 0%$(/.

6) 2ei/#tlessness in Satellites: W/0* % %* '+ '* +,/ % +%(0--'(0; /0 '-- 70-- /'+0-7 0'g/(-0++. T/'+ +(%(0 '+ %--09 “W0'g/(-0++*0++.” “W0'g/(-0++*0++.” I* %* %0-0$%('*g %0-0$%('*g ++(0; (/0 %%$0*( 0'g/( )7 %* )60( '+ *)( 0,%- () '(+ ($,0 0'g/(. I7 (/0 ++(0 '+ 7$00 7%--'*g; %-+,)$(09 7)$0+ ,+( 0 0,%- () 0$) %*9 )60( %0%$ () 0 0'g/(-0++. E!lanation: T) 0?-%'* 0?-%'* (/0 0'g/(-0++ '* +%(0--'(0+; +,)+0 %* )60( )7 0'g/( ’ '+ +,+0*909 '* (/0 $))7 )7 %* 0-0&%()$  0%*+ +$'*g %-%*0 Case 3: Elevator at rest S,)+0 %* 0-0&%()$ '+ '* (/0 +(%(0 )7 $0+(; N0()*’+ +0)*9 +0)*9 -% )7 )(')*; %0-0$%(')* '+ 0,%- () 0$); (/0

$0+,-(%*( 7)$0+ )* (/0 )60( %$0 0,%- () 0$); F   %

F> B  g  O F>   g

O$

Case (: Elevator wit# %onstant velo%it, I7 (/0 &0-)'( '+ )*+(%*(; (/0* (/0 %0-0$%(')* '+ 0$). T/0 %%$0*( 0'g/( '+ 0,%- () %(,%- 0'g/(. C%+0 3 E-0&%()$ %0-0$%('*g %0-0$%('*g ,%$9 S,)+0 %* 0-0&%()$ '+ )&'*g ,%$9 '(/ %0-0$%(')* %0-0$%(')* %’; %-)*g )+'('&0 B %?'+. A)$9'*g () +0)*9 -%; (/0

,*%-%*09 7)$0 ( F> B  g T/0$07)$0

( F>

Bg

( F> )

)

)

%('*g )* '(. C%

C% Pg

T/0 %%$0*( 0'g/( )7 %* )60( '+ )$0 (/%* '(+ $0+( 0'g/(. Case +: Elevator a%%eleratin/ 'ownwar' S,)+0 %* 0-0&%()$ '+ )&'*g 9)*%$9 '(/ %0-0$%(')* %’; %-)*g *0g%('&0 B%?'+. A)$9'*g () (/0 +0)*9

(

-% )7 )(')*; (/0 ,*%-%*09 7)$0  g B F> T/0$07)$0 O$

) %('*g )* '(;

(  g B F> ) C  % ( F> ) C  g B  %

T/0 %%$0*( 0'g/( )7 %* )60( '+ -0++ (/%* '(+ $0+( 0'g/( W/0* %  g T/0*

F>   g B  g %0%$+ () /%&0 *) 0'g/(: '( %0%$+ () 0 “0'g/(-0++*0++”. “0'g/(-0++*0++”. F>   A*9 (/0 )60( %0%$+

4) Arti&i%ial /ravit,: W/0* (/0 %+($)*%,(+ %$0 (/0'$ +%0$%7( (/0 70-- '*)*&0*'0*( '* (/0'$ /%0$+. T) +)-&0 (/'+ $)-0 %* %$('7''%g$%&'( %* 0 $0%(09 '* (/0 +%0$%7(  +'**'*g '(; %),( (/0 %?'+


S,)+0 % +%0$%7( /%+ () /%0$+  % (,**0- )7 -0*g(/ 2 0(0$+. L0( ,+ %-,-%(0 (/0 7$0,0* () $)&'90 %$('7''%- g$%&'( W0 8*) (/%(; %  C

4 N 2 R T2

%5 C 4 N2 R

ν

2

% 5

W/0$0 “R” '+ (/0 /%-7 -0*g(/ )7 (,**0-;

2

4 N R

C S2

% 5 4 N 2 R

I7 % )9 7%--+ 7$00-; (/0* %   g T/0$07)$0;

1

g

2N

R

C S

C S

1

.#  I+05 2

2N

1E 

C S

ν  .15# $0&.  +0 H0*0; %+($)*%,( 700- 0%+ '* +%0 $%7( ,*90$ . $0&.  '* %( % 9'+(%*0 )7 1 0(0$ %% 7$) )'*( )7 %?'+ )7 $)(%(')*

E5UATIONS 3) F  G

1)

g9 g0

1 2



7) g  G

()

$ 2

( R 0 B 9)

2 0

R

G g/

6)

R 0 M 0

g R 20

g



( R0  /)

/ 

 

÷ R 0 3  M0   4)    4  (  R 30 )   

R 20

38) g /  G

 

+) g /  g  1 B 2

 M 0

2

M0

( R0  /)

2

33) g 9  G

-) g 9  g 0  1 B

)

M 0

( R 0 B 9)

1

g

2

R

3() & 

2

9 

÷

R 0

 

2GM R

DI"ENSION PH9SICAL 5UANTIT9  S9";OL

G$%&'(%(')*%G$%&'(%(')*%- )*+(%*( G C M%++ )7 0%$(/

F$0,0*

1 2

2

 M 0

LT B2 L2 M B1 L3 T B2

3  M0

[ M]

g R 02

D0*+'( )7 0%$(/

M L T  L =  MB1 L3 2  M  B2

F $ 2 

DI"ENSION

G

    4  (  R 30 )    g R

 L

 LT  B2

A0-0$%(')* A0-0$%(')* 9,0 () g$%&'( g  G

[ L] M 0 2 0

R

B2

C [ M]

N..28gB2.

8g

8g.B3.

1 2

1 2

C  T B1

 MB1 L3 T B2 [ M ] C  LTB2 2 L


T 

C  M LB3

3

 

UNIT

2

+0B1 )$ /0$(

.+0B2.

SHORT 5UESTIONS 5)< 3 Is t#ere no /ravit, in s!a%e= N) '( '+ *)( )$$0(. )$$0(. I7 (/0$0 0$0 *) g$%&'( g$%&'( '* +%0; (/0 +%0 +%0 +/,((-0 ),-9 *)( 0 %-0 () )$'( (/0 E%$(/; E%$(/; (/0 ))* ),-9 *)( )$'( (/0 E%$(/; %*9 (/0 E%$(/ ),-9 ),-9 *)( )$'( (/0 S,*. T/0 $0%+)* 0 (0*9 () (/'*8 )7 (/0$0 0'*g 0'*g *) g$%&'( '* +%0 '+ (/%( 0 /%&0 +00* )&'0+ )7 (/0 %+($)*%,(+ %+($)*%,(+ 0'*g @0'g/(-0++@. T/0 %$0*>( %(,%-- %(,%-- 0'g/(-0++; (/0 %$0 +('-- 0'*g ,--09 9)*  g$%&'( ,( (/0 %*9 (/0 +%0 +/,((-0 %$0 '* % )*+(%*( +(%(0 )7 7$007%-- %$),*9 (/0 E%$(/. E%$(/. S) (/0 +00 () 0 0'g/(-0++ %+ % $0+,-( )7 (/0 7%--'*g B 6,+( %+ ), ),-9 +00 0'g/(-0++ '7 ), 0$0 '* %* 0-0&%()$ /0* (/0 %-0 $)80. 5) < ( Are .G0 an' ./0 are t#e sa$e t#in/) N); '( '+ *)( )$$0(. W/0$0 W/0$0 “G” '+ (/0 ,*'&0$+%,*'&0$+%- g$%&'(%(')*%g$%&'(%(')*%- )*+(%*( %*9 “g” '+ (/0 %0-0$%(')* %0-0$%(')* 9,0 () (/0 7)$0 7)$0 )7 g$%&'( %*9 '(+ &%-,0 )7 .#++ 9)* '+ )*- ($,0 )* (/'+ -%*0(. I( '+ *)( % ,*'&0$+%- )*+(%*(. 5)  + How wo*l' t#e wei/#t o& t#e >o', var, w#en it is ta?en to t#e %enter o& t#e eart#= T/0 0'g/( )7 (/0 )9 0)0+ 0$) %( (/0 0*(0$ )7 (/0 0%$(/ 0%,+0 (/0 %0-0$%(')* 9,0 () g$%&'( '+ 0$) %( (/0 0*(0$ )7 (/0 0%$(/. 5)- How t#e wei/#t o& a >o', var, as it is ta?en &ro$ t#e eart# to $oon= 2#at will >e t#e e&&e%t on its $ass= A+ (/0 7)$0 )7 g$%&'( '+ '*&0$+0- $))$(')*%$))$(')*%- () (/0 +,%$0 (/0 9'+(%*0; (/0 0'g/( )7 (/0 )9 '-- 90$0%+0+.

A+ (/0 )9 %( (/0 +,$7%0 )7 ))* '+ 7),*9 () 0

1

 1

(/ )* (/0 0'g/( )* (/0 +,$7%0 )7 0%$(/. g  C   g 0 .T/0 %++   

)7 % )9 '-- 0 ,*%770(09. ,*%770(09. 5) <1 How t#e val*e o& /ravitational &or%e >e a&&e%te' i& t#e 'istan%e >etween two $asses >e%o$es #alve'=

W0 8*) (/%(; F  G

F>  G

1  2 2

$

1 2

 $   2

2

F>  G

4 1 2 $ 2

'*$0%+0+ 7),$ ('0+; '7 (/0 9'+(%*0 0(00* 0(00* () %++0+ 0)0+ F>  4 F ; (/'+ +/)+ (/%( (/0 g$%&'(%(')*%- 7)$0 '*$0%+0+ /%-&09. 5) <6 Is t#e /ravitational &or%e >etween t#e eart# an' s*n is sa$e all t#e ti$e &or ,ear= N); (/0 g$%&'(%(')*%g$%&'(%(')*%- 7)$0 0(00* 0(00* +,* %*9 0%$(/ '+ *)( +%0 +%0 %-- (/0 ('0 ('0 )7 % 0%$. 5) <4 How t#e wei/#tlessness is to >e %ontrolle' in satellites= T/0 0'g/(-0++*0++ '* +%(0--'(0+ '+ () 0 )*($)--09  $)9,'*g %$('7''%- %0-0$%(')* 9,0 () g$%&'(. 5)   H) (/0 g$%&'(%(')*%- 7)$0 ),-9 0 %770(09 '7 (/0 0%/ %++ 0)0 9),-0.

FG

1 2 2

$

F>  G

21 22

F>  G

2

$

41  2 $ 2

F>  4F ; T/'+ +/)+ (/%(

(/0 g$%&'(%(')*%- 7)$0 '*$0%+0+ 7),$ ('0+; '7 (/0 0%/ %++ 0)0 9),-0. 5) <7 Co$$ent on @/@ variation '*e to #ei/#t an' 'e!t#)

T/0 &%-,0 )7 @g@ 90$0%+0+ 

 2/  R   0

 2/  1 B R ÷ .A*9 %g%'* '( 0   9  9  g  g 1 B ('0. K0%,+0  ÷.  R  9 R 0  0 

('0+ %+ /0'g/( '*$0%+0+. K0%,+0; g /  g

90$0%+0 ()%$9+ (/0 0*(0$ )7 0%$(/ %( 90(/ 

5) < 38 I& an, one /oes awa, &ro$ t#e eart# s*r&a%e e*al to t#e eart# ra'i*s a>ove t#e eart# s*r&a%e) T#e val*e o& @/@ will >e%o$e iB one &o*rt# iiB one ei/#t iiiB one nint#) I7 %* )*0 g)0+ %% 7$) (/0 0%$(/ +,$7%0 0,%- () (/0 0%$(/ $%9',+ %)&0 (/0 0%$(/ +,$7%0. T/0 &%-,0 )7 @g@ '- 0)0 90$0%+0+ 90$0%+0+  )*0 7),$(/.

Prof: Najeeb Mughal , Composed by Tarvesh Kumar Page 

5) < 33 2#at will >e val*e o& @/@ at t#e %enter o& t#e eart#= Answer: T/0 &%-,0 )7 @g@ (/0 0*(0$ )7 0%$(/ 0)0+ 0$). . K0%,+0; R 0 9

 

g9 C g  1 B

9 

÷

R 0

=

A*9; g 9

5)  3( State Newtons *niversal /ravitational Law) T/0 %g*'(,90 )7 7)$0 )7 %(($%(')* '* (/0 ,*'&0$+0 0(00* () %++0+ '+ 9'$0(- $))$(')*%- %*9 '*&0$+0- $))$(')*%- () (/0 (/0 +,%$0 )7 (/0 9'+(%*0 9'+(%*0 0(00* (/0'$ (/0'$ 0*(0$+. I( '+ 8*)* %+ %+ N0()*>+ ,*'&0$+%- g$%&'(%(')*%g$%&'(%(')*%-%. Fµ

5) <3+

1  2 $ 2

I& t#e $ass o& t#e eart# >e%o$es nine ti$es an' its ra'i*s t#ree ti$es) 2#at will >e t#e val*e o& /=

W0 8*) (/%(; g C

G M0 R 20

 .#+02. G  M0

g> C

( 3R )

2

0

G  M0

g> C

 ( R 02 )

g> C

G M0

( R ) 2 0

g>  g ; (/'+ +/)+ (/%( *) 0770( )* () %0-0$%(')* 9,0 () g$%&'(; g$%&'(; '7 +,/ /%*g0 (%80+ -%0. 5) < 3- Is ./0 is /ravit,) I( '+ *)( )$$0( (/%( “g” '+ (/0 0770( )7 (/0 7)$0 )7 g$%&'(. G$%&'( '+ % 7)$0; -'((-0 “g” '+ %* %0-0$%(')* %,+09  g$%&'(. g$%&'(. 5) < 31 How wo*l' t#e /ravitational &or%e >e a&&e%te' i& t#e one $ass >e%o$e 'o*>le an' t#e 'istan%e >etween t#e$ also 'o*>le=

W0 8*) (/%(; F  G F> C G

F> C G

F> 

1  2 $ 2 2 1  2

( 2$ )

2

2 1  2 4 ( $ )

F>  G

)$

2

1 2 2 ( $ )

2

1

F ; (/'+ +/)+ (/%( 7)$0 0)0 /%-&09 2 Cal%*lation o& $ass o& s*n: "at#e$ati%al Derivation: W0 8*) 8*) (/%( 0%$(/ '+ )$'('*g %$),*9 (/0 +,*. T/0 9'+(%*0 0(00* +,* %*9 0%$(/
W0 8*) (/%(; F C

G M+ 

B B B B B B B B B< 1=

R 2

F% FC

 4 N2 R   T2   

B B B B B B B B B <2=

K )%$'*g 0 <1= %*9 <2=



 4 N 2 R   T2   

 4 N2 R 3  G T2   

C

G M+  R 2

C M+

Prof: Najeeb Mughal , Composed by Tarvesh Kumar Page "

2

M+ C M+ C

11 3

4 <3.1 <3.14 4= <1. <1.4U1E U1E = ."3 ."3 U1E

B11

1.3E4 U 1E .4 U 1E

<35.2 <35.25U 5U24U 24UEU EUE= E=

2

35 4

M + C 1. U 1E 3E g

5*estion +6B: Na$e t#e 'i&&erent la,ers o& t#e eart#= Answer: T/0 )$0; %*(-0 %*9 (/0 $,+(.

5*estion +4B: 2#at is t#e a%%eleration '*e to /ravit, on t#e $oon=

O* (/0 +,$7%0 )7 (/0 ))*; (/0 9'+(%*0 () (/0 0*(0$ )7 %++ '+ (/0 +%0 %+ (/0 $%9',+ $  1."4 ? 1    1 "4  . T/0 %0-0$%(')* %0-0$%(')* 9,0 () g$%&'( )* (/0 +,$7%0 )7 (/0 ))* %* 0 7),*9 ,+'*g (/0 7)$,-%

g  

1.2 +2

T/0 %0-0$%(')* %0-0$%(')* 9,0 () g$%&'( )* (/0 +,$7%0 )7 (/0 ))* '+ 1.2 + 2.

Prof: Najeeb Mughal , Composed by Tarvesh Kumar Page #

F)$ M)$0 N)(0+ O7 C/0'+($; C/0'+($; P/+'+; E*g-'+/ %*9 O(/0$ S,60(+ V'+'( .0,+(%%9.-)g+)(.) .*)(0+*)(0+47$00.).&,

Chapter 6 Gravitation

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