7 Gravitation

8/9/2019 7 Gravitation

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GRAVITATION1. KEPLERS LAWS

Keplers First Law (Law of Orbits): Each planet moves in an elliptical orbit with the Sun at one focus.

Keplers Second Law (Law of Areas): The speed of planet varies in such a wa that the radius vector drawn from

the

Sun to a planet sweeps out e!ual areas in e!ual times.

Thus the law states that the aerial velocit of the planet

is constant.

The orbit of a planet around the sun is shown in fi". the

areas# A $ A and A are swept b the radius vector in

e!ual% & '

times. So$ accordin" to (epler )s second law$

A * A * A% & '

Also$ the planet covers une!ual distances S $S and S in% & '

e!ual times due to the variable speed of the planet.

+a,imum distance is covered in a "iven time when planet is closet to the Sun.

-hen the planet is closest from the sun$ its velocit and the inetic ener" of the planet is ma,imum.

-hen the planet is farthest from the Sun$ its velocit and the inetic ener" is minimum.

/owever$ the total ener" of the planet remains constant.

Keplers Third Law (Law of Periods)

The s!uare of the period of revolution of a planet around the Sun is proportional to the cube of the semi0ma1or

a,is of its elliptical orbit.

A2 is the ma1or a,is and 34 is the minor a,is. AO or O2 is called semi0ma1or a,is.

5et$ T * 6eriod of revolution of planet around Sun.

R * len"th of semi0 ma1or a,is

Accordin" to (epler )s third law$

T&R

'or T

&=(R

'

-here ( is a constant for all planets.

5et

Tand T be the periods of an two planets around the Sun.

5et R and R be the len"ths of their respective semi 7 ma1or a,es% &

T& R'

Then$%

%

& &

Since different planets are at different distances from the Sun$ therefore$ their time periods are different.

Proof of Keplers Third Law 3onsider the motion of a planet around the Sun.

5et m and + represent the masses of the planet and Sun respectivel.

5et us assume that the planet moves around the Sun in a circular orbit. 5et rbe the radius of the orbit.

The "ravitational force between the planet and the Sun provides the necessar centripetal force to the planet.

G+m

mr&

r&

G+ & & 8 &r'& 8

'

r & r

T T

%

&

T R'

&&


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G+ T r

Or T& r'G+


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2. UNIVERSAL LAW OF GRAVITATION

Ever particle of matter in the universe attracts ever other particle

with a force which is directl proportional to the product of their

masses and inversel proportional to the s!uare of the distance between

them. The force of attraction between an two particles in the universe

is nown as force of "ravitation.

The force of "ravitational attraction between the two bodies acts alon" the line 1oinin" their centres. This force

is mutual.

3onsider two bodies of masses

mand m with their centres separatel b a distance r.

5et 9 be the force :in ma"nitude; of "ravitational attraction between the two bodies. Accordin" to Newton)s

law of "ravitation$

:i; 9 m%m

&

m m 3ombinin" both the factors$ we "et 9

% &

r&

:ii;9

%

r&

Or 9G

m%m

&

r&

< :i;

-here G is a constant of proportionalit nown as "ravitational constant

It has the same value everwhere in the universe. So$ it is a universal constant of "ravitation.

In SI$ the value of G is :=.=> ? %@7%%Nm& "7&;.

If m * m * % unit and r * % unit$ then from e!uation :%;$ 9 * G.% &

So$ the universal "ravitational constant :G; is numericall e!ual to the force of attraction between two bodies$each of unit mass$ separated b unit distance.

In SI$ the "ravitational constant is numericall e!ual to the force of attraction between two bodies$ each of mass

one ilo"ram$ separated b a distance of one metre.

Special Features of Graitational Force

!" The "ravitational forces between two bodies constitute an action and reaction pair$ i.e.$ the force is e!ual in

ma"nitude but opposite in direction.

#" The "ravitational force between two bodies does not depend upon the nature of the intervenin" medium.

$" The "ravitational force between two bodies does not depend up the presence or absence of other bodies.

%" The "ravitational force is e,tremel small in the case of li"ht bodies. /owever$ it becomes appreciable in the

case of massive bodies. As an illustration$ althou"h the distance between the Sun and Earth is et thecorrespondin" "ravitational force is of the order of %@&>N.

&ector For' of ewtonss Law of Graitation

3onsider two particles A and of masses m and m respectivel.% &

5et$ r%&

* displacement vector from A to 2$

r&%

* displacement vector from 2 to A.

% &


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9&%

* "ravitational force e,erted on 2 b A.


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9%& * "ravitational force e,erted on A b 2.

In vector notation$ Newton)s law of "ravitation is written as follows

m m9 =7 G

% &rB%& & &%

&%< :%;

-here rB&%

is a unit vector pointin" from 2 to A. The ne"ative si"n indicates that the direction of

9%&

that of rB&%

.

The ne"ative si"n also shows that the "ravitational force is attractive in nature.

is opposite to

Similarl$9

m m= 7 G

% &rB

&% & %&

%&

-here rB%&

is a unit vector pointin" from A to 2.

Also$ r&

* r&

&% %&

9 m m=G % &rB&% & &%

&%

E!uatin" :%; and :&;$

9%&= 7 9

&%

3. ACCELERATION DUE TO GRAVITY OF THE EARTH

5et us consider a bod of mass m lin" on the surface of the Earth of mass + and radius R. 5et gbe the value

of acceleration due to "ravit on the free surface of Earth."

F G+m GM

.......:%;Then$m R

&

m

R&

&ariation ofg with Altitude (ei*ht)

Suppose the bod is taen to a hei"ht Ch) above the surface of Earth where the value of acceleration due to

"ravit isg .

Then$

g GM

h DRhF&< :&;

whereDRhFis the distance between the centres of bod and Earth.

4ividin" :&; b :%;$ we "et

g G+ R&

g R&

h

h

g DRhF&

G+Or

g h &

R % R

5et us now derive an e,pression forg when h R.

g R&

R&

g % h 7&

r

r

r

h

h


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h or

h %

g h & h&

g h&

R

R

% R

&

% R

% R

R


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Since h R$ therefore$ hHR is ver small as compared to %.

E,pandin" the ri"ht hand side of the above e!uation b 2inomial theorem and ne"lectin" the s!uare an d

hi"her

hpowers of $ we "et

R

gh% 7

&hg R

g g % 7&h

Or h R

&hOr gh g 7 gR

gh7 g 7 &h

g or g 7

gR

h

&h

gR

/ere$ Dg 7 ghF"ives the decrease in the value of ".

Since the value of " at a "iven place on the Earth is constant and R is also constant Therefore g 7 ghh .

Thus$ the value of acceleration due to "ravit decreases with increase in hei"ht above the surface of Earth.

Note !: The fractional decrease in the value of " is "iven bDgIg hF

&h

.g g

Note #: The percenta"e decrease in the value of " is&h%@@ .

g

Loss of +ei*ht at ei*ht h(,,-;

g g % 7&h

-e now that$ h R

&h & mghmg % mg 7h R

R

Or mgh7 mg 7 & mgh

R

Or mg 7 mgh

& mgh

R

5oss in wei"ht *& mgh

R

&ariation ofg with .epth

Assume the Earth to be a homo"eneous sphere :havin" uniform densit; of radius R and mass +.

5et pbe the mean densit of Earth.

5et a bod be lin" on the surface of Earth where the value of acceleration due to "ravit is Cg).

mg 7


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G+ Then g

R&

Or g

G8

pR'

r

'

R&

Or g 8

p R rG'

< :%;

5et the bod be now taen to a depth d below the free surface of Earth where the value of acceleration due

to "ravit isgd.

/ere$ the force of "ravit actin" on the bod is onl due to the inner solid sphere of radius :R 7 d;

g G+

d DR 7

dF&

where + is the mass of the inner solid sphere of radius :R 7 d;."

G

8p R 7 d rOr d

D F

DR7 dF& '

Or gd8p GDR 7 dFr

'


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R


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-ei"ht of a bod at the centre of

Earth. " " d

At a depth d below the free surface of Earth$d

% 7 R

At the centre of Earth$ d * R

Rg g % 7 @ R

If Cm) is the mass of a bod lin" at the centre of earth$ then its wei"ht * m" * @

/o'parison of ei*ht and .epth For The Sa'e /han*e in 0g

As we have seen above$ the value of " decreases as we "o above the surface of Earth or when we "o below

the surface of Earth.

This can be taen to mean that value ofg is ma,imum on the surface of Earth.

Now gh g

% 7

&h

R

and g

dg

%

7

d

R

-hen

ghg

d$ then % 7&h

% 7d

R R

7&h7

d

R R

Or &hd

Or d &h

Thus$ the value of acceleration due to "ravit at a hei"ht h is same as the value of acceleration due to "ravit

at a depth d :* &h;. 2ut this is true onl if h is ver small.

4. GRAVITATIONAL POTENTIAL ENERGY

The force of "ravit is a conservative force and we can calculate

the potential ener" of a bod arisin" out of this force$ called

the "ravitational potential ener".

Suppose$ a bod of mass m is placed at 6 in the

"ravitational field of a bod of mass +. 5et r be the

distance of 6 from the centre O of the bod of mass +.

In order to determine the "ravitational potential ener" ofthis

sstem$ let us calculate the wor done in movin" mass m from infinit to 6.

-hen the mass m is at A$ the "ravitational force of attraction on it due to mass + is "iven b 9 G+m

.x

2

-hen the mass m moves from A to 2$ i.e. throu"h distance dx$ then wor done will be

d- 9dx G +m

dxx

&

d

d


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If - is the total wor done b the "ravitational field when a bod of mass m is moved from infinit to a

distance r

from O$ then


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- r G + m

dxG+m

r %dx

x

&

x&

r

x7& %

r

r

- G+m x7&

dx G+ m 7 G+ m

%

7 G+m%

7

%

Or 7 & % x r

Or - 7

G+m

r

This wor done is e!ual to the "ravitational potential ener" J of mass m.

Gravitational potential ener"$ J 7 G+m

r

Now$ J * 7G+m

*7

G+ mr r

Gravitational potential ener" * "ravitational potential ? mass The "ravitational potential ener" of mass m in the "ravitational field of another mass is ne"ative because

wor is done b "ravitational field and not a"ainst it in brin"in" the mass m from infinit to the point under

consideration.

The "ravitational potential ener" of mass m at a hei"ht h above the surface of the Earth is "iven b

J 7G+mKKKKKKKKKKKKKKKKKKKKKL

The distance between the mass m and the centre of the Earth is :R M h;.

R M h

J7G+m G+

m h 7%

7 % R

%M h

*

R

R

R

Since

h%$ so e,pandin" the ri"ht hand side of the above e!uation b 2inomial Theorem and ne"lectin"

R

hs!uares and hi"her powers of

$ we "etR

J 7G+m

% 7h

R R

Or J 7G+m

R

G+

G+mh

R&

2ut g :acceleration due to "ravit;R&

J 7G+m

mghR

G+m2ut 7

R* "ravitational potential ener" of mass m at the surface of the Earth.

Accordin" to convention$ the "ravitational potential ener" at the surface of the Earth is taen to be ero.

J * mgh

Note: The surface of Earth is an e!uipotential surface because the "ravitational potential at all points on the

surface of Earth is the same.


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5. ESCAPE SPEED

Escape velocit is defined as the least velocit with which a bod must be thrown verticall upwards in orderthat

it ma 1ust escape the "ravitational pull of the Earth.

-hen we throw a bod verticall upwards with certain velocit$ the bod returns to the Earth)s surface after

some time. /owever$ when the bod is thrown with a velocit e!ual to the escape velocit$ the bod overcomes the

Earth)s "ravitational pull and also the resistance of the Earth)s atmosphere. This bod never returns to the

surface of the Earth a"ain.

12pression for 1scape &elocit3

3onsider a bod of mass m lin" at a distancex from the centre of the Earth and let + be the mass of the Earth.

Accordin" to Newton)s law of "ravitation$ the "ravitational force 9 of attraction between the bod and the

Earth is "iven b

9 G +m

x&

5et d- be the wor done is raisin" a bod throu"h a small distance dx$ then the total wor - done inraisin" the bod from the surface of the Earth to infinit is "iven b

W d-

G M md,

x

&

/ere$ R is the radius of the Earth.

Or - *

G+m

%

dx

G+m is a constant !uantit

R x

x

7& %

7&

- GMmR

x dxG+m 7 & %

R

x7%

% % %

G+m %

Or - * G+m

7%7 G+m

x

* 7 G+m 7 R R

@

R R

% & 5et ve

be the escape velocit$ then inetic ener" imparted to the bod *&

mve

%mv

&G+m

Or& &G+

&

e R

&G+

ve

R

Or v


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&gR

g D&RF

g4


15/22

/ere$ 4 is the diameter of the Earth.

9rom e!uation :%;$ ve

& G

8p R' r :where r is the mean densit of Earth;

R '


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ve R

Pp G r.' das.

4otion of A Satellite

:i; All satellites toda "et into orbit b ridin" on a rocet Orb ridin" in the

car"o ba of the space shuttle.

:ii; A rocet is aimed strai"ht up at first which "ets the rocet throu"h the

thicest part of the atmosphere most !uicl. This minimises fuel

consumption.

:iii; Once the rocet reaches e,tremel thin air :at %' m;$ the rocet)s

navi"ational sstem fires small rocets. This puts the launch vehicle into

horiontal position.

:iv; The rocets are now a"ain fired to ensure separation between the launch

vehicle and the satellite. Thus$ the satellite is put on the path described in

the fli"ht plan.

Note:

:a; -e consider the "ravitational force onl between the satellite and the Earth. The distribution effect of the

"ravitational force of other bodies is i"nored.

:b; The centre of mass of the Earth)s satellite sstem is at the centre of the Earth.Orbital &elocit3

Orbital velocit :v ; is the velocit which is "iven to an artificial Earth)s satellite a few hundred ilometers

above the Earth)s surface so that it ma start revolvin" round the Earth.

E,pression for orbital velocit

5et m * mass of satellite$ r * radius of the circular orbit of satellite

h * hei"ht of the satellite above the surface of Earth$ R * radius of Earth$

v * orbital velocit$ + * mass of the Earth

@

@


17/22

The centripetal

force

m v&

r

re!uired b the satellite to eep movin" in a circular orbit is produced b the

"ravitational force G+m

between the satellite and the Earth.r

&

m v& G +m Then$

@=

r r&

Or v&=

G+@

r

G+Or v@ = < :i;r

2ut r =R +h

v@ =G+

R +h

< :ii;

5et ghbe the value of acceleration due to "ravit at a hei"ht h above the free surface of the Earth .

G+mThen$

mgh =

(R +h )&

Or G+ =gh

(R +h

)&g (R +h )

&

9rom e!uation :ii;$ v@ = R +h

Or v@= g

h (R +h

)


18/22

Substitutin" values$ v@= .P% =8@@ %@@@ ms

7%

==&>P8@@@ m s

%=>&'. = m s

%= >. m s

7%.. :iv;

9or an orbit close to the surface of the Earth$ escape velocit$ ve

* &"R * & vo

Ti'e Period of a Satellite (T)

The period of revolution of a satellite is the time taen b the satellite to complete one revolution round the

Earth.

T =ci r cu mfe re n ce o f ci rc ul ar o rb i t

Or T =

&r v

@

orbital velocit


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gR&T

& h =

%H'

7 R

8 & Or

8 &


20/22

This is the relation which "ives the orbital radius of a "eostationar satellite.

Substitutin" values ofg * .P m s7&$ T * % da * P=8@@ s$ R * =.'P %@=

m$ we "et& &

%H'&%H'

gR T h =

(P=8@ @)

7 R =

.P (=.'P%@

=

)& =8&$&'> 7 =$'P@ ='Q$ PQ>m

8& 8

&

The "eostationar satellite should be taen to a hei"ht of 'Q$PQ> m above the Earth)s surface at the e!uatorand

"iven the re!uired orbital velocit in the horiontal

direction.

5ses of Artificial Satellites

!" To stud various phenomena in the outer re"ions of the Earth)s atmosphere.

#" To stud various phenomena connected with sea.

$" 9or communication purposes.

%" 9or weather forecastin".

1ner*3 of An Orbitin* Satellite

% & The inetic ener" of a satellite movin" with orbital velocit v is ( E * mv "

Therefore$

Gm+ E

" ("E *

&:RE

Mh;

3onsiderin" the "ravitational potential ener" at infinit to be ero$ the potential ener" at distance :RE

Mh;

from

the centre of the earth is 6.E =Gm+

E.

:RE

Mh;

The (.E is positive whereas the 6.E is ne"ative. /owever$ in ma"nitude the (.E. is half the 6.E$ so that the

total E

is (.E M6.E =Gm+

E

&:RE

Mh;

The total ener" of circularl orbitin" satellite is thus ne"ative$ with the potential ener" bein" ne"ative but

twice the ma"nitude of the positive inetic ener".

-hen the orbit of a satellite becomes elliptic$ both theK.E. and.E. var.

The total ener"$ which remains constant$ is ne"ative in the circular orbit case.

If the total ener" is positive Or ero$ the ob1ect escapes to infinit.

Satellites are alwas at finite distance from the earth and hence their ener"ies cannot be positive or ero.

Geostationar3 or S3nchronous Satellite

A "eostationar satellite is so named because it appears to be stationar to an observer on the Earth.

This satellite is also named as snchronous satellite because the an"ular speed of the satellite is snchronised

with the an"ular speed of the Earth about its a,is.

-hen such a satellite is used for communication purposes$ it is also nown as communication satellite.

3onditions for a satellite to appear stationar to an observer on the Earth

!" It should be at a hei"ht of nearl '=$@@@ m above the e!uator.

#" It should revolve in an orbit which is concentric and coplanar with the e!uatorial plane. So$ the plane of

the orbit of the satellite is normal to the a,is of rotation of the Earth.

$" The sense of rotation of the satellite should be the same as that

of the Earth about its own a,is. So$ the satellite should orbit

&


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around the Earth from west to east as does the Earth. Its orbital

velocit is nearl '.% m s7%.

%" The orbital period of the satellite$ i.e. the time taen b the

satellite to complete one revolution around the Earth should be

the same as that of the Earth about its own a,is. i.e. &8 hours.


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+ei*htlessness

-ei"ht of an ob1ect is the force with which the earth attracts it.

-hen an ob1ect is in free fall$ it is wei"htless and this phenomenon is usuall called the phenomenon

of wei"htlessness.

In a satellite around the earth$ ever part and parcel of the satellite has an acceleration towards the center of

the earth which is e,actl the value of the earth)s acceleration due to "ravit at that position.

Thus in a satellite everthin" inside it is in a state of free fall. This is 1ust as if we were fallin" towards the

earth from a hei"ht.

Therefore$ in a manned satellite$ people inside e,perience no "ravit. Gravit for us defines the vertical

direction and thus for them there are no horiontal or vertical directions$ all directions are the same.

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