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Gravitational potential energy (U)

6.3 Gravitational potential energy and gravitational potential (S p. 21!)

The gravitational potential energy(U) of abody of mass mat a point in a gravitational

field is defined as the negative of work done by

the gravitational force to bring the body frominfinity to that point.

Go to

More to Know 12More to Know 12

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Object is moved dxtowards earth

ork done (dW) ! F dx! dxxmM

G

E

2

Note: "ince the directions of gravitational

force and the displacement are the same#

the work done is positive.

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6.3 Gravitational potential energy and gravitational potential (S p. 216)

$ove object fromx! to r

r

mMGdx

x

mMG

FdxWU

EEr

r

==

==

2

)(P#nalGravitatio

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%. &ve denotes U at is 'ero (highest)and decreases for closer to earth

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. nit for U* jo+le (,)

-. r! RE*

habove s+rface*

ncrease in U! U%& Uo ! mgh

/. 0elationship between Uand F

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E

E

R

mMGU =o

)(1 hR

mMGU

E

E

+=

== r FdrUdrdUFor

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Gravitational potential (V)

6.3 Gravitational potential energy and gravitational potential (S p. 21$)

The gravitational potential(V) at a point in agravitational field is the work done by the

gravitational force to bring a +nit mass from

infinity to that point.

r

GM

m

UV E==

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6.3 Gravitational potential energy and gravitational potential (S p. 21%)

%. V1 at is 'ero

1 scalar

1 +nit* , kg1%

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. VPQ& work done by gravitational force inbringing a +nit mass from P to Q(independent of

path)

===

21 r

GM

r

GMVVVV EEQPPQ

Note: "ince the directions of force and displacement is

opposite in this definition# the work done (per +nit mass) is

negative.

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-. 2ll points at same distance from earth3scentre have same V

The s+rface where all points on it has the

same gravitational potential is known as anequipotential ur!a"e.

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1 4o work done if move object on the samee5+ipotential s+rface

1 Gravitational field e5+ipotential s+rfaces

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Go toMore to Know 1#More to Know 1#

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$%a&ple 6$%a&ple 6

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'otential gradient

6.3 Gravitational potential energy and gravitational potential (S p. 22')

U! ork done ! F r

mV ! 1Fgr! 1mg r

g! 1 V6r

r 7

drdV

g =

potential

gradient

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'otential gradient

6.3 Gravitational potential energy and gravitational potential (S p. 22')

0elationship between V#gand r

Note: The gravitational field

strength (g) is act+ally avector and its val+e sho+ld be

negative to represent its

direction. n section 8.# g is

positive since we consider itsmagnit+de only.

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$%a&ple #$%a&ple #

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1"

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1!

Sign of work

The sign of work depends on the direction of

force and displacement.

1. If their directions are the same, then

positive work is done.

2. If their directions are opposite, then

negative work is done.eturn to

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16

Other definitions of U

1. The work done by an external force to

bring the body from infinity to that point.

2. The work done by the gravitational force

to bring the body from that point to infinity.

eturn to

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1$

Sign of gravitational potential energy

The gravitational potential energy (U) of two

particles at infinite separation is defined as

zero by convention. ence, U m!st be negative

or zero (at infinity).

eturn to

e%te%t


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1%

=

=

r

r

FdrU

FdrUasrittenealso*an

eturn to

e%te%t

6.3 Gravitational potential energy and gravitational potential (S p. 21$)

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1&

"#$can also be defined as

the work done by an externalforce in bringing a !nit massfrom $to #.

eturn to

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i i l i l d i i l i l ( )

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Equipotential surfaces

around the earthThe e%!ipotential s!rfacesaro!nd the earth areimaginary spherical shells

with the same centre at theearth&s centre.

eturn to

e%te%t


6 3 G i i l i l d i i l i l (S 21&)

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$'$' The dashed lines in the fig!re represent thee%!ipotential lines aro!nd the earth. The

gravitational potential is as shown for each of

the e%!ipotential lines.

6.3 Gravitational potential energy and gravitational potential (S p. 21&)

6 3 G it ti l t ti l d it ti l t ti l (S 21&)

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$'$' (a) (i) hich one of the points (or points) has thehighest gravitational potential *xplain yo!ranswer.

(ii) +alc!late the work done by the gravitationalfield in bringing a spacecraft of mass --- kg

() from /to +0 (1) from + to 2.(b) (i) The e%!ipotential lines, which are givenevery -. 3 -45 kg6, are not e%!ally spaced.*xplain why.

(ii) +alc!late the distances /7and 7+.( 89 :.4 3 -6; kg61m10mass of earth 9 :.- 3 -1< kg)

=ol!tion



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*olution:*olution:

(a) (i) +he pointA has the highest gravitational potential.

Gravitational potential (V) ,

Sin*e the distan*e o-A-ro the earth is the greatest/ theval0e is the least negative or the highest.

(ii) (1) or done y gravitational -ield to ring spa*e*ra-t -roAto C

, m(VA4 VC) , ! ''' 54".' 1'$4 (4!.' 1'$)7 , !.' 1'1'8

(2) or done y gravitational -ield to ring spa*e*ra-t -roC toD

, m( VC4 VD) , ' (-or VC, VD)


r

GME


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*olution ("ont+d):*olution ("ont+d):

(b) (i) +he e90ipotential lines are not e90ally spa*ed e*a0se thegravitational potential does not vary linearly ith r0tvaries inversely ith r.

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( ) ( )

( )

1'&'%/1''$1

1''"%1''!

1''61'$6

1'&3%1'!"

1''61'$6:1'''1

1''"

1''61'$6

y(ii)

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$

2"11$

$

2"11

====

=

=

=

==

=

==

=

.rrBC.rrAB

..

..r

..

..r.

.

..r

VGMr,

rGMV

r

GMV

CBBA

C

BA

A

EA

A

EA

E

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6 3 Gravitational potential energy and gravitational potential (S p 221)

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*olution:*olution:

(a) (i) +he gravitational -ield strength at a point in a gravitational-ield is the gravitational -or*e a*ted on a 0nit ass at that

point.

(ii) +he gravitational potential (V) at a point in a gravitational-ield is the or done y the gravitational -or*e to ring a 0nitass -ro in-inity to that point.


2strength-ieldnalGravitatio

r

GM=

gradient)(Potentialstrength-ieldonalGreavitati

)(potentialnalGravitatio

==

=

drdVr

GMV

6 3 Gravitational potential energy and gravitational potential (S p 221)

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*olution ("ont+d) :*olution ("ont+d) :

eturn to

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( )

E

E

E

E

E

E

EE

E

E

E

E

gR

R

gR

R

GMV

g

gR

GM

mgmg

R

mGMm

RrR

GM

===

=

===

==

2

2

2

2

s0r-a*esearth;on thepotentialnalGravitatio.iss0r-a*esearth;on thestrength-ieldnalGravitatio

eight-or*enalgravitatio

s0r-a*e/searth;on theasso-on the()

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