alphy ch06c
TRANSCRIPT
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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.
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More to Know 12More to Know 12
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (S p. 21!)
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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More to Know 13More to Know 13
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (S p. 216)
$ove object fromx! to r
r
mMGdx
x
mMG
FdxWU
EEr
r
==
==
2
)(P#nalGravitatio
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (S p. 216)
%. &ve denotes U at is 'ero (highest)and decreases for closer to earth
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Gravitational potential energy (U)
6.3 Gravitational potential energy and gravitational potential (S p. 216)
. nit for U* jo+le (,)
-. r! RE*
habove s+rface*
ncrease in U! U%& Uo ! mgh
/. 0elationship between Uand F
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More to Know 15More to Know 15
E
E
R
mMGU =o
)(1 hR
mMGU
E
E
+=
== r FdrUdrdUFor
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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (S p. 21$)
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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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (S p. 21%)
%. V1 at is 'ero
1 scalar
1 +nit* , kg1%
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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (S p. 21%)
. 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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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (S p. 21%)
-. 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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Gravitational potential (V)
6.3 Gravitational potential energy and gravitational potential (S p. 21%)
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!
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
e%te%t
6.3 Gravitational potential energy and gravitational potential (S p. 21!)
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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
e%te%t
6.3 Gravitational potential energy and gravitational potential (S p. 21!)
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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
6.3 Gravitational potential energy and gravitational potential (S p. 216)
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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 $to #.
eturn to
e%te%t
6.3 Gravitational potential energy and gravitational potential (S p. 21%)
i i l i l d i i l i l ( )
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Equipotential surfaces
around the earthThe e%!ipotential s!rfacesaro!nd the earth areimaginary spherical shells
with the same centre at theearth&s centre.
eturn to
e%te%t
6.3 Gravitational potential energy and gravitational potential (S p. 21%)
6 3 G i i l i l d i i l i l (S 21&)
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$'$' The dashed lines in the fig!re represent thee%!ipotential lines aro!nd the earth. The
gravitational potential is as shown for each of
the e%!ipotential lines.
6.3 Gravitational potential energy and gravitational potential (S p. 21&)
6 3 G it ti l t ti l d it ti l t ti l (S 21&)
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$'$' (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
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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*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)
6.3 Gravitational potential energy and gravitational potential (S p. 21&)
r
GME
6 3 G it ti l t ti l d it ti l t ti l (S 21&)
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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.
eturn to
e%te%t
6.3 Gravitational potential energy and gravitational potential (S p. 21&)
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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.
6.3 Gravitational potential energy and gravitational potential (S p. 221)
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
e%te%t
6.3 Gravitational potential energy and gravitational potential (S p. 221)
( )
E
E
E
E
E
E
EE
E
E
E
E
gR
R
gR
R
GMV
g
gR
GM
mgmg
R
mGMm
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===
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s0r-a*esearth;on thepotentialnalGravitatio.iss0r-a*esearth;on thestrength-ieldnalGravitatio
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