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8/11/2019 Note - Chapter 13
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Chapter 13
GRAVITATION
13.1 Newtons Law of Universal Gravitation
Everyparticle withmassm1in the universe
attractsevery other particle with massm2with a
force that is directly proportional to the product of
their masses and inversely proportional to the square
of the distance rbetween them.
2
21
r
mGmF =
G = 6.67 x 10 11! !2
2
kg
Nm Universal Gravitational
Constant
r= distance from the center of mass of one particle to
the center of mass of the other particle.
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13.2 Weight
The attractive force exerted by the Earth on an object
is called thegravitational force
r
Fg or
r
W . This forceis called the weightof the object, and its directionis
toward the center of the Earth. The weightof an
object of mass mon the Earth has a magnitudeequal
to:
r
Fg =Weight=GM
Em
RE2
The quantity2
E
E
R
MG appears so often that it is given
the name gr
or simply g. This quantity is also called
thegravitational fieldgenerated by the Earth at
locations near its surface. The gravitational field gr
is
a vectorquantity with a directiontoward the center of
the Earth, and with a magnitudedefined as
2
E
E
R
M
Ggg ==
r
using
G = 6.67x10-11
Nm2/kg
2
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ME= 5.98x1024
kg (mass of the Earth)
RE= 6.37x106m (radius of the Earth)
yields a value of g = 9.8 m/s2. One can thus say thatthe weightof an object of mass mon the surface of
the Earth is
r
W = mr
g
whereg
r
= the acceleration due to gravity (or the
gravitational field), always directed toward the center
of the Earth.
Mass and weight are thus related quantities. The
magnitudeof a bodys weight Wis directly
proportional to its mass m.
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13.3 Gravitational Potential Energy
1.Object of mass mat a heightynear the Earths
surface,y
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If y
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Escape Speed:
With what initial speedmust an object of mass mbe
thrown vertically upwardso it escapes from the
Earth?Neglect air friction effects.
Using conservation of mechanical energy yields
ffii UkEUkE +=+
( )!
"="mGM
mR
mGM
mv E
E
E 22
02121
E
E
escR
GM2=! =
sec2.11 km for the Earth.
13.4 The Motions of Satellites
Many artificial satellites have nearly circular orbits
around the Earth. The orbits of the planets around the
Sun are also nearly circular.
Consider the motion of a satellite of mass min acircularorbit of radius raround the Earth.
F"# =m$
2
r
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GMEm
r2
=
m"2
r
v =GM
E
r
(circular orbit)
use also
T
r!"
2=
where Tis the period of revolution of the satellite
around the Earth, then
GME
r=
2" r
T
#
$%
&
'(2
T2=
4"2
GME
#
$%
&
'(r3
(circular orbit)
These equations indicate that larger orbits
correspond to slower speeds and longer periods.
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13.5 Keplers (1571-1630) Laws and Planetary
Motion:
Geometry of an ellipse.
=a
R Aphelion distance
( )eaRa
+= 1
=pR Perihelion distance
eaRp != 1
Kepplers Laws of Planetary Motion:
1.All planets move in elliptical orbitshaving the Sun
at one focus.
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2.A line joining any planet to the Sun sweeps out
equal areas in equal time intervals. ! Results for
conservation of angular momentum of orbiting
planets
!"=L = constant
L =mr2"
dA =1
2r2d"
r2d"=2 dA
dt
dmrL !2
= so(dt
dA
m
L 2=
dA
dt
=
L
2m
Note that since the only force acting on the planet is
the gravitational force exerted by the sun, and the line
of action of this gravitational force passes through the
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center of the sun, then the torque of the gravitational
force acting on the planet about the sun is 0=! .
Hence
" =
dL
dt=0
ThusL = constant
==
m
L
dt
dA
2constant
!equal areas in equal time intervals.
3.The square of the period of revolution of any
planet about the Sun is proportional to the cube ofthe planets mean distance from the Sun. Newton
showed that for an elliptic orbit, the planets orbit
radius r should be replaced by the semi-major axis
a of the elliptical orbit.
T2=
4"2
GMS
#
$%
&
'(a
3
(elliptical orbit)
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Energy Considerations in Planetary and Satellite
Motion.
The mechanical energyEof a planet orbiting arounda star or a satellite orbiting around a planet is always
(a) Constant and
(b) Negative (bound).
In general, if the mechanical energy E is such that
E(r) < 0"elliptical orbit
E(r) = 0" parablic orbit
E(r) > 0"hyperbolic orbit
#
$%
&%
For an elliptical orbit with semi-major axis a , the
mechanical energyEbecomes
E="G M m
2a (elliptical orbit)
Lets use conservation of energy and show that the
speed of an object in an elliptical orbit satisfies
!"
#$%
&'=ar
GM 12
( (elliptical orbit)
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Proof: Use conservation of mechanical energy
gUkEE +=
r
GMmm
a
GMm!=
! 2
2
1
2"
a
GM
r
GM
22
2
!=
"
"(r) = GM 2
r#1a
$
%& '
()
Here r is the distance of the orbiting body from
the central body whose mass isM.
13.7 Apparent Weight and the Earths Rotation
Gravitation Near the Earths Surface
Assume Earth is spherical, homogeneous, and at
rest (not spinning) Then, the gravitational force (true
weight wo = mgo) acting on an object of mass mis
wo=
GMEm
RE
2
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2
E
Eo
R
mGMmg =
22 8.9
sm
R
GMg
E
Eo ==
But the values ofgovary with latitudebecause:
1.Earths crust is not uniform!density variations
provide information about oil prospects.
2.Earth is not a sphere. Earths equatorial radius
is 21 km longer than the radius at the poles. At
the north and south poles, you are closer to
Earths center sogoshould be higher.
3.
Earths rotation effectleads to a difference in g o of about 0.034
2s
m . Lets show this:
Consider someone at the equator holding a spring
scale with a body of mass mhanging from it. Each
scale applies a tension forcer
F to the body hanging
from it, and the reading on the scale is the magnitudeF of this force. If the observer is unaware of the
Earths rotation, then he thinks that the scale reading
equals the weight of the body because he thinks the
body is in equilibrium. But the body is rotating with
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the Earth and is notprecisely in equilibrium. Thus
the magnitude of the forcer
F is the objects apparent
weight W. Consider the Earth spinning with
"=2#T. Then
F"# =ma
mgo"F=m#
2R
E
r
F = apparent weight = mg
Eo Rmmgmg 2
!="
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Eo Rgg 2
!=" E
RT
2
2!"
#$%
&=
'
( )( )[ ]
( )62
2
10370.6606024
4!=
"
2034.0
s
mggo
=!
That is, the actual acceleration due to gravity at the
equator when the Earth is spinning is smaller by
0.034 2s
m than when the Earth is stationary. Spinning
Earth reduces g slightly.
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13.8 Black Holes
A black holeis an object that exerts a gravitational
force on other objects but does not allow any light ofits own to escape from it.
A body of massMwill act as a black hole if its radius
Ris less than or equal to a certain critical radius
called the Schwarzschild radiusRswhich equals
Rs=
2GM
c2
The surface of the sphere with radius Rssurrounding
a black hole is called the event horizon. Since light
cant escape from within that sphere, we cant see
events occurring inside. All that an observer outsidethe event horizon can know about a black hole are
its mass(from its gravitational effects on other
bodies
itselectric charge(from the electric forces it
exerts on other charged bodies)
its angular momentum(because a black hole
tends to drag space and everything in thatspace around with it).
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