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Chapter 13
Gravity
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 2
Gravity
Kepler’s Laws
Newton’s Law of Gravity
Gravitational Potential Energy
The Gravitational Field
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 3
Characteristics of Gravity
Acts along a line connecting the center of mass of the two
bodies.
It is a central force which implies that it is a conservative
force.
It is only attractive. There are no repulsive gravitational
forces, i.e. there is no anti-gravity.
Its magnitude is proportional to the two masses
g 1 2F α m m
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 4
Characteristics of Gravity
1 2g 2
12
m mF α
r
1 2g 2
12
m mF = G
r
g 2
1F α
r
The strength of the gravitational force decreases with
distance
Therefore
The value of G was determined from the data of
Henry Cavendish: G = 6.67x10-11 Nm2/kg2
The gravitational force is
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 5
Characteristics of Gravity
Gravity is the weakest of all the forces. It takes large
masses to produce a gravitational force that is
noticeable.
Gravity acts most on the grand astronomical scale
where large masses are found.
However, it is the exact balance of the positive and
negative charges that effectively cancels the long range
electric fields that allows gravity to be observed even at
these large astronomical distances.
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 6
Characteristics of Gravity
Gravity is a vector quantity
ˆ�
1 212 122
12
m mF = G r
r
� �
12 21F = - F
ˆ ˆ12 21r = - r
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 7
Early Astronomers
Tycho Brahe (1546-1601)
• He gathered observational data over a twenty year
period of measuring planet positions.
Johannes Kepler (1571-1630)
• Kepler worked with Brahe
• He inherited Brahe’s data when he died.
• Kepler deduced his three laws from this data
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 8
Kepler’s Laws
1. Law of Orbits - ellipse
2. Law of Areas - equal area in equal times
3. Law of Periods -
22 3
s
4πT = r
GM
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 9
Kepler’s 1st Law
Perigee Apogee
(Closest approach) (Farthest approach)
Elliptical
shaped orbit
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 10
Construction of an Ellipse
r1 +r2 = 2a
a = semi major axis
b = semi minor axis
F are the focal points
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 11
Kepler’s 2nd Law
Equal areas are swept out in equal times
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 12
Derivation of Kepler’s 2nd Law
� �
� �
12dA = r × vdt
r × mvdA = dt
2m
dA L=
dt 2m
F is a central force so L
is conserved and
therefore
dA= constant
dt
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 13
rava = rpvp
Conservation of Angular Momentum
L = mrvsinφ = constant, therefore rvsinφ is constant with r,
v and φ all changing but the product of the three remaining
constant in value
Evaluate the
expression at φ = 900
where sinφ = 1, at
apogee and perigee
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 14
Kepler’s LawsVariation of periods with mean orbital radius
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 15
Kepler’s 3rd Law
22 3
s
4πT = r
GM
The Period Equation
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 16
Kepler’s 3rd Law
22 3
s
4πT = r
GM
Astronomical Unit = Mean radius of earth orbit around sun
1 AU = 1.50 x 10-11 m = 93.0 x 106 mi
For orbits around the Sun 2 3
T (Years) = R (AU)
For other applications replace Ms in the equation above
with the mass of the body that the object is orbiting
around.
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 17
The Determination of G
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 18
The Determination of G
According to legend, i.e. one text book author copying the
historical aspects of physics from the authors that came before
him, Henry Cavendish made the first accurate measurement of
G in 1745.
It turns out that the legend is incorrect. In fact, Cavendish
wasn’t even interested in measuring the value of G. Henry
Cavendish wanted to measure the density of the earth.
75 years later another researcher was able to squeeze a value of
G out of Cavendish’s old data.
Moral of the story - take good notes while you’re alive and you
might still be making discoveries after you’re dead.
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 19
The Cavendish Experiment(minus the environmental shielding)
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 20
The Cavendish Experiment
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 21
Actual Cavendish Experiment Data
Each tiny square is 1 minute.
Scr
e en
Def
lect
ion
- 1
.5m
aw
ay
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 22
Gravitational Potential Energy
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 23
Bound State - Unbound State
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 24
EGM mU(r) = -
r
Escape Speed of a Projectile
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 25
Gravitational Field
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 26
Preview of Electric Field Calc.-EPII
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 27
g-Field of a Thin Spherical Shell
ˆ�
�
2
GMg = - r; r > R
r
g = 0; r < R
g outside the mass distribution
is the same as if all the mass
was concentrated at the CM.
Requires uniform mass
distribution or a mass
distribution that only varies
with r.
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 28
Gravitational Field Inside a Hollow Sphere
�
gF = 02
1 1 1
2
2 2 2
1 2
2 2
1 2
1
2
1
m A r= =
m A r
m m=
r r
Gmg =
r
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 29
g-Field Distribution - Solid Sphere
≤
3 34 43 3
3
3
3
2 2 3 3
M' M=
πr πR
rM' = M
R
GM' GM r GMg(r) = - = - = - r
r r R R
for r R
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 30
Homework Problems
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 31
Gravitational force on m and G field when m is gone
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 32
Force on mass m at distances: 3a, 1.9a and 0.9 a
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 33
What must their speed be if they are to orbit their common
center under their mutual gravitational attraction?
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 34
Extra Slides
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Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 35