Universal Gravitation

1
Chapter 8

• Universal Gravitation

2
Drill

• Calculate ac Fc of a 25 kg ball rotation at the
  end of a 4.0 m rope at 20.0 revolutions per
  second.

3
Planetary Motion

• Galileo
• Tycho Brahe
• Johannes Kepler
• Isaac Newton

4
Keplers Laws of Planetary Motion
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1) The paths of planets are ellipses, with the
sun as one focus
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2) An imaginary line from the sun to any planet
sweeps out equal areas in equal time intervals.
Planets move faster when closer to the sun
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3) The square of the ratio of the periods of any
two planets the cube of the ratio of their
orbits average radii
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( )2( )3
TA rA TB rB
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DrillPlanet Orbital PeriodName Rad. (km)
(s)Mo 2.0 x 108 4.0 x 106Smit 4.0 x
1010 ?Solve for ?
10
DrillPlanet Orbital PeriodName Rad. (km)
(s)Smo 2.0 x 108 4.0 x 106Mit
? 4.0 x 109 Solve for ?
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Universal Gravitation

• Isaac Newton
• Henry Cavendish
• Michael Faraday
• Albert Einstein

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Gravitational Force

• An attractive force that exist between all objects

13
Gravitational Force

• Found to directly proportioned to the masses of
  the two objects

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Gravitational Force
Fg ? m
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Gravitational Force

• Found to inversely proportioned to the distance2
  between two objects

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Gravitational Force
1 d2
Fg ?
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Universal Law of Gravitation
mAmB d2
Fg G
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Gravitational Constant

• G
• 6.67 x 10-11 Nm2/kg2

19
Calculate the gravitational force between 5.0 Mg
6.0 kg objects whose centers are 3.0 mm apart
20
The gravitational force between 6.0 Mg 50.0 kg
objects is 2.22 x 10-2 N. Calculate the distance
between them.
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mEmo d2
Fg G Fg mog
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Thus
mEmo d2
mog G
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mEmo d2
mog G d r, thus
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GmE r2
g
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GmE r2
g or
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Mass of the Earth
gr2 G
mE
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DrillrE 6.37 x 103 kmSolve for the mass of
the Earth
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Centripetal Force
m4p2r T2
Fc
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Fg Fc
msmp mp4p2r r2 T2
G
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msmp mp4p2r r2 T2
G
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ms 4p2r r2 T2
G
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T2ms 4p2r3
G
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Therefore
( )
4p2 Gms
T2 r3
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Keplers Constant for Solar System
( )
4p2 Gms
k
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Earths mean orbital radius is 1.50 x 107 km
while its period 365.25 days. Calculate the Suns
orbital constant in m3/s2
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The moons mean orbital radius is 4.00 x 105 km
while its period 28 days. Calculate the Earths
orbital constant in m3/s2
37
or
( )
4p2 Gms
T2 r3
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Mass of the Sun
4p2r3 GT2
ms
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mEmo d2
Fg G Fc mov2/r
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Thus
mEmo mov2 r2 r
G
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mEmo mov2 r2 r
G
Thus
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mE v2 r 1
G
Take sq. rt. of both sides
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Orbital Velocity
GmE r
v
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v
(g)r
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Calculate the velocity of a object orbiting at
50.0 m around a 5.0 x 106 Mg object
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r3 GmE
Ts 2p
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Calculate the period of a object orbiting at 50.0
m around a 5.0 x 106 Mg object
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( )
rE d
a g
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Calculate the velocity of a satelite orbiting at
620 km above the Earths surface
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Gravitational Field

• The space in which the force of gravity is
  apparent

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Measure of Mass

• Use a balance

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Inertial Mass

• minertial Fnet/a

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Gravitational Mass
Fgr2 GmE
mg
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Warped Space

• ?

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Review
56
List Keplers Laws of Planetary Motion
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What is the formula for Keplers 3rd Law
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Planet Orbital PeriodName Radius (km)
(s)Two 2.0 x 108 4.0 x 106Twit ?
2.0 x 109Solve for ?
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What is the formula for the Universal Law of
Gravitation
60
The masses of Earth moon are 5.98 x 1024 kg
1.0 x 1024 kg respectively. The average radius of
the moons orbit is 4.0 x 106 km. Calculate the
force of gravity between the Earth the moon.
61
Orbital Velocity
GmE r
v
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Calculate the velocity of the moon.
63
Calculate the ac Fc of a merry-go-round with a
radius of 50.0 m spinning at 1 revolution every 4
seconds.
64
A car is driven off a 2.0 km cliff at 180 km /hr.
Calculate tair, maximum vV, dH
65
A catapult launches a 250 kg rock at 100.0 m/s at
37o from horizontal. Calculate tup, dV, tair,
dH
66
Drill

• A 3200 kg carousel with a diameter of 20.0 m is
  spinning at 1 revolution every 5 seconds.
• Calculate ac Fc

67
Important Formulas
68
( )2( )3
Orbital Formula
TA rA TB rB
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Universal Law of Gravitation
mAmB d2
Fg G
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Orbital Velocity
GmE r
v
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Orbital Velocity
v
(g)r