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Universal Gravitation

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Title: 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
5
1) The paths of planets are ellipses, with the
sun as one focus
6
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
7
3) The square of the ratio of the periods of any
two planets the cube of the ratio of their
orbits average radii
8
( )2( )3
TA rA TB rB
9
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 ?
11
Universal Gravitation
  • Isaac Newton
  • Henry Cavendish
  • Michael Faraday
  • Albert Einstein

12
Gravitational Force
  • An attractive force that exist between all objects

13
Gravitational Force
  • Found to directly proportioned to the masses of
    the two objects

14
Gravitational Force
Fg ? m
15
Gravitational Force
  • Found to inversely proportioned to the distance2
    between two objects

16
Gravitational Force
1 d2
Fg ?
17
Universal Law of Gravitation
mAmB d2
Fg G
18
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.
21
mEmo d2
Fg G Fg mog
22
Thus
mEmo d2
mog G
23
mEmo d2
mog G d r, thus
24
GmE r2
g
25
GmE r2
g or
26
Mass of the Earth
gr2 G
mE
27
DrillrE 6.37 x 103 kmSolve for the mass of
the Earth
28
Centripetal Force
m4p2r T2
Fc
29
Fg Fc
msmp mp4p2r r2 T2
G
30
msmp mp4p2r r2 T2
G
31
ms 4p2r r2 T2
G
32
T2ms 4p2r3
G
33
Therefore
( )
4p2 Gms
T2 r3
34
Keplers Constant for Solar System
( )
4p2 Gms
k
35
Earths mean orbital radius is 1.50 x 107 km
while its period 365.25 days. Calculate the Suns
orbital constant in m3/s2
36
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
38
Mass of the Sun
4p2r3 GT2
ms
39
mEmo d2
Fg G Fc mov2/r
40
Thus
mEmo mov2 r2 r
G
41
mEmo mov2 r2 r
G
Thus
42
mE v2 r 1
G
Take sq. rt. of both sides
43
Orbital Velocity
GmE r
v
44
v
(g)r
45
Calculate the velocity of a object orbiting at
50.0 m around a 5.0 x 106 Mg object
46
r3 GmE
Ts 2p
47
Calculate the period of a object orbiting at 50.0
m around a 5.0 x 106 Mg object
48
( )
rE d
a g
49
Calculate the velocity of a satelite orbiting at
620 km above the Earths surface
50
Gravitational Field
  • The space in which the force of gravity is
    apparent

51
Measure of Mass
  • Use a balance

52
Inertial Mass
  • minertial Fnet/a

53
Gravitational Mass
Fgr2 GmE
mg
54
Warped Space
  • ?

55
Review
56
List Keplers Laws of Planetary Motion
57
What is the formula for Keplers 3rd Law
58
Planet Orbital PeriodName Radius (km)
(s)Two 2.0 x 108 4.0 x 106Twit ?
2.0 x 109Solve for ?
59
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
62
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
69
Universal Law of Gravitation
mAmB d2
Fg G
70
Orbital Velocity
GmE r
v
71
Orbital Velocity
v
(g)r
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