Title: Universal Gravitation
1Chapter 8
2Drill
- Calculate ac Fc of a 25 kg ball rotation at the
end of a 4.0 m rope at 20.0 revolutions per
second.
3Planetary Motion
- Galileo
- Tycho Brahe
- Johannes Kepler
- Isaac Newton
4Keplers Laws of Planetary Motion
51) The paths of planets are ellipses, with the
sun as one focus
62) 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
73) 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
9DrillPlanet Orbital PeriodName Rad. (km)
(s)Mo 2.0 x 108 4.0 x 106Smit 4.0 x
1010 ?Solve for ?
10DrillPlanet Orbital PeriodName Rad. (km)
(s)Smo 2.0 x 108 4.0 x 106Mit
? 4.0 x 109 Solve for ?
11Universal Gravitation
- Isaac Newton
- Henry Cavendish
- Michael Faraday
- Albert Einstein
12Gravitational Force
- An attractive force that exist between all objects
13Gravitational Force
- Found to directly proportioned to the masses of
the two objects
14Gravitational Force
Fg ? m
15Gravitational Force
- Found to inversely proportioned to the distance2
between two objects
16Gravitational Force
1 d2
Fg ?
17Universal Law of Gravitation
mAmB d2
Fg G
18Gravitational Constant
19Calculate the gravitational force between 5.0 Mg
6.0 kg objects whose centers are 3.0 mm apart
20The 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
22Thus
mEmo d2
mog G
23 mEmo d2
mog G d r, thus
24 GmE r2
g
25 GmE r2
g or
26Mass of the Earth
gr2 G
mE
27DrillrE 6.37 x 103 kmSolve for the mass of
the Earth
28Centripetal Force
m4p2r T2
Fc
29Fg Fc
msmp mp4p2r r2 T2
G
30 msmp mp4p2r r2 T2
G
31 ms 4p2r r2 T2
G
32 T2ms 4p2r3
G
33Therefore
( )
4p2 Gms
T2 r3
34Keplers Constant for Solar System
( )
4p2 Gms
k
35Earths mean orbital radius is 1.50 x 107 km
while its period 365.25 days. Calculate the Suns
orbital constant in m3/s2
36The moons mean orbital radius is 4.00 x 105 km
while its period 28 days. Calculate the Earths
orbital constant in m3/s2
37or
( )
4p2 Gms
T2 r3
38Mass of the Sun
4p2r3 GT2
ms
39 mEmo d2
Fg G Fc mov2/r
40Thus
mEmo mov2 r2 r
G
41 mEmo mov2 r2 r
G
Thus
42 mE v2 r 1
G
Take sq. rt. of both sides
43Orbital Velocity
GmE r
v
44v
(g)r
45Calculate the velocity of a object orbiting at
50.0 m around a 5.0 x 106 Mg object
46 r3 GmE
Ts 2p
47Calculate the period of a object orbiting at 50.0
m around a 5.0 x 106 Mg object
48( )
rE d
a g
49Calculate the velocity of a satelite orbiting at
620 km above the Earths surface
50Gravitational Field
- The space in which the force of gravity is
apparent
51Measure of Mass
52Inertial Mass
53Gravitational Mass
Fgr2 GmE
mg
54Warped Space
55Review
56List Keplers Laws of Planetary Motion
57What is the formula for Keplers 3rd Law
58Planet Orbital PeriodName Radius (km)
(s)Two 2.0 x 108 4.0 x 106Twit ?
2.0 x 109Solve for ?
59What is the formula for the Universal Law of
Gravitation
60The 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.
61Orbital Velocity
GmE r
v
62Calculate the velocity of the moon.
63Calculate the ac Fc of a merry-go-round with a
radius of 50.0 m spinning at 1 revolution every 4
seconds.
64A car is driven off a 2.0 km cliff at 180 km /hr.
Calculate tair, maximum vV, dH
65A catapult launches a 250 kg rock at 100.0 m/s at
37o from horizontal. Calculate tup, dV, tair,
dH
66Drill
- A 3200 kg carousel with a diameter of 20.0 m is
spinning at 1 revolution every 5 seconds. - Calculate ac Fc
67Important Formulas
68( )2( )3
Orbital Formula
TA rA TB rB
69Universal Law of Gravitation
mAmB d2
Fg G
70Orbital Velocity
GmE r
v
71Orbital Velocity
v
(g)r