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SATELLITES AND SATELLITES AND GRAVITATIONGRAVITATION
John Parkinson John Parkinson ©©
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SATELLITES
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Newton’s Law of Gravitation
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rMGM
F
M1 M2
r
F F
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CIRCULAR MOTION
rF
CENTRIPETAL FORCEr
mvF
2
2mrF
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SATELLITES
r m
v
Equation of Motion r
mvF
2
r
mv
r
GMm 2
2
M
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SATELLITE VELOCITY
v
r mM
r
mv
r
GMm 2
2
r
GMv
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SATELLITE VELOCITY
v
R m
For an orbit CLOSE to the surface
F = mg
r
mvF
2
r
mvmg
2
v = √ r g
v = √ 6.4x106 x 10 = 8000 ms-1
v = 8 km/s
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SATELLITE PERIOD
r mω
Equation of Motion
2mrF
T
2
M
2
2
2
4
Tmr
r
GMm
GMrT3
2
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GE0SYNCHRONOUS COMMUNICATIONS SATELLITE
TO REMAIN OVER ONE PLACE ON THE EARTH’S SURFACE, THE PERIOD HAS TO
BE THE SAME AS THE EARTH’S DAY.
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COMMUNICATIONS SATELLITE
THIS GIVES A RADIUS OF 42 000 km
r
2mrF 2
2
2
4
Tmr
r
GMm
32
2
4GMT
r 32
22411
4
)360024(104.61067.6
r
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VG
Definition The gravitational potential at a point is the work done in moving unit mass (1kg) from infinity to that point.
But potential energy at infinity is zero – no attraction.
Hence the gravitational potential at the point really measures the absolute potential energy that 1 kilogram has at that point.
M1 kgr
r
GMVG
FOR A RADIAL FIELD
UNITS J kg-1
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The gravitational potential at a point measures the potential energy per kilogram at that point.
Mm kgr
r
GMVG
Joules
The potential energy of a mass of m kilograms is given by:
Ep = mVG
r
GMmEP
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g
Relationship between Field Strength and
Potential
r
Vg
At any point Field Strength = POTENTIAL GRADIENT
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VG = -40 MJkg-1
VG = -10 MJkg-1
A
How much energy is needed to move a 200 tonne spaceship from A to B?
B
POTENTIAL DIFFERENCE, ΔVG = - 10 – ( - 40 ) = 30 MJ kg-1
EP = m ΔVG = 200 000 kg x 30 MJ kg-1
ENERGY NEEDED = 6 TJ
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TOTAL ENERGY OF A SATELLITE
m
v
r r
GMmEP
r
GMVG
M
r
mv
r
GMm 2
2
r
GMmmvEK 22
1 2
r
GMm
r
GMmEEE KPtotal 2
r
GMmEtotal 2
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ORBITAL DECAY
m
v
rr
GMmEK 2
M r
GMmEtotal 2
r
GMmEP
When a satellite enters the atmosphere, drag heats it up. It falls to a lower orbit and SPEEDS UP.
The decrease in total energy is only half of the decrease in potential energy.
As r decreases, the potential energy becomes numerically larger but decreases overall as it is
more negative.
The kinetic energy is numerically equal to the total energy, but is positive, so increases.
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