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1 SATELLITES AND SATELLITES AND GRAVITATION GRAVITATION John Parkinson John Parkinson © ©

1 SATELLITES AND GRAVITATION John Parkinson © 2 SATELLITES

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Page 1: 1 SATELLITES AND GRAVITATION John Parkinson © 2 SATELLITES

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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

221

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.