Physics problem on variable gravitation field !!!

[From: ] [author: ] [Date: 11-08-09] [Hit: ]

38 * 10^6 mr = distance from the center of the Earth to the satelliteFg = 6.67 * 10^-11 * (5.98 * 10^24 * m2) ÷ r^2Assuming the orbit is circular, Force centripetal = ( mass of satellite * velocity^2) ÷ radius velocity = 6,500 m/sForce centripetal = (m2 * 6500^2) ÷ rForce of Universal gravitation = Force centripetal6.67 * 10^-11 * (5.......

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What is the altitude above Earth's surface of a satellite with an orbital speed of 6.5 km/s?

The universal gravitational force equation
Fg = 6.67 * 10^-11 * (m1 * m2) ÷ r^2
m1 = mass of Earth = 5.98 * 10^24 kg
m2 = mass of satellite
Radius of Earth = 6.38 * 10^6 m
r = distance from the center of the Earth to the satellite

Fg = 6.67 * 10^-11 * (5.98 * 10^24 * m2) ÷ r^2

Assuming the orbit is circular,
Force centripetal = ( mass of satellite * velocity^2) ÷ radius
velocity = 6,500 m/s

Force centripetal = (m2 * 6500^2) ÷ r

Force of Universal gravitation = Force centripetal
6.67 * 10^-11 * (5.98 * 10^24 * m2) ÷ r^2 = (m2 * 6500^2) ÷ r

Divide both sides by m2
(6.67 * 10^-11 * 5.98 * 10^24) ÷ r^2 = 6500^2 ÷ r

Multiply both sides by r
(6.67 * 10^-11 * 5.98 * 10^24) ÷ r = 6500^2

r = 9.44 * 10^6
r = distance from the center of the earth to the satellite

Radius of Earth = 6.38 * 10^6 m

Altitude = 9.44 * 10^6 – 6.38 * 10^6 = 3.06 *10^6 m

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the acceleration of gravity can be calculated as GM/R^2
At the surface of the earth R is the radius of the earth
and we know that the gravity is g
so g = GM/R^2
at any other height then acceleration is GM/(R+h)^2
= GM/R^2 * R^2/(R+h)^2
= g * (R/(R+h) )^2

So what does this mean.
It means that if h is SIGNIFICANT compared to R then gravity can't be assumed to be constant.
Significant in this context is only a matter of how much error is acceptable.
If you require an answer to 1% accuracy then this is valid up to a height h = 1/200 R above the earths surface.

If you need an answer more accurate then you need to consider the variation at much lower altitudes.

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