AQA A2 Physics Homework Help - Keplers Laws

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CLJD
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I've got to make a presentation on Kepler's Laws of Planetary Motion, which is fine, but the it says "use information from your research to show by calculation: the orbital height of geostationary satellite; the angular velocity and period of a low polar satellite at 1000km above the earth; and the velocity of both satellites"
I really don't understand how to do any of this - even what calculations to use, can anyone help?
Thank you
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Gr0mit
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From Newton's law of gravitation, we know the force due to two point masses is given by GMm/r^2. G is the universal gravitation constant, M is the mass of the larger object (most of the time) and m is the mass of the second object. r is the distance between the two objects as well. As satellites move in circular orbits, it is given there is a centripetal force acting on the satellite. This force acts towards the centre and is given by mv^2/r. Here v is the linear velocity.

The main point to consider is the centripetal force acting on the satellites is the gravitation force acting on the satellite. Therefore we can now equate the formula for Newton's law of gravitation to the formula of circular motion. This gives GMm/r^2 = mv^2/r. Doing a bit of rearranging, hopefully you will find the r's and m's cancel. Making v^2 the subject gives v^2=GM/r. Just to recap, m is the mass of the satellite orbiting the earth and M is the mass of the earth.

v is the linear velocity and is equal to 2πr/T. Where T is the time period (ie time for the satellite to make one complete osccilation around the earth) and π is, well π (3.142). We can substitute this expression for v. Now we have (2πr/T)^2=GM/r. Expanding gives GM/r = 4π^2 r^2 / T^2. Rearranging we get r^3/T^2 = GM/4π^2.

The ratio r^3/T^2 is known as Keplers 3rd Law. From the equation in italics above, you can see if the time period is known, then the radius of the orbit can be calculated. Now I can answer the part on geostationary and polar orbit.

For a Geostationary orbit, the time period is exactly 24 hours, so I can substitute this into the italic equation and find the radial orbit because GM/4π^2 is constant.

For a polar orbit, you know the radius of the orbit so you can work out the time period of the orbit using the italic equation I have stated again.
Angular velocity is known as the rate of change of angle and has equation 2π/T. You can also work out this as well now.

The velocity (linear velocity) of both satelites can be worked by 2πr/T (as previously stated), where r is the radius of the orbit of the satellite.

I don't know if you are an A2 student, but angular velocity, Newtons law of gravitation and centipetal force/acceleration is part of further mechanics. Hope this helps and sorry for the long post.
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