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Old Apr 1st 2009, 05:32 PM   #1
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Motion in a central force

Hello, I have a problem with 2 questions of an exercie about an
circular motion in a central force, which can help me are welcome;

We 're interested in an artificial satellite, mass $\displaystyle m $, in circular orbit of radius $\displaystyle R $ around the Earth mass $\displaystyle M_E $ and radius $\displaystyle R_E $.

I have already shown that the motion of the satellite was uniform by the Newton's laws of motion.

Now I'd like some help to answer the following question :
Show that the mechanical energy "E" of the satellite is related to its kinetic energy $\displaystyle E_k$ by E =- $\displaystyle E_k$.
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Old Apr 1st 2009, 07:56 PM   #2
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You know that the orbital velocity of a satellite is given by -
V = sqrt(GM/r), where M is the mass of the earth and r is the distance of the satellite from the centre of the earth. If mass of the satellite be m then its kinetic energy

K = (1/2)mV^2 = (1/2)GMm/r ..........(1)

Next, its potential energy is given by

U = - GMm/r ..............................(2)

Mechanical energy (E) is the sum of the two-

E = K + U = - (1/2)GMm/r ..............(3)

From eq (1) and (3) it is clear that E= -K
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