a particle of mass m, in a circular orbit of radius r around a planet of mass M,

Question

a particle of mass m, in a circular orbit of radius r around a planet of mass M, has a mechanical energy E=-GMm/2r. Suppose that particle encounters very slight friction due to space dust. The friction is so slight that the particle essentially remains in a circular orbit, but the radius of the orbit slowly changes. The friction will cause it to lose energy, i.e., go to a more negative energy. Since the energy is E=-GMm/2r, this means that the particle must decrease its orbital radius due to the friction. But the kinetic energy of the particle (again given in your book) is K=+GMm/2r. Thus a decrease in radius means an increase in kinetic energy. The conclusion is that friction, a force directed against the motion of the particle, speeds up the particle's motion! Either show what is wrong with this argument, or explain how it's possible for a force opposite to the direction of the velocity to increase the velocity.

Explanation / Answer

whats wrong with this argument is that is says friction speeds up the particle ie increases its kinetic energy but what exactly happening is that particle's kinetic energy is coming from decrease in its potential energy! here is how: the speed of particle rotating in orbit at r distance is given by the force equation centripetal force=gravitaional force mv^2/r=GMm/r^2 1.as the friction acts on particle, its speed decreases (initially) which decreases the centripetal force. 2.So there is net force on particle towards the center of planet due its potential energy decreases and kinetic energy increases. 3.this decrease in potential energy increases the kinetic energy of particle. and note that the change is potential energy is not exactly equal to change in kinetic energy because a part of it is wasted in friction too! So in this process the friction is not the source of kinetic energy. It is just a stimulus which triggers the conversion of potential energy int kinetic energy and takes a share of potential energy for itself also.

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