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If the kinetic energy of an object in circular motion due to a gravitational force is given by:

$$E_{kinetic} = \frac{GM_{object}M_{planet}}{2X_{radius}},$$

then why does radius decrease when there is a resistive force acting on the body when it is in orbit? Would the resistive force not cause the kinetic energy to decrease and thereby cause the radius to increase (because they are inversely proportional according to the equation)?

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The equation is only valid for circular orbits, it basically means that if you have a circular orbit, the smaller the radius the faster the motion will be.

Once friction starts acting the equation stops being valid. This is because the orbit is no longer circular. If we were to suddenly stop friction the object would start orbiting in an elliptical fashion, not circular, because the speed is no longer enough to support a circular orbit.

In an extreme example, supose friction is so large that the object practically stops moving. In such a case if you stop the friction the motion will be directed towards the center of the earth in a degenerate "elliptical" orbit (a line).

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  • $\begingroup$ Whilst it is certainly true that the orbits will not be exactly circular the question asked about the increase in kinetic energy even though a frictional force acts. For slowly decaying orbits which are approximately circular the equation for kinetic energy is reasonable in giving an order of magnitude. I think what the OP missed was that in a lower orbit the gravitational potential decrease more than the kinetic energy increases. An extreme (non circular orbit) case is when a spacecraft enters the Earth’s atmosphere. $\endgroup$ – Farcher Oct 27 '18 at 17:31
  • $\begingroup$ I agree that in most realistic cases there is increase in kinetic energy due to the decrease in potential energy, but I do not think it is to satisfy that equation. But I reversed my down vote. $\endgroup$ – Wolphram jonny Oct 27 '18 at 17:41
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The total energy of a body in orbit $-\frac{GMm}{2R}$ is the sum of the kinetic energy $\frac{GMm}{2R}$ and the gravitational potential energy $-\frac{GMm}{R}$.

As the total energy of the body decreases due to frictional forces the radius of the orbit decreases and two energy changes are taking place.
The kinetic energy of the object increases but the gravitational potential energy decreases by more. The difference between these two changes is equal to the loss of energy of the body due to frictional forces.

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