Understanding Circular Orbits and Central Forces and Explanation

[fahd]

central forces-please chek answer!

hello...
i have this question in my classical mechanics book and was wondering if some one could help me?

the question says that there is a particle of mass "m" that is moving under an attractive central force F=k/r^5 where k<0
show that the particles orbit is circular and this particle aslo passes through the force centre?


Answer) what i did was that i said that in order for the orbit to be circular, the minimum potential ebergy should be equal to the total energy at that point,,
finding the equilibrium points, the only equilibrium point that made the potential energy minimum was r=0 and this value gave the minimum value of p.e as infinity...is this approach right! please help...also please tell me how do we know which central force leads to what kind of an orbit!
please replyn soon..
thanks!

 

[siddharth]

Fahd, One way to do this, is to show that for a particle to move in a circular orbit passing through the force center, the attractive force must be of the form k/r^5.

If the particle is moving in a circle of radius $R$ passing through the force center , then the distance of the particle $r$ to the force center as a function of $\theta$ will be $r=2R \cos\theta$ (From the property that an angle in a semicircle is a right angle) where $\theta$ is the polar angle

Once you have this relation, can you figure out how you can find the Force as a function of $r$? (Hint: Try using the differential equation which relates $u$ and $F$ where $u=\frac{1}{r}$ and F is the force.

If you solve for F, you will get F as k/r^5.