Central Force Problem: Nature of Orbit when Force is Halved | Homework Help

[neelakash]

Homework Statement

A particle moves in a circular orbit under the action of a force
f(r)=-(k/r^2).If k is suddenly reduced to half its value, what would be the nature of the orbit?

Homework Equations

e=sqrt[1+(2*L^2*E)/(mk^2)]

The Attempt at a Solution

My attempt:
Clearly,the particle moves under attractive central force.Now,for the circular orbit,eccentricity e=0 and as the motion is bound,the energy is negative.
If k is reduced to k/2, eccentricity changes to
e=1+(8*L^2*E)/(mk^2)=6*L^2*E/(mk^2)

Since e becomes negative as E is negative,no motion is possible.

Am I correct?

 

[berkeman]

I didn't check your math, but your solution for e being negative sounds reasonable. But that doesn't mean "no motion is possible". It means that something happens to the formerly circular motion of the particle. Think about it in a real-life physical sense. The particle is moving around in a circle, the central force is suddenly cut in half, describe how the particle moves next...

 

[m2003]

Your solution is on the right track, but there are a few things to consider. Firstly, the central force in this problem is not purely attractive, as the negative sign indicates. This means that the force is actually repulsive at very small distances. Secondly, the eccentricity of a circular orbit is always 0, regardless of the value of the force or energy.

In this case, if k is halved, the force will also be halved, which means that the particle will experience a weaker force. This will cause the orbit to widen, but it will still remain circular since the eccentricity is 0. The energy of the particle will also decrease, but it will still be negative, indicating that the particle is still in a bound orbit.

Therefore, the nature of the orbit will be a wider circular orbit with the same eccentricity of 0, due to the halving of the force. This can also be seen in the equation you provided, where the eccentricity is directly proportional to the energy and inversely proportional to the force.

I hope this helps clarify the concept of central force and its effect on orbits. Keep up the good work in your studies as a scientist!