I am having trouble in applying the orbital speed, $v=\sqrt{\frac{GM}{r}}$ in the following problem stated below.Usually we have one satellite orbitting one planet.But in this case there are two satellites orbitting in different orbits,thus i am confused how to crack it.The question is:

Two satellites named $\alpha$ and $\beta$ both of mass m are orbiting around a large mass $M$ in circular orbits having radius $R_1$ and $2R_1$ respectively. Initially,the satellites are connected by a massless string. After some time,the string is cut. Find the ratio of velocities of the satellite $\alpha$ before to after cutting the string, i.e. how much will the velocity of $\alpha$ increase/decrease after cutting the string?

Anyone with some intuitive hints?

  • $\begingroup$ @Ghoster edited. Now can u help? $\endgroup$
    – Rafi Islam
    Oct 1 at 4:17
  • 2
    $\begingroup$ It seems like a poorly posed question. If you released them like that (and even that would depend on where and how you released them), the inner one would immediately try to pull ahead of the outer one, resulting in a very complex spiral motion that I'm sure you'd need a computer to solve. $\endgroup$
    – RC_23
    Oct 1 at 7:44

1 Answer 1


This is a very badly worded and very unclear question.

Two satellites named α and β both of mass m are orbiting around a large mass M in circular orbits having radius R1 and 2R1 respectively. Initially, the satellites are connected by a massless string.

We are not told how long the string is, or whether or not it is taut. We can guess that the point of the string is to keep the satellites orbiting at the same angular speed. But in this case the satellites are not in free-fall circular orbits (because if they were they would have different angular speeds) so there must be some additional force acting on them other than gravity. This additional force could just be the tension in the string. But equally well, the satellites could be in powered orbits, in which case they could be initially orbiting at any speed at all.

How much will the velocity of α increase/decrease after cutting the string?

The velocity of α will not change instantaneously, because it has inertia. Its velocity may change over time, but that change depends on the forces acting on it. Is it only acted on by gravity - in which case it will go into an elliptical orbit - or is there some other force acting on it that constrains it to follow a circular orbit ? The question does not give us enough information to construct a sensible answer.

  • $\begingroup$ I would dispute the content that "they could be orbiting at any speed at all". If we assume that the string connecting the two satellites is inextensible (which seems "reasonable" in the context of a problem like this), then they have to be orbiting with the same angular velocity; and if the orbits of each satellite are circular, then string must be vertical (otherwise the string would apply a force in the wrong direction.) Under this assumption, it is possible to solve for the shared angular velocity by drawing free-body diagrams for each mass. $\endgroup$ Oct 1 at 12:34
  • $\begingroup$ @MichaelSeifert Well, you could interpret the question like that and assume that the string is taut and vertical and the only forces acting on the satellites are gravity and the tension in the string and they just happen to be orbiting at exactly the right angular speed to keep the string vertical. But we are not told any of that. And if the satellites are instead in powered orbits then they could be orbiting at any speed at all. In any case, the question is still hugely ambiguous and unclear. $\endgroup$
    – gandalf61
    Oct 1 at 17:47

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