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Two stars of masses $M$ and $2M$ move in circular motion about their common centre of mass. Which of the following statements is true?

A. Both stars move with the same radius.

B. Both stars move with the same speed.

C. Both stars move with the same angular velocity.

D. Such a motion is not possible.

Since the gravitational force contributes to centripetal force of the rotation, consider mass $M$, we have $$\frac{2GM^2}{R^2} = \frac{Mv^2}{R}$$ where $R$ is the distance between the two masses.

I have no idea how to show they have the same angular velocity.

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    $\begingroup$ It seems I asked similar question before. $\endgroup$
    – Idonknow
    Commented Sep 28, 2016 at 6:58

1 Answer 1

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The root of the problem with differing angular velocities lies in that the center of gravity would not exhibit straight-line motion. To see this, let's call the origin the center of gravity at the starting time. At this starting time (which is arbitrary), the stars must have the origin along the line between them (or else the origin would not lie along the line containing their center of gravity, which would be a contradiction of my definition of origin). The origin must continue to lie along the line between them, and therefore they must rotate around the origin at the same rate and the angular velocities must be equal.

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