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Two spherical bodies of different radii and different masses are separated by a certain distance in free space.

  • If they attract each other only due to gravitational force only, will they meet at their center of mass ?

  • Why or why not?

  • What would be the relevant equations to find their final meeting position (along the line joining their centers) ?

Intuitively it feels that they should meet at their center of mass, however I am not able to mathematically verify it.

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    $\begingroup$ Think about it. If you have two billiard balls, can their centers of mass really coincide? $\endgroup$
    – Chet Miller
    Commented Mar 1, 2017 at 12:10
  • $\begingroup$ @ChesterMiller Yes, I understood. However, how to calculate the meeting point of the two spheres (where they touch) ? $\endgroup$
    – user2531998
    Commented Mar 1, 2017 at 12:49

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You aren't able to verify it because it isn't true.

For example, take two spherical masses, each of mass 1 kg. Let their centers lie at $(-1,0)$ and $(1,0)$. Obviously, the center of mass of the system is at $(0,0)$.

Now what if one of the spheres had a radius of $1.5$ units and the other a radius of $0.5$ units?

enter image description here

I'll leave this confirmation up to you!

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  • $\begingroup$ So how to calculate the point where the surfaces of the two spheres touch? $\endgroup$
    – user2531998
    Commented Mar 1, 2017 at 12:41
  • $\begingroup$ @user2531998 You'll have to minimise the distance between the centers of the spheres, while conserving the center of mass of the system. In the above example, the minimum distance is 2m (because of the fact that the spheres cant go through each other). The center of mass is at (0,0) $\endgroup$
    – praeseo
    Commented Mar 2, 2017 at 6:57

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