Can two stationary masses repel each other through gravitational forces alone?

[Bob not Alice]

Hi folks,

A system of two masses, stationary with respect two each other, is the starting point. For convenience I'll specify one mass as being much larger than the other. The only force acting between them is their mutual gravitational attraction. At time zero they are an arbitrary distance apart. Some time later they are further apart. Possible or not?

It's actually quite possible. Hint: the smaller mass splits into two during the manoeuvre. Springs, or similar mechanical devices, are allowed.

Just a bit of fun. ;)

Bob.

 

[mfb]

Possible or not?

Sure, many possible solutions.

Spoiler

Let them attract each other, at the point where they nearly touch, activate a repulsive spring that splits the smaller mass into two pieces that rapidly move away from each other until they are far away. Find some way to avoid a collision of the spring with the larger mass. Wait. After a while, combine the two parts again.
You trade energy needed by the spring for kinetic energy and/or energy in the gravitational potential.

Alternatively: use a rocket. That is some sort of splitting as well but I guess it is not the intended solution as you lose the mass.

Alternatively: send bullets in a hyperbolic trajectory around the more massive object, catch them again. If you do this with sufficient mass, you get a net repulsion.

Alternatively: directly start splitting the smaller mass, position one half behind the larger mass and close to it, wait a bit, then combine the two smaller masses again. Probably needs some third piece or gyroscopes to avoid a collision.

 

[Drakkith]

A system of two masses, stationary with respect two each other, is the starting point. For convenience I'll specify one mass as being much larger than the other. The only force acting between them is their mutual gravitational attraction. At time zero they are an arbitrary distance apart. Some time later they are further apart. Possible or not?

It's actually quite possible. Hint: the smaller mass splits into two during the manoeuvre. Springs, or similar mechanical devices, are allowed.

The wording in your question implies that there are only ever two objects, meaning that they can't be split. Please clarify.

 

[mfb]

Oh, I have a trick solution. How is distance measured? Between their closest points? Between centers of mass? Between geometrical center for balls? For (1) and (3) here is another solution:

Spoiler

Have the center of mass of the larger object close to the point with the largest initial distance to the smaller mass, and have a small hollow tube in the larger mass where the smaller and lighter mass can pass through instead of impacting. Then just wait. The distance between the geometric centers will increase beyond its initial value after a while.

 

[Bob not Alice]

Hi mfb,

Good stuff. The rocket concept isn't really what I was looking for as it involves a permanent mass loss to the system. Neither was the "bounce", hence my stipulation the "the only force acting between them is their mutual gravitational attraction". That was meant in reference to the two initial masses, which I may not have stated explicitly enough.

Spoiler

Your third solution is closest to my own which involved splitting the smaller mass into two equal masses following elliptical orbits.

With the same initial conditions as before I believe it should also be possible to end up with the two masses orbiting each other! Extra hint: The smaller mass has to end up spinning. I'll not say more so that anyone who wants to offer a solution can do so...

Not news for most here, but I think there is something to be taken away from this game. Without knowing you can temporarily split the smaller mass it's not possible to get from the initial state to the final state and if the only two observations one could make were the initial and final states then one would cry "fraud". And this is just in the world of classical physics.

Bob.

 

[mfb]

Which bounce? None of my ideas involve the masses touching each other in any way.

I don't see limitations if you allow arbitrary rearrangements of components of the smaller mass.