Start with a complete n prize payload, complete with orbital insertion banger. Attach that to the end of the spoke of a wheel, spin it as fast as it will go, then release it. How far will it go? Not far enough!

Add weights One on the end of each of ends of the rest of the spokes. Spin it up
as before, and just before release pull all of the extra weights in to the middle. How far will it go? further but still, not far enough! Why did it go further? Because like a figure skater spins faster when he pulls his arms close to his body, the wheel will spin faster when the weights are drawn in. ( The conservation of angular momentum.).

Suppose the next step is to use rockets to make the wheel spin faster, and to move the weights in, in a quarter of a turn. How far will it go? No?

Suppose we now apply this to a rail gun. Two loops of track. One with the projectile on it, that can be opened at the right time to launch the projectile. The other made like oversized camera shutters, so that its circumference can be reduced when needed, and carnying the weights. Power up both loops then, at the right moment, switch to using the second loop as a generator and, draw the masses in. Feed the energy released to the projectile. How far will it go?

What about the shards that fly off as the thing
disintegrates, how far will they go?

All you are proposing is a centripetal catapult,
widely know to exist, but the materials to build it
don't exist. If they did, spinning it up to speed
would not be the problem.

At the moment the payload detaches, the asymmetric forces on the centre bearing will be considerable; unless a counterweight is simultaneously released from the other side of the "centrifuge". And it will have to have the same mass and momentum as the payload, so it's not going to be easy to stop.

Assume a large wheel, its axis parallel to the surface of your planet. The payload and the "sacrificial" mass are attached to opposite points on the circumference. Spin the wheel up to the requisite speed; release both masses simultaneously, one to ascend into orbit, the other to impact with an efficient shock-absorbing system, possibly a pile of sand.

Theoretically, multiple balanced payloads can be impelled sequentially into orbit.

Then spin the wheel down again.

Suggestion; when you build the prototype and launch the payload, make sure you are some distance away.

//Shirley...// umm right, one of those circly words. That cuts the 1ft diam wheel to less than a million rpm.

//one to ascend into orbit, the other to impact with an efficient shock-absorbing system, possibly a pile of sand// Been awhile since Orbital Mechanics 101 has it ? Just wait'til you have a payload you want in the opposite orbit and send them both out sideways.